Biomarkers and molecular mechanisms of Amyotrophic Lateral Sclerosis

Ashok Chakraborty; Anil Diwan; Ashok Chakraborty; Anil Diwan

doi:10.3934/Neuroscience.2022023

AIMS Neuroscience

2022, Volume 9, Issue 4: 423-443. doi: 10.3934/Neuroscience.2022023

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Mini review

Biomarkers and molecular mechanisms of Amyotrophic Lateral Sclerosis

Running title:Running Title: ALS Detection and its Mechanism of Development

Ashok Chakraborty ^,,
Anil Diwan

AllExcel, Inc. Shelton, CT, USA

Received: 03 July 2022 Revised: 29 October 2022 Accepted: 02 November 2022 Published: 10 November 2022

Amyotrophic lateral sclerosis (ALS) is a fatal neurodegenerative disease in adults involving non-demyelinating motor disorders. About 90% of ALS cases are sporadic, while 10–12% of cases are due to some genetic reasons. Mutations in superoxide dismutase 1 (SOD1), TAR, c9orf72 (chromosome 9 open reading frame 72) and VAPB genes are commonly found in ALS patients. Therefore, the mechanism of ALS development involves oxidative stress, endoplasmic reticulum stress, glutamate excitotoxicity and aggregation of proteins, neuro-inflammation and defective RNA function. Cholesterol and LDL/HDL levels are also associated with ALS development. As a result, sterols could be a suitable biomarker for this ailment. The main mechanisms of ALS development are reticulum stress, neuroinflammation and RNA metabolism. The multi-nature development of ALS makes it more challenging to pinpoint a treatment.

Keywords:

Citation: Ashok Chakraborty, Anil Diwan. Biomarkers and molecular mechanisms of Amyotrophic Lateral Sclerosis[J]. AIMS Neuroscience, 2022, 9(4): 423-443. doi: 10.3934/Neuroscience.2022023

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Abstract

Abbreviations

AD:

Alzheimer's Disease;

Ataxin-2:

ATXN2 Gene-Product;

ALS:

Amyotrophic Lateral Sclerosis;

HSP:

Hereditary spastic paralysis;

AMPA:

α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid;

LDL:

Low Density Lipoprotein;

AOA2:

Ataxia with Oculomotor Apraxia Type 2;

MMP-2:

Metalloproteinase-2;

fALS:

Familial ALS;

MMP-9:

Metalloproteinase-9;

sALS:

Sporadic ALS;

NfL:

Neurofilament Light Chain;

APOE:

Apolipoprotein E;

NMDA:

α-Amino-3-Hydroxy-5-Methyl-4-Isoxazole-propionic Acid;

Ataxin-2:

ATXN2 Gene-Product;

PD:

Parkinson's Disease;

CCR2:

C-C Chemokine Receptor Type 2;

pNfH:

Phosphorylated Neurofilament Heavy Chain;

c9orf72:

Chromosome 9 Open Reading Frame 72;

Optn:

Optineurin;

CMT type 4 J:

Charcot-Marie-Tooth disease type 4;

OX40:

CD134 (TNFRSF4);

CRP:

C-Reactive Protein;

PLS:

Primary lateral sclerosis;

CSF:

Cerebrospinal Fluid;

RBP:

RNA-binding protein;

DFT:

Frontotemporal Dementia;

RNA:

Ribonucleic acid;

ELISA:

Enzyme-Linked Immunosorbent Assay;

ROS:

Reactive Oxygen Species;

EMMPRIN:

Extracellular Matrix Metalloproteinase Inducer;

SCAR1:

Spinocerebellar Ataxia, Autosomal Recessive 1;

FDA:

U.S. Food and Drug Administration;

SMA:

Spinal muscular atrophy;

FIG 4:

FIG4 Phosphoinositide 5-Phosphatase;

SOD:

Superoxide Dismutase 1;

FTD:

Frontotemporal disorder;

TDP-43:

TAR DNA-Binding Protein 43;

FTLD:

Frontotemporal lobar degeneration;

TARDBP:

TAR DNA Binding Protein;

FUS:

Fused in Sarcoma;

TNF-α:

Tumor Necrosis Factor-α;

GM-CSF:

Granulocyte Macrophage Colony Stimulating Factor;

VAPB:

Vesicle-associated Membrane Protein-associated Protein B;

HDL:

High-density lipoprotein;

VCP:

Valosin Containing Protein;

HMGB1:

High Density Lipoprotein;

WASP:

Wiskott–Aldrich syndrome protein;

APOE:

Apolipoprotein E;

wr-CRP:

Wide Range C-Reactive Protein

1. Introduction

Let $\mathcal{A}$ denote the class of functions of the normalized form

$\begin{align} f(z) = z+\sum\limits^{\infty}_{k = 2}a_{k}z^{k}, \end{align}$

(1.1)

which are analytic in the unit open disk

$\mathbb{D} = \{z\in\mathbb{C}:|z| < 1\}$

and all coefficients are complex numbers. Let $\Phi$ denote the set of analytic function with positive real part on $\mathbb{D}$ with

$\phi(0) = 1,\ \ \phi^{\prime}(0) > 0$

and $\phi(z)$ maps $\mathbb{D}$ onto a region starlike with respect to 1 and symmetric with respect to the $x$ -axis. And, the function $\phi(z)$ has a series expansion of the form

$\phi(z) = 1+\sum\limits^{\infty}_{k = 1}A_{k}z^{k},$

where all coefficients $A_k(k\geq 1)$ are real number and $A_1 > 0$ . Also let $\mathcal{U}$ denote the class of Schwartz functions, which is analytic in $\mathbb{D}$ satisfying

$u(0) = 0 \ \; \; \text{and}\; \; \; |u(z)| < 1.$

In 1970, Robertson ^[1] introduced the concept of quasi-subordination. For two analytic functions $f_1(z)$ and $f_2(z)$ , the function $f_1(z)$ is quasi-subordinate to $f_2(z)$ in $\mathbb{D}$ , denoted by

$f_1(z)\prec _{q}f_2(z), z\in\mathbb{D},$

if there exists a Schwarz function $u(z)\in\mathcal{U}$ and an analytic function $h(z)$ with

$|h(z)|\leq 1$

such that

$f_1(z) = h(z)f_2(u(z)).$

Observe that when

$h(z) = 1,$

then

$f_1(z) = f_2(u(z))$

and it is said that $f_1(z)$ is subordinate to $f_2(z)$ and written

$f_1(z)\prec f_2(z)$

in $\mathbb{D}$ . Also notice that if

$u(z) = z,$

then

$f_1(z) = h(z)f_2(z)$

and it is said that $f_1(z)$ is majorized by $f_2(z)$ and written

$f_1(z)\ll f_2(z)$

in $\mathbb{D}$ . Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization. For works related to early study of the quasi-subordination concept, see ^[2,3,4].

In order to further explore the concept of quasi-subordination, some researchers have extended the construction of function classes and obtained some geometric properties of function classes. In 2012, Mohd and Darus generalized Ma-Minda starlike and convex classes in ^[5] and defined the generalized starlike class $\mathcal{S}^*_q(\phi)$ and the generalized convex class $\mathcal{C}_q(\phi)$ by using quasi-subordination, as below

$\begin{align*} \mathcal{S}^*_q(\phi)& = \left\{f(z)\in \mathcal{A}:\frac{zf^{\prime}(z)}{f(z)}-1\prec_q\phi(z)-1,\phi(z)\in\Phi,z\in\mathbb{D}\right\},\\ \mathcal{C}_q(\phi)& = \left\{f(z)\in \mathcal{A}:\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\prec_q\phi(z)-1,\phi(z)\in\Phi,z\in\mathbb{D}\right\}. \end{align*}$

And, they defined the following function class (also see ^[6])

$\mathcal{M}_q(\alpha;\phi) = \left\{f(z)\in \mathcal{A}:(1-\alpha)\frac{zf^{\prime}(z)}{f(z)}+\alpha\left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)-1\prec_q\phi(z)-1,\phi(z)\in\Phi,\alpha\geq0,z\in\mathbb{D}\right\}.$

In 2015, El-Ashwah et al. ^[7] introduced the generalized starlike class $\mathcal{S}^*_q(\mu; \phi)$ of complex order and the generalized convex class $\mathcal{C}_q(\mu; \phi)$ of complex order as follows,

$\begin{align*} \mathcal{S}^*_q(\mu;\phi)& = \left\{f(z)\in \mathcal{A}:1+\frac{1}{\mu}\left(\frac{zf^{\prime}(z)}{f(z)}-1\right)\prec_q\phi(z),\phi(z)\in\Phi,\mu\in\mathbb{C}\backslash \{0\} , z\in\mathbb{D}\right\},\\ \mathcal{C}_q(\mu;\phi)& = \left\{f(z)\in \mathcal{A}:1+\frac{1}{\mu}\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\prec_q\phi(z),\phi(z)\in\Phi,\mu\in\mathbb{C}\backslash \{0\},z\in\mathbb{D}\right\}. \end{align*}$

In 2020, Ramachandran et al. ^[8] defined the class $\mathcal{M}^*_q(\alpha, \beta, \lambda; \psi)$ by using quasi-subordination. The function $f(z)\in \mathcal{A}$ is in the class $\mathcal{M}^*_q(\alpha, \beta, \lambda; \phi)$ if

$\left(\frac{zf^\prime (z)}{f(z)}\right)^{\alpha}\left[(1-\lambda)\frac{zf^\prime (z)}{f(z)}+\lambda\left(1+\frac{zf^{\prime \prime} (z)}{f(z)}\right)\right]^{\beta}-1\prec_q\phi(z)-1,$

where $\phi(z)\in\Phi$ and $0\leq\alpha, \beta, \lambda\leq1$ . Many authors have studied various function subclasses defined by quasi-subordination. For example, Vays et al. ^[9], Altinkaya et al. ^[10], Goyal et al. ^[11] and Choi et al. ^[12] studied bi-univalent functions using quasi-subordination. Shah et al. ^[13] and Aoen et al. ^[14] introduced meromorphic functions using quasi-subordination. Karthikeyan et al. ^[15] studied Bazilević function using quasi-subordination. Shah et al. ^[16] studied non-Bazilević function using quasi-subordination. And, there are some function subclasses of linear and nonlinear operators (such as, hohlov operator ^[17], difference operator ^[18] and derivative operator ^[19]) using quasi-subordination.

Recently, some researchers have begun to generalize close-to-convex function classes by using quasi-subordination relationship. In 2019, Gurmeet Singh et al. ^[20] introduced the subclass of bi-close-to-convex function defined by quasi-subordination. In 2023, Aoen et al. ^[21] introduced the class of generalized close-to-convex function with complex order written as $\mathcal{K}_q(\gamma; \phi, \psi)$ . This class were defined as below

$\mathcal{K}_q(\gamma;\phi,\psi) = \left\{f(z)\in \mathcal{A}:\frac{1}{\gamma}\left(\frac{zf^{\prime}(z)}{g(z)}-1\right)\prec_q\psi(z)-1, g(z)\in\mathcal{S}^*_q(\phi),\phi(z),\psi(z)\in\Phi,\gamma\in\mathbb{C}\backslash\{0\},z\in\mathbb{D}\right\}.$

In order to denote a new function class, we need to introduce the following function subclasses.

