Some features of geological structure of the Shikotan Island (Lesser Kuril Arc)—A view from "Space"

Evgeny P Terekhov; Anatoly V Mozherovsky; Evgeny P Terekhov; Anatoly V Mozherovsky

doi:10.3934/geosci.2024042

AIMS Geosciences

2024, Volume 10, Issue 4: 907-917. doi: 10.3934/geosci.2024042

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Research article Special Issues

Some features of geological structure of the Shikotan Island (Lesser Kuril Arc)—A view from "Space"

Evgeny P Terekhov ,
Anatoly V Mozherovsky ^,

V.I. Il'ichev Pacific Oceanological Institute, Far Eastern Branch, Russian Academy of Sciences, Vladivostok, Russia

Received: 28 June 2024 Revised: 14 October 2024 Accepted: 05 November 2024 Published: 18 November 2024

In this paper, for the first time, we used Google Earth, an easily accessible method for obtaining geological information to study Shikotan Island (Lesser Kuril Arc). Google Earth flight mode made it possible to examine coastal cliffs on the island, which are inaccessible for hiking or walking, whereas 3-D visualization mode helped us study topographic features, tectonic dislocations, and sediment layering hidden by vegetation and soil, thereby significantly expanding understanding of the geologic structure of the island. Researchers conducting studies in the northwestern part of the island (Tat'yana Cape) discovered a previously unknown structure—a dike field. In the southern part, two thrust faults were identified: An unnamed peak and Tomari Mountain, previously considered a volcano. In the southwestern part of Shikotan Island, there are four unknown volcanic peaks. Together with the Notoro Volcano, they mark the rim of an interpreted caldera of a paleovolcano, which could have been the main unknown source of tuffaceous material for the Mesozoic-Cenozoic deposits of the Matakotanskaya, Malokuril'skaya, and Zelenovskaya Suites. It has been shown that the gabbroid massif of Tsunami Bay (northeastern part of the island) is an autochthonous (local) formation, and not allochthonous, that is, brought from the Pacific Ocean, as evidenced by an intrusive contact with the rocks of the Malokuril'skaya Suite. Despite these positive results, analysis of satellite images of Shikotan Island unexpectedly has not confirmed the existence of the Central Shikotan thrust fault, the largest previously mapped tectonic structure on the island. This work confirms that Google Earth is a very useful tool for geological research in remote areas.

Keywords:

Citation: Evgeny P Terekhov, Anatoly V Mozherovsky. Some features of geological structure of the Shikotan Island (Lesser Kuril Arc)—A view from 'Space'[J]. AIMS Geosciences, 2024, 10(4): 907-917. doi: 10.3934/geosci.2024042

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Abstract

1. Introduction

Let $\mathcal{A}_{p}$ denote the class of analytic and $p$ -valent functions $f(z)$ with the next form

$\begin{equation} f(z) = z^{p}+\sum\limits^{\infty}_{n = 1}a_{n+p}z^{n+p}, \; \; \; \; (p\in\mathbb{N} = \{1, 2, \cdots\}) \end{equation}$

(1.1)

in the open unit disk $\Delta = \{z\in\mathbb{C}: \mid z\mid < 1\}$ .

For $f\in\mathcal{A}_{p}$ , its $q$ -derivative or the $q$ -difference $\mathcal{D}_{q}f(z)$ is given by

$\begin{equation*} \mathcal{D}_{q}f(z) = [p]_{q}z^{p-1}+\sum\limits^{\infty}_{n = 1}[n+p]_{q}a_{n+p}z^{n+p-1}, \; \; \; \; (0 \lt q \lt 1), \end{equation*}$

where the $q$ -derivative operator $\mathcal{D}_{q}f(z)$ (refer to ^[13] and ^[14]) of the function $f$ is defined by

$\mathcal{D}_{q}f(z): = \left\{\begin{array}{ll} \frac{f(z)-f(qz)}{(1-q)z}, \; \; &(z\neq0; 0 \lt q \lt 1), \\ f'(0), \; \; &(z = 0) \end{array}\right.$

provided that $f'(0)$ exists, and the $q$ -number $[n]_{q}$ is just $[\chi]_{q}$ when $\chi = n\in\mathbb{N}$ , here

$[\chi]_{q} = \left\{\begin{array}{ll} \frac{1-q^{\chi}}{1-q}\; \; \; &\mbox{for}\; \; \chi\in\mathbb{C}, \\ \sum^{\chi-1}_{k = 0}q^{k}\; \; \; &\mbox{for}\; \; \chi = n\in\mathbb{N}. \end{array}\right.$

Note that $\mathcal{D}_{q}f(z)\longrightarrow f'(z)$ when $q\longrightarrow1_{-}$ , where $f'$ is the ordinary derivative of the function $f$ .

Consider the generalized Bernardi integral operator $\mathcal{J}^{\eta}_{p, q}:\mathcal{A}_{p}\longrightarrow\mathcal{A}_{p}$ with the next form

$\begin{equation} \mathcal{J}^{\eta}_{p, q}f(z) = \frac{[p+\eta]_{q}}{z^{\eta}}\int^{z}_{0}t^{\eta-1}f(t)d_{q}t, \; \; (z\in\Delta, \Re\eta \gt -1\; \text{and}\; f\in\mathcal{A}_{p}). \end{equation}$

(1.2)

Then, for $f\in\mathcal{A}_{p}$ , we obtain that

$\begin{equation} \mathcal{J}^{\eta}_{p, q}f(z) = z^{p}+\sum\limits^{\infty}_{n = 1}L^{\eta}_{p, q}(n)a_{n+p}z^{n+p}, \; \; (z\in\Delta), \end{equation}$

(1.3)

where

$\begin{equation} L^{\eta}_{p, q}(n