Definition 1.1. Let

$\alpha\in[0,1],\ \ \mu\in\mathbb{C}\backslash\{0\}.$

Also let $\phi(z)\in\Phi$ . A function $f(z)\in\mathcal{A}$ given by $(1.1)$ is said to be in the class $\mathcal{M}_{q}(\alpha, \mu; \phi)$ if the following condition is satisfied

$\frac{1}{\mu}\left[(1-\alpha)\frac{zf^{\prime}(z)}{f(z)}+\alpha\frac{(zf^{\prime}(z))^{\prime}}{f^{\prime}(z)}-1\right] \prec_q\phi(z)-1,\ \ \ z\in\mathbb{D}.$

Example 1.2. Let

$\alpha\in[0,1],\ \ \mu\in\mathbb{C}\backslash\{0\},\ \ \phi(z)\in\Phi.$

The function

$f(z):\mathbb{D}\rightarrow \mathbb{C}$

defined by the following

$\frac{1}{\mu}\left[(1-\alpha)\frac{zf^{\prime}(z)}{f(z)}+\alpha\frac{(zf^{\prime}(z))^{\prime}}{f^{\prime}(z)}-1\right] = z[\phi(z)-1]$

belongs to the class $\mathcal{M}_{q}(\alpha, \mu; \phi)$ .

Remark 1.3. There are some suitable choices of $\alpha, \mu$ which would provide some classical subclasses of analytic functions.

(1) By taking $\mu = 1$ in Definition 1.1, we have

$\mathcal{M}_{q}(\alpha,1;\phi)\equiv\mathcal{M}_{q}(\alpha;\phi)$

which is introduced by Mohd et al. ^[5].

(2) By taking $\alpha = 0$ in Definition 1.1, we have

$\mathcal{M}_{q}(0,\mu;\phi)\equiv\mathcal{S}^*_{q}(\mu;\phi)$

which is introduced by El-Ashwah et al.^[7]. Specially, for $\mu = 1$ we have

$\mathcal{M}_{q}(0,1;\phi)\equiv\mathcal{S}^*_{q}(\phi)$

which is introduced and studied by Mohd et al. ^[5].

(3) By taking $\alpha = 1$ in Definition 1.1, we have

$\mathcal{M}_{q}(1,\mu;\phi)\equiv\mathcal{C}_{q}(\mu;\phi)$

which is introduced by El-Ashwah et al. ^[7]. Specially, for $\mu = 1$ we have

$\mathcal{M}_{q}(1,1;\phi)\equiv\mathcal{C}_{q}(\phi)$

which is introduced and studied by Mohd et al. ^[5].

Now we define a generalization class of close-to-convex function by using quasi-subordination relationship.

Definition 1.4. Let

$\alpha\in[0,1],\beta\in[0,1],\ \ \mu\in\mathbb{C}\backslash\{0\},\gamma\in\mathbb{C}\backslash\{0\}.$

Also let

$\psi(z)\in\Phi,\ \ g(z)\in \mathcal{M}_{q}(\alpha,\mu;\phi).$

A function $f(z)\in\mathcal{A}$ given by $(1.1)$ is said to be in the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ if the following condition is satisfied

$\frac{1}{\gamma}\left[(1-\beta)\frac{zf^{\prime}(z)}{g(z)}+\beta\frac{(zf^{\prime}(z))^{\prime}}{g^{\prime}(z)}-1\right] \prec_q\psi(z)-1,\ \ \ z\in\mathbb{D}.$

Example 1.5. Let

$\alpha\in[0,1],\ \ \beta\in[0,1],\ \ \mu\in\mathbb{C}\backslash\{0\},\ \ \gamma\in\mathbb{C}\backslash\{0\},\ \ \phi(z)\in\Phi,\ \ g(z)\in \mathcal{M}_{q}(\alpha,\mu;\phi).$

The function

$f(z):\mathbb{D}\rightarrow \mathbb{C}$

defined by the following

$\frac{1}{\gamma}\left[(1-\beta)\frac{zf^{\prime}(z)}{g(z)}+\beta\frac{(zf^{\prime}(z))^{\prime}}{g^{\prime}(z)}-1\right] = z[\psi(z)-1]$

belongs to the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ .

Remark 1.6. There are some suitable choices of $\alpha, \beta, \mu, \gamma$ which would provide the following subclasses of the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ .

(1) By taking $\beta = 0$ in Definition 1.4, the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ reduces to the new subclass $\mathcal{K}_{q}(\alpha, \mu, \gamma; \phi, \psi)$ which is the class of generalized close-to-convex function satisfied by

$\frac{1}{\gamma}\left(\frac{zf^\prime(z)}{g(z)}-1\right)\prec_q\psi(z)-1,\ \ \ g(z)\in\mathcal{M}_{q}(\alpha,\mu;\phi),\phi(z),\psi(z)\in\Phi,z\in\mathbb{D}.$

Specially, for $\alpha = 1, \mu = 1$ in the class $\mathcal{K}_{q}(\alpha, \mu, \gamma; \phi, \psi)$ , we have

$\mathcal{H}_{q}(\gamma;\phi,\psi) = \left\{f(z)\in \mathcal{A}:\frac{1}{\gamma}\left(\frac{zf^\prime(z)}{g(z)}-1\right)\prec_q\psi(z)-1,g(z)\in\mathcal{C}_{q}(\phi),\phi(z),\psi(z)\in\Phi,z\in\mathbb{D}\right\};$

for $\alpha = 0, \mu = 1$ in the class $\mathcal{K}_{q}(\alpha, \mu, \gamma; \phi, \psi)$ , we have

$\mathcal{K}_{q}(0,1,\gamma;\phi,\psi)\equiv \mathcal{K}_{q}(\gamma;\phi,\psi)$

which is introduced and studied by Aoen et al. ^[21]. Also, for $\gamma = 1$ in the class $\mathcal{K}_{q}(\gamma; \phi, \psi)$ , we have the class $\mathcal{K}_{q}(\phi, \psi)$ which is introduced and studied by Aoen et al. ^[21].

(2) By taking $\beta = 1$ in Definition 1.4, the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ reduces to the new subclass $\mathcal{C}^*_{q}(\alpha, \mu, \gamma; \phi, \psi)$ which is the class of generalized quasi-convex function satisfied by

$\frac{1}{\gamma}\left(\frac{(zf^{\prime}(z))^\prime}{g^\prime(z)}-1\right)\prec_q\phi(z)-1,\ \ \ g(z)\in\mathcal{M}_{q}(\alpha,\mu;\phi),\phi(z),\psi(z)\in\Phi,z\in\mathbb{D}.$

Specially, for $\alpha = 1, \mu = 1$ in the class $\mathcal{C}^*_{q}(\alpha, \mu, \gamma; \phi, \psi)$ , we have

$\mathcal{C}^*_{q}(\gamma;\phi,\psi) = \left\{f(z)\in\mathcal{A}: \frac{1}{\gamma}\left(\frac{(zf^{\prime}(z))^\prime}{g^\prime(z)}-1\right)\prec_q\phi(z)-1,g(z)\in\mathcal{C}_{q}(\phi),\phi(z),\psi(z)\in\Phi,z\in\mathbb{D}\right\};$

for $\alpha = 0, \mu = 1$ in the class $K_{q}(\alpha, \mu, \gamma; \phi, \psi)$ , we have

$\mathcal{L}_{q}(\gamma;\phi,\psi) = \left\{f(z)\in\mathcal{A}: \frac{1}{\gamma}\left(\frac{(zf^{\prime}(z))^\prime}{g^\prime(z)}-1\right)\prec_q\phi(z)-1,g(z)\in\mathcal{S}^*_{q}(\phi),\phi(z),\psi(z)\in\Phi,z\in\mathbb{D}\right\}.$

Studying the theory of analytic functions has been an area of concern for many researchers. The study of coefficients estimate is a more special and important field in complex analysis. For example, the bound for the second coefficient $a_2$ of normalized univalent functions readily yields the growth and distortion bounds for univalent functions. The coefficient functional $|a_3-\mu a_2^2|$ (that is, Fekete-Szegö problem) also naturally arises in the investigation of univalency of analytic functions. There are now many results of this type in the literature, each of them dealing with coefficient estimate for various classes of functions. In particular, some authors start to study the coefficient estimates for various classes using quasi-subordination. For example, Arikan et al. ^[22] and Marut et al. ^[23] studied the Fekete-Szegö problem for some function subclasses using quasi-subordination. Aoen et al. ^[24] and Ahman et al. ^[25] obtained the results on coefficient estimates for various subclasses using quasi-subordination. The purpose of this paper is to study some properties of the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ and some of its subclasses, such as the integral expression, the first two coefficient estimate problems and Fekete-Szegö problem. Our results are new in this direction and they give birth to many corollaries.

In order to derive our main results, we have to recall here the following lemmas.

Lemma 1.7. Let $f(z)\in \mathcal{C}_{q}(\phi)$ , then

$\begin{align} f(z) = \displaystyle {\int}_0^z\exp\left(\displaystyle {\int}_0^t\frac{h(\xi)[\phi(u(\xi))-1]}{\xi}d\xi\right)dt, \end{align}$

(1.2)

where

$|h(z)|\leq1,\ \ u(z)\in\mathcal{U},\ \ \phi(z)\in\Phi.$

Proof. Since

$f(z)\in\mathcal{C}_{q}(\phi),$

then there exist two analytic functions $h(z), u(z)$ with

$|h(z)|\leq1,\ \ |u(z)| < 1,\ \ u(0) = 0$

such that

$\begin{align} \frac{zf^{\prime\prime}(z)}{f^{\prime}(z)} = h(z)[\phi(u(z))-1]. \end{align}$

(1.3)

By substitution, the Eq (1.3) can be reduced to a first-order differential equation. According to the method of solving the first-order differential equations, we can obtain the general solution of the equation. That is,

$\begin{align} f^{\prime}(z) = \exp\left(\displaystyle {\int}_0^z\frac{h(t)[\phi(u(t))-1]}{t}dt\right). \end{align}$

(1.4)

Integrating both sides of Eq (1.4), we get (1.2). Thus, the proof of Lemma 1.7 is complete.□

Lemma 1.8. ^[21] Let $f(z)\in\mathcal{S}_q^*(\phi)$ , then

$f(z) = z\exp\left(\displaystyle {\int}_0^z\frac{h(\xi)[\phi(u(\xi))-1]}{\xi}d\xi\right),$

where

$|h(z)|\leq1,\ \ u(z)\in\mathcal{U},\ \ \phi(z)\in\Phi.$

Lemma 1.9. ^[26] Let

$\varphi(z) = c_0+\sum\limits^{\infty}_{k = 1}c_kz^k$

be an analytic function in $\mathbb{D}$ with $|\varphi(z)|\leq1$ , then

$|c_0|\leq1,\ \ \ |c_1|\leq1-|c_0|^2.$

Lemma 1.10. ^[27] Let

$t(z) = \sum\limits^{\infty}_{k = 1}t_kz^k$

be an analytic function in $\mathbb{D}$ with $|t(z)| < 1$ , then

$|t_1|\leq1,|t_2-\mu t^2_1|\leq\max\{1,|\mu|\},$

where $\mu\in\mathbb{C}.$ The result is sharp for the functions

$t(z) = z\; \; \; \text{or} \; \; t(z) = z^2.$

2. Integral expressions

In this section, we discuss the integral expressions for the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ and some of its subclasses by using methods for solving differential equations.

Theorem 2.1. Let

$\alpha\in[0,1],\ \ \beta\in[0,1],\ \ \mu\in\mathbb{C}\backslash\{0\},\ \ \gamma\in\mathbb{C}\backslash\{0\},$

the function $f(z)\in\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ be given by (1.1). Then,

(ⅰ) If $\beta\neq0$ , then

$\begin{align} f(z) = \frac{1}{\beta}\displaystyle {\int}_0^z\frac{[g(t)]^{1-\frac{1}{\beta}}}{t}\left(\displaystyle {\int}_0^t[g(\xi)]^{\frac{1}{\beta}-1}g^{\prime}(\xi) [1+\gamma h(\xi)\left(\psi(u(\xi))-1\right)]d\xi\right)dt. \end{align}$

(2.1)

(ⅱ) If $\beta = 0$ , then

$f(z) = \displaystyle {\int}_0^z\frac{g(t)}{t}[1+\gamma h(t)\left(\psi(u(t))-1\right)]dt,$

where

$|h(z)|\leq1,\ \ u(z)\in\mathcal{U},\ \ \psi(z)\in\Phi,\ \ g(z)\in \mathcal{M}_{q}(\alpha,\mu;\phi).$

Proof. Since $f(z)\in\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ , then there exist two analytic functions $h(z), u(z)$ with

$|h(z)|\leq1,\ \ |u(z)| < 1,\ \ u(0) = 0$

such that

$\frac{1}{\gamma}\left[(1-\beta)\frac{zf^{\prime}(z)}{g(z)}+\beta\frac{(zf^{\prime}(z))^{\prime}}{g^{\prime}(z)}-1\right] = h(z)[\psi(u(z))-1].$

Then, we have

$\left(zf^\prime(z)\right)^{\prime} = \frac{(\beta-1)g^\prime(z)}{\beta g(z)}zf^\prime(z)+ \frac{\left[1+\gamma h(z)(\psi(u(z))-1)\right]g^{\prime}(z)}{\beta}.$

Let

$zf^\prime(z) = F(z),$

then we have

$F^\prime(z) = \frac{(\beta-1)g^\prime(z)}{\beta g(z)}F(z)+\frac{\left[1+\gamma h(z)(\psi(u(z))-1)\right]g^{\prime}(z)}{\beta}.$

Then, the above equation is a first-order nonhomogeneous linear differential equation. According to the method of solving first-order linear differential equations, we can obtain the general solution of the equation. That is,

$\begin{align} f^{\prime}(z) = \frac{1}{\beta}\frac{\left[g(z)\right]^{1-\frac{1}{\beta}}}{z}\displaystyle {\int}_0^z\left[g(t)\right]^{\frac{1}{\beta}-1}g^{\prime}(t) \left[1+\gamma h(t)\left(\psi(u(t))-1\right)\right]dt. \end{align}$

(2.2)

Integrating both sides of Eq (2.2), we get (2.1). Thus, the proof of Theorem 2.1 is complete. □

By taking $\beta = 1$ in Theorem 2.1, we obtain the following result.

Corollary 2.2. Let the function $f(z)\in \mathcal{C}^*_{q}(\alpha, \mu, \gamma; \phi, \psi)$ be given by (1.1). Then

$f(z) = \displaystyle {\int}_0^z\frac{1}{t}\left(\displaystyle {\int}_0^tg^{\prime}(\xi)[1+\gamma h(\xi)\left(\psi(u(\xi))-1\right)]d\xi\right)dt,$

where

$|h(z)|\leq1,\ \ u(z)\in\mathcal{U},\ \ \psi(z)\in\Phi,\ \ g(z)\in \mathcal{M}_{q}(\alpha,\mu;\phi).$

According to Lemmas 1.7 and 1.8 and Corollary 2.2, we can obtain the following two results.

Corollary 2.3. Let the function $f(z)\in \mathcal{C}^*_{q}(\gamma; \phi, \psi)$ be given by (1.1). Then

$f(z) = \displaystyle {\int}_0^z\frac{1}{t}\left(\displaystyle {\int}_0^t[1+\gamma h(\xi)\left(\psi(u(\xi))-1\right)] \exp\left(\displaystyle {\int}_0^\xi\frac{h_1(\zeta)[\phi(u_1(\zeta))-1]}{\zeta}d\zeta\right)d\xi\right)dt,$

where

$|h(z)|\leq1,\ \ |h_1(z)|\leq1,u(z),\ \ u_1(z)\in\mathcal{U},\ \ \phi(z),\psi(z)\in\Phi.$

Corollary 2.4. Let the function $f(z)\in \mathcal{L}_{q}(\gamma; \phi, \psi)$ be given by (1.1). Then

$\begin{equation*} \begin{aligned} f(z) = \displaystyle {\int}_0^z\frac{1}{t}\left(\displaystyle {\int}_0^t[1+\gamma h(\xi)\left(\psi(u(\xi))-1\right)][1+h_1(\xi)(\phi(u_1(\xi))-1)] \exp\left(\displaystyle {\int}_0^\xi\frac{h_1(\zeta)[\phi(u_1(\zeta))-1]}{\zeta}d\zeta\right)d\xi\right)dt,\end{aligned} \end{equation*}$

where

$|h(z)|\leq1,|h_1(z)|\leq1,\ \ u(z),u_1(z)\in\mathcal{U},\ \ \phi(z),\psi(z)\in\Phi.$

3. Coefficient estimates problem

In this section, we obtain the first two coefficient estimate and Fekete-Szegö problem for the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ and some subclasses of this class by using algebraic operations, fundamental inequalities of analytic functions.

In addition to special statements, suppose the Taylor series expression for the following functions, as follows

$\begin{align*} f(z)& = z+\sum\limits_{k = 2}^{\infty}a_kz^k,\ \ \ \ g(z) = z+\sum\limits_{k = 2}^{\infty}b_kz^k,\\ \phi(z)& = 1+\sum\limits_{k = 1}^{\infty}A_kz^k(A_1\in\mathbb{R},A_1 > 0),\ \ \ \ \psi(z) = 1+\sum\limits_{k = 1}^{\infty}B_kz^k(B_1\in \mathbb{R},B_1 > 0),\\ \varphi(z)& = c_0+\sum\limits_{k = 1}^{\infty}c_kz^k,\ \ \ \ h(z) = h_0+\sum\limits_{k = 1}^{\infty}h_kz^k,\\ u(z)& = \sum\limits_{k = 1}^{\infty}u_kz^k,\ \ \ \ v(z) = \sum\limits_{k = 1}^{\infty}v_kz^k. \end{align*}$

In order to derive our main results, we have to discuss the first two coefficient estimates and Fekete-Szegö problem for the class $\mathcal{M}_{q}(\alpha, \mu; \phi)$ .

Theorem 3.1. Let $\alpha\in[0, 1], \mu\in\mathbb{C}\backslash\{0\},$ the function $f(z)\in\mathcal{M}_{q}(\alpha, \mu; \phi)$ be given by (1.1). Then

$\begin{align} |a_2|&\leq \frac{|\mu|A_1}{1+\alpha}, \end{align}$

(3.1)

$\begin{align} |a_3|&\leq\frac{|\mu|A_1}{2(1+2\alpha)}\max\left\{1,\left|\frac{\mu(1+3\alpha)}{(1+\alpha)^2}A_1+\frac{A_2}{A_1}\right|\right\}, \end{align}$

(3.2)

and for any $\eta\in\mathbb{C}$ ,

$\begin{align} |a_3-\eta a_2^2|\leq \frac{|\mu|A_1}{2(1+2\alpha)}\max\left\{1,\left|MA_1-\frac{A_2}{A_1}\right|\right\} , \end{align}$

(3.3)

where

$M = \frac{\mu[2\eta(1+2\alpha)-(1+3\alpha)]}{(1+\alpha)^2}.$

Proof. If $f(z)\in\mathcal{M}_{q}(\alpha, \mu; \phi)$ , according to Definition 1.1, there exist analytic functions $\varphi(z)$ and $u(z)$ , with

$|\varphi(z)|\leq 1,\ \ u(0) = 0\; \; \text{and}\; \; |u(z)| < 1$

such that

$\begin{align} \frac{1}{\mu}\left[(1-\alpha)\frac{zf^{\prime}(z)}{f(z)}+\alpha\frac{(zf^{\prime}(z))^{\prime}}{f^{\prime}(z)}-1\right] = \varphi(z)[\phi(u(z))-1]. \end{align}$

(3.4)

By substituting the Taylor series expression for the function $f(z)$ to the left of the above expression, we have

$\frac{zf'(z)}{f(z)} = 1+a_2z+(2a_3-a_2^2)z^2+\cdots,$

$\frac{(zf'(z))'}{f'(z)} = 1+2a_2z+2(3a_3-2a_2^2)z^2+\cdots.$

Thus we get the following expression

$\begin{align} \frac{1}{\mu}\left[(1-\alpha)\frac{zf^{\prime}(z)}{f(z)}+\alpha\frac{(zf^{\prime}(z))^{\prime}}{f^{\prime}(z)}-1\right] = \frac{1}{\mu}\left\{(1+\alpha)a_2z+[2(1+2\alpha)a_3-(1+3\alpha)a_2^2]z^{2}+\cdots\right\}. \end{align}$

(3.5)

And by substituting the power series expression of the functions $\varphi(z), \phi(z), u(z)$ to the right of (3.4), we can get the following expression

$\begin{align} \varphi(z)[\phi(u(z))-1]& = (c_0+c_1z+c_2z^2+\cdots)\left[A_1(u_1z+u_2z_2^2+\cdots)+A_2(u_1z+u_2z_2^2+\cdots)^2+\cdots\right]\\ & = A_{1}c_{0}u_{1}z+[A_{1}c_{1}u_{1}+c_{0}\left(A_{1}u_{2}+A_{2}u_1^2\right)]z^{2}+\cdots. \end{align}$

(3.6)

By substituting (3.5) and (3.6) into (3.4) and comparing the coefficients of the same power terms on both sides, we can get

$\begin{align} a_{2} = \frac{\mu A_{1}c_{0}u_{1}}{1+\alpha}, \end{align}$

(3.7)

$a_{3} = \frac{\mu A_1}{2(1+2\alpha)} \left[c_1u_1+c_0\left(u_2+\frac{A_2}{A_1}u_1^2\right)+\frac{(1+3\alpha)\mu}{(1+\alpha)^2}A_1c_0^2u_1^2\right].$

Further,

$\begin{align} a_{3}-\eta a_{2}^2 = \frac{\mu A_1}{2(1+2\alpha)} \left[c_1u_1+c_0\left(u_2+\frac{A_2}{A_1}u_1^2\right)-\frac{2\eta(1+2\alpha)-(1+3\alpha)}{(1+\alpha)^2}\mu A_1c_0^2u_1^2\right]. \end{align}$

(3.8)

Applying Lemmas 1.9 and 1.10 to $(3.7)$ , we obtain

$|a_{2}|\leq\frac{|\mu|A_1}{1+\alpha}.$

Since $\varphi(z)$ is analytic and bounded in $\mathbb{D}$ , using ^[28], for some

$y,|y| < 1:|c_0|\leq1,c_1 = (1-c_0^2)y.$

Replacing the value of $c_1$ as defined above, we get

$\begin{align} a_{3}-\eta a_{2}^2 = \frac{\mu A_1}{2(1+2\alpha)}\left[yu_1+c_0\left(u_2+\frac{A_2}{A_1}u_1^2\right)-\left(\frac{2\eta(1+2\alpha)-(1+3\alpha)}{(1+\alpha)^2}\mu A_1u_1^2+yu_1\right)c_0^2\right]. \end{align}$

(3.9)

If $c_0 = 0$ , then applying Lemmas 1.9 and 1.10 to $(3.9)$ , we obtain

$|a_{3}-\eta a_{2}^2|\leq\frac{|\mu| A_1}{2(1+2\alpha)}.$

If $c_0\neq0$ , let

$\begin{align} G(c_0) = yu_1+c_0\left(u_2+\frac{A_2}{A_1}u_1^2\right)-\left(\frac{2\eta(1+2\alpha)-(1+3\alpha)}{(1+\alpha)^2}\mu A_1u_1^2+yu_1\right)c_0^2, \end{align}$

(3.10)

which is a polynomial in $c_0$ and have analytic in $|c_0|\leq1$ .

According to Maximum modulus principle, we get

$\max|G(c_0)| = \max\limits_{0\leq\theta\leq2\pi}|G(e^{i\theta})| = |G(1)|.$

Thus

$\begin{align} |a_{3}-\eta a_{2}^2|\leq\frac{|\mu| A_1}{2(1+2\alpha)}\left|u_2-\left(\frac{2\eta(1+2\alpha)-(1+3\alpha)}{(1+\alpha)^2}\mu A_1-\frac{A_2}{A_1}\right)u_1^2\right|. \end{align}$

(3.11)

Applying Lemma 1.10 to (3.11), we can conclude (3.3). For $\eta = 0$ in (3.3), we have (3.2).

Let

$\frac{1}{\mu}\left[(1-\alpha)\frac{zf^{\prime}(z)}{f(z)}+\alpha\frac{(zf^{\prime}(z))^{\prime}}{f^{\prime}(z)}-1\right] = \phi(z)-1$

$\frac{1}{\mu}\left[(1-\alpha)\frac{zf^{\prime}(z)}{f(z)}+\alpha\frac{(zf^{\prime}(z))^{\prime}}{f^{\prime}(z)}-1\right] = z[\phi(z^2)-1].$

then the results of (3.1)–(3.3) are sharp. Thus, the proof of Theorem 3.1 is complete. □

Remark 3.2. (1) For $\mu = 1$ in Theorem 3.1, we can obtain the result which is Theorem 2.10 in ^[5].

(2) For $\mu = 1, \alpha = 0$ and $\mu = 1, \alpha = 1$ in Theorem 3.1, we can obtain the results which are Theorems 2.1 and 2.4 in ^[5], respectively.

(3) For $\alpha = 0$ and $\alpha = 1$ in Theorem 3.1, we improve the results which are Theorems 2.1 and 2.7 in ^[7], respectively.

Theorem 3.3. Let

$\alpha\in[0,1],\ \ \beta\in(0,1],\ \ \mu\in\mathbb{C}\backslash\{0\},\ \ \gamma\in\mathbb{C}\backslash\{0\},$

the function $f(z)\in\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ be given by (1.1). Then

$\begin{align} |a_2|\leq& \frac{|\mu|A_1}{2(1+\alpha)}+\frac{|\gamma|B_1}{2(1+\beta)}, \end{align}$

(3.12)

$\begin{align} |a_3|\leq&\ \frac{|\mu|A_1}{6(1+2\alpha)}\max\left\{1,\left|\frac{\mu(1+3\alpha)}{(1+\alpha)^2}A_1+\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{3(1+2\beta)}\max\left\{1,\frac{|B_2|}{B_1}\right\}\\&+ \frac{|\mu\gamma|(1+3\beta)}{3(1+\alpha)(1+\beta)(1+2\beta)}A_1B_1, \end{align}$

(3.13)

and for any $\tau\in\mathbb{C}$

$\begin{align} |a_3-\tau a_2^2|\leq& \frac{|\mu|A_1}{6(1+2\alpha)}\max\left\{1,\left|PA_1-\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{3(1+2\beta)}\max\left\{1,\left|QB_1-\frac{B_2}{B_1}\right|\right\}\\&+ \frac{|\mu\gamma[2(1+3\beta)-3\tau(1+2\beta)]|}{6(1+\alpha)(1+\beta)(1+2\beta)}A_1B_1, \end{align}$

(3.14)

where

$P = \frac{\mu[3\tau(1+2\alpha)-2(1+3\alpha)]}{2(1+\alpha)^2},\ \ \ Q = \frac{3\tau\gamma(1+2\beta)}{4(1+\beta)^2}.$

Proof. If $f(z)\in\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ , then there exist analytic functions $h(z)$ and $v(z)$ , with

$|\varphi(z)|\leq 1,\ \ v(0) = 0\; \; \text{and}\; \; |v(z)| < 1$

such that

$\begin{align} \frac{1}{\gamma}\left[(1-\beta)\frac{zf^{\prime}(z)}{g(z)}+\beta\frac{(zf^{\prime}(z))^{\prime}}{g^{\prime}(z)}-1\right] = h(z)[\psi(v(z))-1]. \end{align}$

(3.15)

By substituting the Taylor series expression for the functions $f(z), g(z)$ to the left of the above expression, we have

$\begin{align*} \frac{zf'(z)}{g(z)}& = 1+(2a_2-b_2)z+(3a_3-b_3+b_2^2-2a_2b_2)z^2+\cdots,\\ \frac{(zf'(z))'}{g'(z)}& = 1+2(2a_2-b_2)z+2(9a_3-3b_3+4b_2^2-8a_2b_2)z^2+\cdots. \end{align*}$

Thus we get the following expression

$\begin{align} &\frac{1}{\gamma}\left[(1-\beta)\frac{zf^{\prime}(z)}{g(z)}+\beta\frac{(zf^{\prime}(z))^{\prime}}{g^{\prime}(z)}-1\right] \\ & = \frac{1}{\gamma}\left\{(1+\beta)(2a_2-b_2)z+\left[(1+2\beta)(3a_3-b_3)+(1+3\beta)(b_2^2-2a_2b_2)\right]z^{2}+\cdots\right\}. \end{align}$

(3.16)

Similar to the proof of Theorem 3.1, by substituting the power series expression of the functions $h(z), \psi(z), v(z)$ to the right of (3.15), we can get the following expression

$\begin{align} h(z)[\psi(v(z))-1] = B_{1}h_{0}v_{1}z+[B_{1}h_{1}v_{1}+h_{0}(B_{1}v_{2}+B_{2}v_1^2)]z^{2}+\cdots. \end{align}$

(3.17)

By substituting (3.16) and (3.17) into (3.15) and comparing the coefficients of the same power terms on both sides, we can get

$\begin{align} a_{2} = \frac{1}{2}\left(\frac{\gamma B_1h_0v_1}{1+\beta}+b_2\right) \end{align}$

(3.18)

and

$a_{3} = \frac{1}{3(1+2\beta)}\left\{(1+2\beta)b_3-(1+3\beta)(b_2^2-2a_2b_2)+\gamma[B_1h_1v_1+h_0(B_1v_2+B_2v_1^2)]\right\}.$

Further,

$\begin{align} a_{3}-\tau a_{2}^2 = &\frac{1}{3}\left(b_3-\frac{3}{4}\tau b_2^2\right)+ \frac{\gamma[2(1+3\beta)-3\tau(1+2\beta)]}{6(1+\beta)(1+2\beta)}B_1h_0v_1b_2\\&+\frac{\gamma B_1}{3(1+2\beta)}\left[h_1v_1+h_0\left(v_2+\frac{B_2}{B_1}v_1^2\right)-\frac{3\tau\gamma(1+2\beta)}{4(1+\beta)^2}B_1h_0^2v_1^2\right]. \end{align}$

(3.19)

Applying Lemmas 1.9 and 1.10 to (3.18) and (3.19), we obtain

$\begin{align} |a_{2}|\leq\frac{1}{2}\left(\frac{|\gamma|B_1}{1+\beta}+|b_2|\right) \end{align}$

(3.20)

and

$\begin{align} |a_{3}-\tau a_{2}^2|\leq& \frac{1}{3}\left|b_3-\frac{3}{4}\tau b_2^2\right|+ \frac{|\gamma[2(1+3\beta)-3\tau(1+2\beta)]|}{6(1+\beta)(1+2\beta)}B_1|b_2|\\&+\frac{|\gamma|B_1}{3(1+2\beta)} \left|h_1v_1+h_0\left(v_2+\frac{B_2}{B_1}v_1^2\right)-\frac{3\tau\gamma(1+2\beta)}{4(1+\beta)^2}B_1h_0^2v_1^2\right|. \end{align}$

(3.21)

According to Theorem 3.1, it follows that

$\begin{align} |b_2|\leq \frac{|\mu|A_1}{1+\alpha} \end{align}$

(3.22)

and, for any complex number $\tau$ , we have

$\begin{align} |b_3-\frac{3}{4}\tau b_2^2|\leq\frac{|\mu|A_1}{2(1+2\alpha)}\max\left\{1,\left|PA_1-\frac{A_2}{A_1}\right|\right\}, \end{align}$

(3.23)

where

$P = \frac{\mu[3\tau(1+2\alpha)-2(1+3\alpha)]}{2(1+\alpha)^2}.$

Similar to the proof of Theorem 3.1, we can also get the following inequality

$\begin{align} \left|h_1v_1+h_0\left(v_2+\frac{B_2}{B_1}v_1^2\right)-\frac{3\tau\gamma(1+2\beta)}{4(1+\beta)^2}B_1h_0^2v_1^2\right|\leq \max\left\{1,\left|QB_1-\frac{B_2}{B_1}\right|\right\}, \end{align}$

(3.24)

where

$Q = \frac{3\tau\gamma(1+2\beta)}{4(1+\beta)^2}.$

By substituting (3.22) into (3.20), we get (3.12). And by substituting (3.22)–(3.24) into (3.21), we can conclude (3.14). For $\tau = 0$ in (3.14), we have (3.13).

The results of (3.12) and (3.13) are sharp for $\beta\neq0$ if

$f(z) = \frac{1}{\beta}\displaystyle {\int}_0^z\frac{[g(t)]^{1-\frac{1}{\beta}}}{t}\left(\displaystyle {\int}_0^t[g(\xi)]^{\frac{1}{\beta}-1}g^{\prime}(\xi) [1+\gamma\left(\psi(\xi)-1\right)]d\xi\right)dt$

and the results of (3.14) and (3.15) are sharp for $\beta = 0$ if

$f(z) = \displaystyle {\int}_0^z\frac{g(t)}{t}[1+\gamma\left(\psi(t)-1\right)]dt$

$f(z) = \displaystyle {\int}_0^z\frac{g(t)}{t}[1+\gamma\left(\psi(t^2)-1\right)]dt.$

Thus, the proof of Theorem 3.3 is complete. □

By taking special values of parameters $\alpha, \beta, \mu$ in Theorem 3.3, we can obtain coefficient estimates for functions belonging to some subclasses of the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ .

Corollary 3.4. Let the function $f(z)\in\mathcal{K}_{q}(\alpha, \mu, \gamma; \phi, \psi)$ . Then

$\begin{align*} |a_2|&\leq \frac{|\mu|A_1}{2(1+\alpha)}+\frac{|\gamma|B_1}{2},\\ |a_3|&\leq\ \frac{|\mu|A_1}{6(1+2\alpha)}\max\left\{1,\left|\frac{\mu(1+3\alpha)}{(1+\alpha)^2}A_1+\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{3}\max\left\{1,\frac{|B_2|}{B_1}\right\}+ \frac{|\mu\gamma|}{3(1+\alpha)}A_1B_1, \end{align*}$

and for any $\tau\in\mathbb{C}$ ,

$\begin{align*} |a_3-\tau a_2^2|\leq& \frac{|\mu|A_1}{6(1+2\alpha)}\max\left\{1,\left|\frac{\mu[3\tau(1+2\alpha)-2(1+3\alpha)]}{2(1+\alpha)^2}A_1-\frac{A_2}{A_1}\right|\right\}\\&+ \frac{|\gamma|B_1}{3}\max\left\{1,\left|\frac{3\tau\gamma}{4}B_1-\frac{B_2}{B_1}\right|\right\}+ \frac{|\mu\gamma(2-3\tau)|}{6(1+\alpha)}A_1B_1. \end{align*}$

Remark 3.5. For $\alpha = \beta = 0, \mu = 1$ in Theorem 3.3 or $\alpha = 0, \mu = 1$ in Corollary 3.4, we obtain the result which is Corollary 3 in ^[21].

Corollary 3.6. Let the function $f(z)\in\mathcal{H}_{q}(\gamma; \phi, \psi)$ . Then

$\begin{align*} |a_2|&\leq \frac{A_1+2|\gamma|B_1}{4},\\ |a_3|&\leq\ \frac{A_1}{18}\max\left\{1,\left|A_1+\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{3}\max\left\{1,\frac{|B_2|}{B_1}\right\}+ \frac{|\gamma|}{6}A_1B_1, \end{align*}$

and for any $\tau\in\mathbb{C}$

$|a_3-\tau a_2^2|\leq \frac{A_1}{18}\max\left\{1,\left|\frac{9\tau-8}{8}A_1-\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{3}\max\left\{1,\left|\frac{3\tau\gamma}{4}B_1-\frac{B_2}{B_1}\right|\right\}+ \frac{|\gamma(2-3\tau)|}{12}A_1B_1.$

Especially, let

$\gamma = 1,\ \ \ \phi(z) = \psi(z) = \frac{1+z}{1-z},$

we can obtain the following result.

Remark 3.7. Let the function

$f(z)\in\mathcal{H}_{q}(1;\frac{1+z}{1-z},\frac{1+z}{1-z}).$

Then

$|a_2|\leq \frac{3}{2},\quad |a_3|\leq\ \frac{5}{3},$

and for any $\tau\in\mathbb{C}$

$|a_3-\tau a_2^2|\leq \frac{1}{9}\max\left\{1,\left|\frac{9\tau-12}{4}\right|\right\}+ \frac{2}{3}\max\left\{1,\left|\frac{3\tau}{2}-1\right|\right\}+ \frac{|2-3\tau|}{3}.$

The sharpness of the estimates is demonstrated by the functions

$f_1(z) = \frac{2z}{1-z}+\log(1-z)$

$f_2(z) = \frac{z}{1-z}+\frac{1}{2}\log(1-z^2),$

and the graph of the functions $f_1(z)$ and $f_1(z)$ are shown as follows,

In and , the three-dimensional coordinate system, coupled with color, is used to represent complex functions. Specifically, the $x$ -axis corresponds to the real part of the variable $z$ , the $y$ -axis to the imaginary part of $z$ , the $z$ -axis indicates the real part of the function, and the color signifies the imaginary part of the function.

Figure 1. The image of

$f_1(z)$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. The image of

$f_2(z)$ .

DownLoad: Full-Size Img PowerPoint

Corollary 3.8. Let the function $f(z)\in\mathcal{C}^*_{q}(\alpha, \mu, \gamma; \phi, \psi)$ . Then

$\begin{align*} |a_2|&\leq \frac{|\mu|A_1}{2(1+\alpha)}+\frac{|\gamma|B_1}{4},\\ |a_3|&\leq\ \frac{|\mu|A_1}{6(1+2\alpha)}\max\left\{1,\left|\frac{\mu(1+3\alpha)}{(1+\alpha)^2}A_1+\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{9}\max\left\{1,\frac{|B_2|}{B_1}\right\}+ \frac{2|\mu\gamma|}{9(1+\alpha)}A_1B_1, \end{align*}$

and for any $\tau\in\mathbb{C}$ ,

$\begin{align*} |a_3-\tau a_2^2|\leq& \frac{|\mu|A_1}{6(1+2\alpha)}\max\left\{1,\left|\frac{\mu[3\tau(1+2\alpha)-2(1+3\alpha)]}{2(1+\alpha)^2}A_1-\frac{A_2}{A_1}\right|\right\}\\&+ \frac{|\gamma|B_1}{9}\max\left\{1,\left|\frac{9\tau\gamma}{16}B_1-\frac{B_2}{B_1}\right|\right\}+ \frac{|\mu\gamma(8-9\tau)|}{36(1+\alpha)}A_1B_1. \end{align*}$

Corollary 3.9. Let the function $f(z)\in\mathcal{C}^*_{q}(\gamma; \phi, \psi)$ . Then

$\begin{align*} |a_2|\leq& \frac{A_1+|\gamma|B_1}{4},\\ |a_3|&\leq\ \frac{A_1}{18}\max\left\{1,\left|A_1+\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{9}\max\left\{1,\frac{|B_2|}{B_1}\right\}+ \frac{|\gamma|}{9}A_1B_1, \end{align*}$

and for any $\tau\in\mathbb{C}$ ,

$|a_3-\tau a_2^2|\leq \frac{A_1}{18}\max\left\{1,\left|\frac{9\tau-8}{8}A_1-\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{9}\max\left\{1,\left|\frac{9\tau\gamma}{16}B_1-\frac{B_2}{B_1}\right|\right\}+ \frac{|\gamma(8-9\tau)|}{72}A_1B_1.$

Corollary 3.10. Let the function $f(z)\in\mathcal{L}_{q}(\gamma; \phi, \psi)$ . Then

$\begin{align*} |a_2|&\leq \frac{2A_1+|\gamma|B_1}{4},\\ |a_3|&\leq\ \frac{A_1}{6}\max\left\{1,\left|A_1+\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{9}\max\left\{1,\frac{|B_2|}{B_1}\right\}+ \frac{2|\gamma|}{9}A_1B_1, \end{align*}$

and for any $\tau\in\mathbb{C}$ ,

$|a_3-\tau a_2^2|\leq \frac{A_1}{6}\max\left\{1,\left|\frac{3\tau-2}{2}A_1-\frac{A_2}{A_1}\right|\right\}+ \frac{|\gamma|B_1}{9}\max\left\{1,\left|\frac{9\tau\gamma}{16}B_1-\frac{B_2}{B_1}\right|\right\}+ \frac{|\gamma(8-9\tau)|}{36}A_1B_1.$

4. Conclusions

In this paper, we introduce the new function class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ , which is a expanded close-to-convex functions defined by quasi-subordination. We mainly study the integral expression, the first two coefficient estimates and Fekete-Szegö problem for this class and some of its subclasses. In the future, we can consider to study other forms of coefficient estimation, such as Milin coefficient eatimate, Zal-cman functional estimate, high order Hankel Determinant estimate for these classes using the concepts dealt with in the paper.

Author contributions

Aoen: conceptualization, methodology, software, investigation, writing—original draft preparation, writing—review and editing, project administration, funding acquisition, visualization; Shuhai Li: conceptualization, methodology, formal analysis, resources; Tula: validation, data curation; Shuwen Li and Hang Gao: supervision. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The present investigation was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region of China under Grant (No. 2025MS01034, 2024MS01014, 2020MS01010), the National Natural Science Foundation of China (Grant No.11561001) and the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant (No. NJYT-18-A14).

Conflicts of interest

The authors declare no conflicts of interest.

Acknowledgments

We acknowledge all our colleagues for their help during the preparation of the manuscript by providing all the relevant information. We are also thankful to Ms. Bethany Pond (Analytical Chemist at AllExcel, Inc.) for editing the manuscript.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Consent for publications

Both the authors have agreed to submit this paper for publication.

Ethical approval

Not applicable.

Conflict of interest

The authors declare no conflict of interests.

References

[1]	Kumar DR, Aslinia F, Yale SH, et al. (2011) Jean-Martin Charcot: the father of neurology. Clin Med Res 9: 46-49. https://doi.org/10.3121/cmr.2009.883
[2]	Swinnen B, Robberecht W (2014) The phenotypic variability of amyotrophic lateral sclerosis. Nat Rev Neurol 10: 661-670. https://doi.org/10.1038/nrneurol.2014.184
[3]	Kaufman David Myland (2007) Peripheral Nerve Disorders. In, Clinical Neurology for Psychiatrists Pages 61-85. ISBN 9781416030744, https://doi.org/10.1016/B978-1-4160-3074-4.10005-0
[4]	Mathis S, Goizet C, Soulages A, et al. (2019) Genetics of amyotrophic lateral sclerosis: a review. J Neurol Sci 399: 217-226. https://doi.org/10.1016/j.jns.2019.02.030
[5]	Cook C, Petrucelli L (2019) Genetic convergence brings clarity to the enigmatic red line in ALS. Neuron 101: 1057-1069. https://doi.org/10.1016/j.neuron.2019.02.032
[6]	Bozzoni V, Pansarasa O, Diamanti L, et al. (2016) Amyotrophic lateral sclerosis and environmental factors. Funct Neurol 31: 7-19. https://doi.org/10.11138/FNeur/2016.31.1.007
[7]	Malek AM, Barchowsky A, Bowser R, et al. (2015) Exposure to hazardous air pollutants and the risk of amyotrophic lateral sclerosis. Environ Pollut 197: 181-6. https://doi.org/10.1016/j.envpol.2014.12.010
[8]	Wang MD, Little J, Gomes J, et al. (2016) Identification of risk factors associated with onset and progression of amyotrophic lateral sclerosis using systematic review and meta-analysis. Neurotoxicology pii: S0161-813X(16)30116-4. https://doi.org/10.1016/j.neuro.2016.06.015
[9]	Appel SH, Zhao W, Beers DR, et al. (2011) The microglial-motoneuron dialogue in ALS. Acta Myol 30: 4-8.
[10]	Pasinelli P, Brown RH (2006) Molecular biology of amyotrophic lateral sclerosis: insights from genetics. Nat Rev Neurosci 7: 710-723. https://doi.org/10.1038/nrn1971
[11]	Czaplinski A, Yen AA, Simpson EP, et al. (2006) Slower disease progression and prolonged survival in contemporary patients with amyotrophic lateral sclerosis: is the natural history of amyotrophic lateral sclerosis changing?. Arch Neurol 63: 1139-1143. https://doi.org/10.1001/archneur.63.8.1139
[12]	Ueno H, Kobatake K, Matsumoto M, et al. (2011) Severe brain atrophy after long-term survival seen in siblings with familial amyotrophic lateral sclerosis and a mutation in the optineurin gene: a case series. J Med Case Re 5: 573. https://doi.org/10.1186/1752-1947-5-573
[13]	Benatar M, Boylan K, Jeromin A, et al. (2016) ALS biomarkers for therapy development: state of the field and future directions. Muscle Nerve 53: 169-182. https://doi.org/10.1002/mus.24979
[14]	Poesen K, Van Damme P (2019) Diagnostic and prognostic performance of neurofilaments in ALS. Front Neurol 9: 1167. https://doi.org/10.3389/fneur.2018.01167
[15]	Zetterberg H, Jacobsson J, Rosengren L, et al. (2007) Cerebrospinal fluid neurofilament light levels in amyotrophic lateral sclerosis: impact of SOD1 genotype. Eur J Neurol 14: 1329-1333. https://doi.org/10.1111/j.1468-1331.2007.01972.x
[16]	Gendron TF, Daughrity LM, Heckman MG, et al. (2017) Phosphorylated neurofilament heavy chain: a biomarker of survival for C9ORF72-associated amyotrophic lateral sclerosis. Ann Neurol 82: 139-146. https://doi.org/10.1002/ana.24980
[17]	Lu CH, Macdonald-Wallis C, Gray E, et al. (2015) Neurofilament light chain: a prognostic biomarker in amyotrophic lateral sclerosis. Neurology 84: 2247-2257. https://doi.org/10.1212/WNL.0000000000001642
[18]	Vu LT, Bowser R (2017) Fluid-based biomarkers for amyotrophic lateral sclerosis. Neurotherapeutics 14: 119-134. https://doi.org/10.1007/s13311-016-0503-x
[19]	Fortea J, Carmona-Iragui M, Benejam B, et al. (2018) Plasma and CSF biomarkers for the diagnosis of Alzheimer's disease in adults with Down syndrome: a cross-sectional study. Lancet Neurol 17: 860-869. https://doi.org/10.1016/S1474-4422(18)30285-0
[20]	Bowser R, Turner MR, Shefner J (2011) Biomarkers in amyotrophic lateral sclerosis: opportunities and limitations. Nat Rev Neurol 7: 631-638. https://doi.org/10.1038/nrneurol.2011.151
[21]	Benatar M, Wuu J, Lombardi V, et al. (2019) Neurofilaments in pre-symptomatic ALS and the impact of genotype. Amyotroph Lateral Scler Frontotemporal Degener 20: 538-548. https://doi.org/10.1080/21678421.2019.1646769
[22]	Chen X, Chen Y, Wei Q, et al. (2016) Assessment of a multiple biomarker panel for diagnosis of amyotrophic lateral sclerosis. BMC Neurol 16: 173. https://doi.org/10.1186/s12883-016-0689-x
[23]	Zhou YN, Chen YH, Dong SQ, et al. (2021) Role of Blood Neurofilaments in the Prognosis of Amyotrophic Lateral Sclerosis: A Meta-Analysis. Front Neurol 12: 712245. https://doi.org/10.3389/fneur.2021.712245
[24]	Steinacker P, Feneberg E, Weishaupt J, et al. (2016) Neurofilaments in the diagnosis of motoneuron diseases: a prospective study on 455 patients. J Neurol Neurosurg Psychiatry 87: 12-20. https://doi.org/10.1136/jnnp-2015-311387
[25]	Chipika RH, Finegan E, Li Hi Shing S, et al. (2019) Tracking a fast-moving disease: longitudinal markers, monitoring, and clinical trial endpoints in ALS. Front Neurol 10: 229. https://doi.org/10.3389/fneur.2019.00229
[26]	Jara JH, Gautam M, Kocak N, et al. (2019) MCP1-CCR2 and neuroinflammation in the ALS motor cortex with TDP-43 pathology. J Neuroinflammation 16: 196. https://doi.org/10.1186/s12974-019-1589-y
[27]	Lu CH, Allen K, Oei F, et al. (2016) Systemic inflammatory response and neuromuscular involvement in amyotrophic lateral sclerosis. Neurol Neuroimmunol Neuroinflamm 3: e244. https://doi.org/10.1212/NXI.0000000000000244
[28]	Lunetta C, Lizio A, Maestri E, et al. (2017) Serum C-reactive protein as a prognostic biomarker in amyotrophic lateral sclerosis. JAMA Neurol 74: 660-667. https://doi.org/10.1001/jamaneurol.2016.6179
[29]	Beuche W, Yushchenko M, Mäder M, et al. (2000) Matrix metalloproteinase-9 is elevated in serum of patients with amyotrophic lateral sclerosis. Neuro Rep 11: 3419-3422. https://doi.org/10.1097/00001756-200011090-00003
[30]	Ilzecka J (2001) EMMPRIN levels in serum of patients with amyotrophic lateral sclerosis. Acta Neurol Scand 124: 424-428. https://doi.org/10.1111/j.1600-0404.2011.01519.x
[31]	Simpson EP, Henry YK, Henkel JS, et al. (2004) Increased lipid peroxidation in sera of ALS patients: a potential biomarker of disease burden. Neurology 62: 1758-1765. https://doi.org/10.1212/WNL.62.10.1758
[32]	Babu GN, Kumar A, Chandra R, et al. (2008) Elevated inflammatory markers in a group of amyotrophic lateral sclerosis patients from northern India. Neurochem Res 33: 1145-1149. https://doi.org/10.1007/s11064-007-9564-x
[33]	Kharel S, Ojha R, Preethish-Kumar V, et al. (2022) C-reactive protein levels in patients with amyotrophic lateral sclerosis: A systematic review. Brain Behav 12: e2532. https://doi.org/10.1002/brb3.2532
[34]	Zhang R, Gascon R, Miller RG, et al. (2006) MCP-1 chemokine receptor CCR2 is decreased on circulating monocytes in sporadic amyotrophic lateral sclerosis (sALS). J Neuroimmunol 179: 87-93. https://doi.org/10.1016/j.jneuroim.2006.06.008
[35]	Liu GT, Hwang CS, Hsieh CH, et al. (2013) Eosinophil-derived neurotoxin is elevated in patients with amyotrophic lateral sclerosis. Mediat Inflamm 2013: 421389. https://doi.org/10.1155/2013/421389
[36]	Kuhle J, Lindberg RLP, Regeniter A, et al. (2009) Increased levels of inflammatory chemokines in amyotrophic lateral sclerosis. Eur J Neurol 16: 771-774. https://doi.org/10.1111/j.1468-1331.2009.02560.x
[37]	Ilzecka J (2011) Granzymes A and B levels in serum of patients with amyotrophic lateral sclerosis. Clin Biochem 44: 650-653. https://doi.org/10.1016/j.clinbiochem.2011.02.006
[38]	Hwang CS, Liu GT, Chang MD, et al. (2013) Elevated serum autoantibody against high mobility group box 1 as a potent surrogate biomarker for amyotrophic lateral sclerosis. Neurobiol Dis 58: 13-18. https://doi.org/10.1016/j.nbd.2013.04.013
[39]	Moreau C, Devos D, Brunaud-Danel V, et al. (2005) Elevated IL-6 and TNF-α levels in patients with ALS: inflammation or hypoxia?. Neurology 65: 1958-1960. https://doi.org/10.1212/01.wnl.0000188907.97339.76
[40]	Baron P, Bussini S, Cardin V, et al. (2005) Production of monocyte chemoattractant protein-1 in amyotrophic lateral sclerosis. Muscle Nerve 32: 541-544. https://doi.org/10.1002/mus.20376
[41]	Zhu Y, Li M, Zhang J, et al. (2022) Association Between CReactive Protein and Risk of Amyotrophic Lateral Sclerosis: A Mendelian Randomization Study. Front Genet 13: 919031. https://doi.org/10.3389/fgene.2022.919031
[42]	Cereda C, Baiocchi C, Bongioanni P, et al. (2008) TNF and sTNFR1/2 plasma levels in ALS patients. J Neuroimmunol 194: 123-131. https://doi.org/10.1016/j.jneuroim.2007.10.028
[43]	Dupuis L, Pradat PF, Ludolph AC, et al. (2011) Energy metabolism in amyotrophic lateral sclerosis. Lancet Neurol 10: 75-82. https://doi.org/10.1016/S1474-4422(10)70224-6
[44]	Sokolowska B, Jozwik A, Niebroj-Dobosz I, et al. (2009) Evaluation of matrix metalloproteinases in serum of patients with amyotrophic lateral sclerosis with pattern recognition methods. J Physiol Pharmacol 60: 117-120.
[45]	Gao J, Wang D, Liu D, et al. (2015) Tumor necrosis factor-related apoptosis-inducing ligand induces the expression of proinflammatory cytokines in macrophages and re-educates tumor-associated macrophages to an antitumor phenotype. Mol Biol Cell 26: 3178-89. https://doi.org/10.1091/mbc.e15-04-0209
[46]	Iłzecka J (2012) Serum soluble OX40 in patients with amyotrophic lateral sclerosis. Acta Clin Croat 51: 3-7.
[47]	Iłzecka J (2009) Serum-soluble receptor for advanced glycation end product levels in patients with amyotrophic lateral sclerosis. Acta Neurol Scand 120: 119-122. https://doi.org/10.1111/j.1600-0404.2008.01133.x
[48]	Iłzecka J (2008) Decreased serum-soluble TRAIL levels in patients with amyotrophic lateral sclerosis. Acta Neurol Scand 117: 343-346. https://doi.org/10.1111/j.1600-0404.2007.00947.x
[49]	Dupuis L, Corcia P, Ferganietal A (2008) Dyslipidemiaisaprotective factor in amyotrophic lateral sclerosis. Neurology 70: 1004-1009. https://doi.org/10.1212/01.wnl.0000285080.70324.27
[50]	Ilzecka J, Stelmasiak Z, Dobosz B (2001) Interleukin-1β converting enzyme/Caspase-1 (ICE/Caspase-1) and soluble APO-1/Fas/CD 95 receptor in amyotrophic lateral sclerosis patients. Acta Neurol Scand 103: 255-258. https://doi.org/10.1034/j.1600-0404.2001.103004255.x
[51]	Iłzecka J (2012) Serum caspase-9 levels are increased in patients with amyotrophic lateral sclerosis. Neurol Sci 33: 825-829. https://doi.org/10.1007/s10072-011-0837-4
[52]	Staats KA, Borchelt DR, Tansey MG, et al. (2022) Blood-based biomarkers of inflammation in amyotrophic lateral sclerosis. Mol Neurodegener 17: 11. https://doi.org/10.1186/s13024-022-00515-1
[53]	Mitchell RM, Simmons Z, Beard JL, et al. (2010) Plasma biomarkers associated with ALS and their relationship to iron homeostasis. Muscle Nerve 42: 95-103. https://doi.org/10.1002/mus.21625
[54]	Robelin L, De Aguilar JLG (2014) Blood Biomarkers for Amyotrophic Lateral Sclerosis: Myth or Reality?. BioMed Res Inter : Article ID 525097. https://doi.org/10.1155/2014/525097
[55]	Keizman D, Rogowski O, Berlineretal S (2009) Low-grade systemic inflammation in patients with amyotrophic lateral sclerosis. Acta Neurol Scand 119: 383-389. https://doi.org/10.1111/j.1600-0404.2008.01112.x
[56]	Lacomblez L, Doppler V, Beucler I, et al. (2002) APOE: a potential marker of disease progression in ALS. Neurology 58: 1112-1114. https://doi.org/10.1212/WNL.58.7.1112
[57]	Wilson ME, Boumaza I, Lacomis D, et al. (2010) Cystatin C: a candidate biomarker for amyotrophic lateral sclerosis. PLoS ONE 5: Article ID e15133. https://doi.org/10.1371/journal.pone.0015133
[58]	Fang F, Kwee LC, Allen KD, et al. (2010) Association between blood lead and the risk of amyotrophic lateral sclerosis. Am J Epidemiol 171: 1126-1133. https://doi.org/10.1093/aje/kwq063
[59]	Roos PM, Vesterberg O, Syversen T, et al. (2013) Metal concentrations in cerebrospinal fluid and blood plasma from patients with amyotrophic lateral sclerosis. Biol Trace Elem Res 151: 159-170. https://doi.org/10.1007/s12011-012-9547-x
[60]	Mantovani S, Garbelli S, Pasini A, et al. (2009) Immune system alterations in sporadic amyotrophic lateral sclerosis patients suggest an ongoing neuroinflammatory process. J Neuroimm 210: 73-79. https://doi.org/10.1016/j.jneuroim.2009.02.012
[61]	Simone IL, Ruggieri M, Tortelli R, et al. (2011) Serum N-acetylaspartate level in amyotrophic lateral sclerosis. Arch Neurol 68: 1308-1312. https://doi.org/10.1001/archneurol.2011.217
[62]	Gaiottino J, Norgren N, Dobson R, et al. (2013) Increased neurofilament light chain blood levels in neurodegenerative neurological diseases. PLoS ONE 8: Article ID e75091. https://doi.org/10.1371/journal.pone.0075091
[63]	Tortelli R, Ruggieri M, Cortese R, et al. (2012) Elevated cerebrospinal fluid neurofilament light levels in patients with amyotrophic lateral sclerosis: a possible marker of disease severity and progression. Eur J Neurol 19: 1561-1567. https://doi.org/10.1111/j.1468-1331.2012.03777.x
[64]	DeJesus-Hernandez M, Mackenzie IR, Boeve BF, et al. (2011) Expanded GGGGCC hexanucleotide repeat in noncoding region of C9ORF72 causes chromosome 9p-linked FTD and ALS. Neuron 72: 245-256. https://doi.org/10.1016/j.neuron.2011.09.011
[65]	Sreedharan J, Blair IP, Tripathietal VB (2008) TDP-43mutations in familial and sporadic amyotrophic lateral sclerosis. Science 319: 1668-1672. https://doi.org/10.1126/science.1154584
[66]	Verstraete E, Kuiperij HB, VanBlitterswijketal MM (2012) TDP-43 plasma levels are higher in amyotrophic lateral sclerosis. Amyotroph Lateral Sc 13: 446-451. https://doi.org/10.3109/17482968.2012.703208
[67]	Noto YI, Shibuya K, Satoetal Y (2011) Elevated CSFTDP-43 levels in amyotrophic lateral sclerosis: specificity, sensitivity, and a possible prognostic value. Amyotroph Lateral Sc 12: 140-143. https://doi.org/10.3109/17482968.2010.541263
[68]	Ganesalingam J, An J, Shaw CE, et al. (2011) Combination of neurofilament heavy chain and complement C3 as CSF biomarkers for ALS. J Neurochem 117: 528-537. https://doi.org/10.1111/j.1471-4159.2011.07224.x
[69]	Boylan KB, Glass JD, Crooketal JE (2013) Phosphorylated neurofilament heavy subunit (pNF-H) in peripheral blood and CSF as a potential prognostic biomarker in Amyotrophic Lateral Sclerosis. J Neurol Neurosur Ps 84: 467-472. https://doi.org/10.1136/jnnp-2012-303768
[70]	Belzil VV, Bauer PO, Prudencio M, et al. (2013) Reduced C9orf72 gene expression in c9FTD/ALS is caused by histone trimethylation, an epigenetic event detectable in blood. Acta Neuropathol 126: 895-905. https://doi.org/10.1007/s00401-013-1199-1
[71]	Rentzos M, Evangelopoulos E, Sereti E, et al. (2012) Alterations of T cell subsets in ALS: a systemic immune activation?. Acta Neurol Scand 125: 260-264. https://doi.org/10.1111/j.1600-0404.2011.01528.x
[72]	Brettschneider J, Petzold A, Süßmuth SD, et al. (2006) Axonal damage markers in cerebrospinal fluid are increased in ALS. Neurology 66: 852-856. https://doi.org/10.1212/01.wnl.0000203120.85850.54
[73]	Ganesalingam J, An J, Bowser R, et al. (2013) PNfH is a promising biomarker for ALS. Amyotroph Lat Scl Fr 14: 146-149. https://doi.org/10.3109/21678421.2012.729596
[74]	Turner MR, Hardiman O, Benatar M, et al. (2013) Controversies and priorities in amyotrophic lateral sclerosis. Lancet Neurol 12: 310-322. https://doi.org/10.1016/S1474-4422(13)70036-X
[75]	Mitchell J, Paul P, Chen HJ, et al. (2010) Familial amyotrophic lateral sclerosis is associated with a mutation in D-amino acid oxidase. Proc Natl Acad Sci USA 107: 7556-7561. https://doi.org/10.1073/pnas.0914128107
[76]	Barker RF, Hopkinson DA (1977) The genetic and biochemical properties of the D-amino acid oxidases in human tissues. Ann Hum Genet 41: 27-42. https://doi.org/10.1111/j.1469-1809.1977.tb01959.x
[77]	Rosen DR, Siddique T, Patterson D, et al. (1993) Mutations in Cu/Zn superoxide dismutase gene are associated with familial amyotrophic lateral sclerosis. Nature 362: 59-62. https://doi.org/10.1038/364362c0
[78]	Deng HX, Hentati A, Tainer JA, et al. (1993) Amyotrophic lateral sclerosis and structural defects in Cu,Zn superoxide dismutase. Science 261: 1047-1051. https://doi.org/10.1126/science.8351519
[79]	Buratti E, Baralle FE (2001) Characterization and functional implications of the RNA binding properties of nuclear factor TDP-43, a novel splicing regulator of CFTR exon 9. J Biol Chem 276: 36337-36343. https://doi.org/10.1074/jbc
[80]	Wang HY, Wang IF, Bose J, et al. (2004) Structural diversity and functional implications of the eukaryotic TDP gene family. Genomics 83: 130-139. https://doi.org/10.1016/S0888-7543(03)00214-3
[81]	Neumann M, Sampathu DM, Kwong LK, et al. (2006) Ubiquitinated TDP-43 in frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Science 314: 130-133. https://doi.org/10.1126/science.1134108
[82]	Zinszner H, Sok J, Immanuel D, et al. (1997) TLS (FUS) binds RNA in vivo and engages in nucleo-cytoplasmic shuttling. J Cell Sci 110: 1741-1750. https://doi.org/10.1242/jcs.110.15.1741
[83]	Ayala YM, Zago P, D'Ambrogio A, et al. (2008) Structural determinants of the cellular localization and shuttling of TDP-43. J Cell Sci 121: 3778-3785. https://doi.org/10.1242/jcs.038950
[84]	Fujii R, Takumi T (2005) TLS facilitates transport of mRNA encoding an actin-stabilizing protein to dendritic spines. J Cell Sci 118: 5755-5765. https://doi.org/10.1242/jcs.02692
[85]	Fujii R, Okabe S, Urushido T, et al. (2005) The RNA binding protein TLS is translocated to dendritic spines by mGluR5 activation and regulates spine morphology. Curr Biol 15: 587-593. https://doi.org/10.1016/j.cub.2005.01.058
[86]	van Blitterswijk M, DeJesus-Hernandez M, Rademakers R (2012) How do C9ORF72 repeat expansions cause amyotrophic lateral sclerosis and frontotemporal dementia. Curr Opin Neurol 25: 689-700. https://doi.org/10.1097/WCO.0b013e32835a3efb
[87]	Suzuki H, Kanekura K, Levine TP, et al. (2009) ALS-linked P56S-VAPB, an aggregated loss-of-function mutant of VAPB, predisposes motor neurons to ER stress-related death by inducing aggregation of co-expressed wild-type VAPB. J Neurochem 108: 973-985. https://doi.org/10.1111/j.1471-4159.2008.05857.x
[88]	Beckman JS, Crow JP (1993) Pathological implications of nitric oxide, superoxide and peroxynitrite formation. Biochem. Soc. Trans 21: 330-334. https://doi.org/10.1042/bst0210330
[89]	ALS Association. https://www.als.org/research/research-we-fund/scientific-focus-areas/genetics.
[90]	Ishikawa H, Yasui K, Oketa Y, et al. (2012) Increased expression of valosin-containing protein in the skin of patients with amyotrophic lateral sclerosis. J Clin Neurosci 19: 522-526. https://doi.org/10.1016/j.jocn.2011.05.044
[91]	Hadano S, Hand CK, Osuga H, et al. (2001) A gene encoding a putative GTPase regulator is mutated in familial amyotrophic lateral sclerosis 2. Nature Genet 29: 166-173. https://doi.org/10.1038/ng1001-166
[92]	Cai H, Shim H, Lai C, et al. (2008) ALS2/alsin knockout mice and motor neuron diseases. Neurodegener Dis 5: 359-366. https://doi.org/10.1159/000151295
[93]	Gros-Louis F, Kriz J, Kabashi E, et al. (2008) Als2 mRNA splicing variants detected in KO mice rescue severe motor dysfunction phenotype in Als2 knock-down zebrafish. Hum Mol Genet 17: 2691-2702. https://doi.org/10.1093/hmg/ddn171
[94]	Skourti-Stathaki K, Proudfoot NJ, Gromak N (2011) Human senataxin resolves RNA/DNA hybrids formed at transcriptional pause sites to promote Xrn2-dependenttermination. Mol Cell 42: 794-805. https://doi.org/10.1016/j.molcel.2011.04.026
[95]	Padhi AK, Kumar H, Vasaikar SV, et al. (2012) Mechanisms of loss of functions of human angiogenin variants implicated in amyotrophic lateral sclerosis. PLoS One 7: e32479. https://doi.org/10.1371/journal.pone.0032479
[96]	Maruyama H, Morino H, Ito H, et al. (2010) Mutations of optineurin in amyotrophic lateral sclerosis. Nature 465: 223-226. https://doi.org/10.1038/nature08971
[97]	Hentati A, Ouahchi K, Pericak-Vance MA, et al. (1998) Linkage of a commoner form of recessive amyotrophic lateral sclerosis to chromosome 15q15-q22 markers. Neurogenetics 2: 55-60. https://doi.org/10.1007/s100480050052
[98]	Stevanin G, Santorelli FM, Azzedine H, et al. (2007) Mutations in SPG11, encoding spatacsin, are a major cause of spastic paraplegia with thin corpus callosum. Nat Genet 39: 366-372. https://doi.org/10.1038/ng1980
[99]	Murmu RP, Martin E, Rastetter A, et al. (2011) Cellular distribution and subcellular localization of spatacsin and spastizin, two proteins involved in hereditary spastic paraplegia. Mol Cell Neurosci 47: 191-202. https://doi.org/10.1016/j.mcn.2011.04.004
[100]	Michell RH, Dove SK (2009) A protein complex that regulates PtdIns(3,5)P2 levels. EMBO J 28: 86-97. https://doi.org/10.1038/emboj.2008.270
[101]	Chow CY, Zhang Y, Dowling JJ, et al. (2007) Mutation of FIG 4 causes neurodegeneration in the pale tremor mouse and patients with CMT4J. Nature 448: 68-72. https://doi.org/10.1038/nature05876
[102]	Zhang Y, Zolov SN, Chow CY, et al. (2007) Loss of Vac14, a regulator of the signaling lipid phosphatidylinositol 3,5-biphosphate, results in neurodegeneration in mice. Proc Natl Acad Sci USA 104: 17518-17523. https://doi.org/10.1073/pnas.0702275104
[103]	Luty AA, Kwok JB, Dobson-Stone C, et al. (2010) Sigma nonopioid intracellular receptor 1 mutations cause frontotemporal lobar degeneration-motor neuron disease. Ann Neurol 68: 639-649. https://doi.org/10.1002/ana.22274
[104]	Al-Saif A, Al-Mohanna F, Bohlega S (2011) A mutation in sigma-1 receptor causes juvenile amyotrophic lateral sclerosis. Ann Neurol 70: 913-919. https://doi.org/10.1002/ana.22534
[105]	Morita M, Al-Chalabi A, Andersen PM, et al. (2006) A locus on chromosome 9p confers susceptibility to ALS and frontotemporal dementia. Neurology 66: 839-844. https://doi.org/10.1212/01.wnl.0000200048.53766.b4
[106]	Münch C, Sedlmeier R, Meyer T, et al. (2004) Point mutations of the p150 subunit of dynactin (DCTN1) gene in ALS. Neurology 63: 724-726. https://doi.org/10.1212/01.WNL.0000134608.83927.B1
[107]	Laird FM, Farah MH, Ackerley S, et al. (2008) Motor neuron disease occurring in a mutant dynactin mouse model is characterized by defects in vesicular trafficking. J Neurosci 28: 1997-2005. https://doi.org/10.1523/JNEUROSCI.4231-07.2008
[108]	Puls I, Jonnakuty C, LaMonte BH, et al. (2003) Mutant dynactin in motor neuron disease. Nat Genet 33: 455-456. https://doi.org/10.1038/ng1123
[109]	Vasko MR, Guo C, Thompson EL, et al. (2011) The repair function of the multifunctional DNA repair/redox protein APE1 is neuroprotective after ionizing radiation. DNA Repair (Amst) 10: 942-952. https://doi.org/10.1016/j.dnarep.2011.06.004
[110]	Belly A, Bodon G, Blot B, et al. (2010) CHMP2B mutants linked to frontotemporal dementia impair maturation of dendritic spines. J Cell Sci 123: 2943-2954. https://doi.org/10.1242/jcs.068817
[111]	Ghazi-Noori S, Froud KE, Mizielinska S, et al. (2012) Progressive neuronal inclusion formation and axonal degeneration in CHMP2B mutant transgenic mice. Brian 135: 819-832. https://doi.org/10.1093/brain/aws006
[112]	Mersiyanova IV, Perepelov AV, Polyakov AV, et al. (2000) A new variant of Charcot-Marie-Tooth disease type 2 is probably the result of a mutation in the neurofilament-light gene. Am J Hum Gene 67: 37-46. https://doi.org/10.1086/302962
[113]	Skvortsova V, Shadrina M, Slominsky P, et al. (2004) Analysis of heavy neurofilament subunit gene polymorphism in Russian patients with sporadic motor neuron disease (MND). Eur J Hum Genet 12: 241-244. https://doi.org/10.1038/sj.ejhg.5201144
[114]	Al-Chalabi A, Andersen PM, Nilsson P, et al. (1999) Deletions of the heavy neurofilament subunit tail in amyotrophic lateral sclerosis. Hum Molec Genet 8: 157-164. https://doi.org/10.1093/hmg/8.2.157
[115]	Giordano G, Cole TB, Furlong CE, et al. (2011) Paraoxonase 2 (PON2) in the mouse central nervous system: a neuroprotective role?. Toxicol Appl Pharmacol 256: 369-378. https://doi.org/10.1016/j.taap.2011.02.014
[116]	Ticozzi N, LeClerc AL, Keagle PJ, et al. (2010) Paraoxonase gene mutations in amyotrophic lateral sclerosis. Ann Neurol 68: 102-107. https://doi.org/10.1002/ana.21993
[117]	Mizuno Y, Fujita Y, Takatama M, et al. (2011) Peripherin partially localizes in Bunina bodies in amyotrophic lateral sclerosis. J Neurol Sci 302: 14-18. https://doi.org/10.1016/j.jns.2010.12.023
[118]	Beaulieu JM, Nguyen MD, Julien JP (1999) Late onset of motor neurons in mice overexpressing wild-type peripherin. J Cell Biol 147: 531-544. https://doi.org/10.1083/jcb.147.3.531
[119]	Corrado L, Carlomagno Y, Falasco L, et al. (2011) A novel peripherin gene (PRPH) mutation identified in one sporadic amyotrophic lateral sclerosis patient. Neurobiol Aging 32: e1-e6. https://doi.org/10.1016/j.neurobiolaging.2010.02.011
[120]	Elden AC, Kim HJ, Hart MP, et al. (2010) Ataxin-2 intermediate-length polyglutamine expansions are associated with increased risk for ALS. Nature 466: 1069-1075. https://doi.org/10.1038/nature09320
[121]	Baker M, Mackenzie IR, Pickering-Brown SM, et al. (2006) Mutations in progranulin cause tau-negative frontotemporal dementia linked to chromosome 17. Nature 442: 916-919. https://doi.org/10.1038/nature05016
[122]	Schymick JC, Yang Y, Andersen PM, et al. (2007) Progranulin mutations and amyotrophic lateral sclerosis or amyotrophic lateral sclerosis-frontotemporal dementia phenotypes. J Neurol Neurosurg Psychiatry 78: 754-756. https://doi.org/10.1136/jnnp.2006.109553
[123]	Oosthuyse B, Moons L, Storkebaum E, et al. (2001) Deletion of the hypoxia-response element in the vascular endothelial growth factor promoter causes motor neuron degeneration. Nat Genet 28: 131-138. https://doi.org/10.1038/88842
[124]	Brockington A (2006) Expression of vascular endothelial growth factor and its receptors in the central nervous system in amyotrophic lateral sclerosis. J Neuropathol Exp Neurol 65: 26-36. https://doi.org/10.1097/01.jnen.0000196134.51217.74
[125]	Lambrechts D, Poesen K, Fernández-Santiago R, et al. (2009) Meta-analysis of vascular endothelial growth factor variations in amyotrophic lateral sclerosis: increased susceptibility in male carriers of the -2578AA genotype. J Med Genet 46: 840-846. https://doi.org/10.1136/jmg.2008.058222
[126]	Lefebvre S, Burlet P, Liu Q, et al. (1997) Correlation between severity and SMN protein level in spinal muscular atrophy. Nat Genet 16: 265-269. https://doi.org/10.1038/ng0797-265
[127]	Veldink JH, Kalmijn S, Van der Hout AH, et al. (2005) SMN genotypesproducing less SMN protein increase susceptibility to and severity of sporadic ALS. Neurology 65: 820-825. https://doi.org/10.1212/01.wnl.0000174472.03292.dd
[128]	Corcia P, Mayeux-Portas V, Khoris J, et al. (2002) Amyotrophic Lateral Sclerosis. Abnormal SMN1 gene copy number is a susceptibility factor for amyotrophic lateral sclerosis. Ann Neurol 51: 243-246. https://doi.org/10.1002/ana.10104
[129]	Blokhuis AM, Groen EJ, Koppers M, et al. (2013) Protein aggregation in amyotrophic lateral sclerosis. Acta Neuropathol 125: 777-794. https://doi.org/10.1007/s00401-013-1125-6
[130]	Arai T, Hasegawa M, Akiyama H, et al. (2006) TDP-43 is a component of ubiquitin-positive tau-negative inclusions in frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Biochem Biophys Res Commun 351: 602-611. https://doi.org/10.1016/j.bbrc.2006.10.093
[131]	Neumann M, Roeber S, Kretzschmar HA, et al. (2009b) Abundant FUS-immunoreactive pathology in neuronal intermediate filament inclusion disease. Acta Neuropathol 118: 605-616. https://doi.org/10.1007/s00401-009-0581-5
[132]	Doi H, Koyano S, Suzuki Y, et al. (2010) The RNA-binding protein FUS/TLS is a common aggregate-interacting protein in polyglutamine diseases. Neurosci Res 66: 131-133. https://doi.org/10.1016/j.neures.2009.10.004
[133]	Ferraiuolo L, Kirby J, Grierson AJ, et al. (2011) Molecular pathways of motor neuron injury in amyotrophic lateral sclerosis. Nat Rev Neurol 7: 616-630. https://doi.org/10.1038/nrneurol.2011.152
[134]	Nishitoh H, Kadowaki H, Nagai A, et al. (2008) ALS-linked mutant SOD1 induces ER stress- and ASK1-dependent motor neuron death by targeting Derlin-1. Genes Dev 22: 1451-1464. https://doi.org/10.1101/gad.1640108
[135]	Oyanagi K, Yamazaki M, Takahashi H, et al. (2008) Spinal anterior horn cells in sporadic amyotrophic lateral sclerosis show ribosomal detachment from, and cisternal distention of the rough endoplasmic reticulum. Neuropathol Appl Neurobiol 34: 650-658. https://doi.org/10.1111/j.1365-2990.2008.00941.x
[136]	Mattiazzi M, D'Aurelio M, Gajewski CD, et al. (2002) Mutated human SOD1 causes dysfunction of oxidative phosphorylation in mitochondria of transgenic mice. J Biol Chem 277: 29626-29633. https://doi.org/10.1074/jbc.M203065200
[137]	Rothstein JD, Martin LJ, Kuncl RW (1992) Decreased glutamate transport by the brain and spinal cord in amyotrophic lateral sclerosis. N Engl J Med 326: 1464-1468. https://doi.org/10.1056/NEJM199205283262204
[138]	Plaitakis A, Constantakakis E (1993) Altered metabolism of excitatory amino acids, N-acetyl-aspartate and N-acetyl-aspartyl-glutamate in amyotrophic lateral sclerosis. Brain Res Bull 30: 381-386. https://doi.org/10.1016/0361-9230(93)90269-H
[139]	Ferrarese C, Sala G, Riva R, et al. (2001) Decreased platelet glutamate uptake in patients with amyotrophic lateral sclerosis. Neurology 56: 270-272. https://doi.org/10.1212/WNL.56.2.270
[140]	Almer G, Vukosavic S, Romero N, et al. (1999) Inducible nitric oxide synthase up-regulation in a transgenic mouse model of familial amyotrophic lateral sclerosis. J Neurochem 72: 2415-2425. https://doi.org/10.1046/j.1471-4159.1999.0722415.x
[141]	Elliott JL (2001) Cytokine upregulation in a murine model of familial amyotrophic lateral sclerosis. Brain Res Mol Brain Res 95: 172-178. https://doi.org/10.1016/S0169-328X(01)00242-X
[142]	Doble A (1996) The pharmacology and mechanism of action of riluzole. Neurology 47: 233S-241S. https://doi.org/10.1212/WNL.47.6_Suppl_4.233S
[143]	Lacomblez L, Bensimon G, Leigh PN, et al. (1996) Amyotrophic Lateral Sclerosis/Riluzole Study Group II. Dose-ranging study of riluzole in amyotrophic lateral sclerosis. Lancet 347: 1425-1431. https://doi.org/10.1016/S0140-6736(96)91680-3
[144]	Mayo Clinic Drug Information. Riluzole (Oral Route)https://www.mayoclinic.org/drugs-supplements/riluzole-oral-route/description/drg-20065853.
[145]	Martin D, Thompson MA, Nadler JV (1993) The neuroprotective agent riluzole inhibits release of Glu and aspartate from slices of hippocampal area CA1. Eur J Pharmacol 250: 473-476. https://doi.org/10.1016/0014-2999(93)90037-I
[146]	Hubert JP, Delumeau JC, Glowinski J, et al. (1994) Antagonism by riluzole of entry of calcium evoked by NMDA and veratridine in rat cultured granule cells: evidence for a dual mechanism of action. Br J Pharmacol 113: 261-267. https://doi.org/10.1111/j.1476-5381.1994.tb16203.x
[147]	Carbone M, Duty S, Rattray M (2012) Riluzole neuroprotection in a Parkinson's disease model involves suppression of reactive astrocytosis but not GLT-1 regulation. BMC Neurosci 13: 38. https://doi.org/10.1186/1471-2202-13-38
[148]	ALZ Forum: Riluzule Therapeutics. https://www.alzforum.org/therapeutics/riluzole
[149]	Abe K, Itoyama Y, Sobue G, et al. (2014) Confirmatory double-blind, parallel-group, placebo-controlled study of efficacy and safety of edaravone (MCI-186) in amyotrophic lateral sclerosis patients. Amyotroph Lateral Scler Frontotemporal Degener 15: 610-617. https://doi.org/10.3109/21678421.2014.959024
[150]	Writing Group; Edaravone (MCI-186).ALS 19 Study Group. Safety and efficacy of edaravone in well defined patients with amyotrophic lateral sclerosis: a randomized, double-blind, placebo-controlled trial. Lancet Neurol (2017) 16: 505-512. https://doi.org/10.1016/S1474-4422(17)30115-1
[151]	Gurney ME, Cutting FB, Zhai P, et al. (1996) Benefit of vitamin E, riluzole, and gabapentin in a transgenic model of familial amyotrophic lateral sclerosis. Ann Neurol 39: 147-157. https://doi.org/10.1002/ana.410390203
[152]	Peters OM, Ghasemi M, Brown RH (2015) Emerging mechanisms of molecular pathology in ALS. J Clin Invest 125: 1767-79. https://doi.org/10.1172/JCI71601
[153]	Cozzolino M, Pesaresi MG, Gerbino V, et al. (2012) Amyotrophic lateral sclerosis: new insights into underlying molecular mechanisms and opportunities for therapeutic intervention. Antioxid Redox Signal 17: 1277-330. https://doi.org/10.1089/ars.2011.4328
[154]	Tortelli R, Zecca C, Piccininni M, et al. (2020) Plasma Inflammatory Cytokines Are Elevated in ALS. Front Neurol 11: 552295. https://doi.org/10.3389/fneur.2020.552295
[155]	Gagliardi D, Meneri M, Saccomanno D, et al. (2019) Diagnostic and Prognostic Role of Blood and Cerebrospinal Fluid and Blood Neurofilaments in Amyotrophic Lateral Sclerosis: A Review of the Literature. Int J Mol Sci 20: 4152. https://doi.org/10.3390/ijms20174152
[156]	Lijmer JG, Mol BW, Heisterkamp S, et al. (1999) Empirical evidence of design-related bias in studies of diagnostic tests. JAMA 282: 1061-6. https://doi.org/10.1001/jama.282.11.1061
[157]	Freischmidt A, Müller K, Ludolph AC, et al. (2013) Systemic dysregulation of TDP-43 binding microRNAs in amyotrophic lateral sclerosis. Acta Neuropathol Com 1: article 42. https://doi.org/10.1186/2051-5960-1-42

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AIMS Neuroscience

Biomarkers and molecular mechanisms of Amyotrophic Lateral Sclerosis

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AIMS Neuroscience

Biomarkers and molecular mechanisms of Amyotrophic Lateral Sclerosis

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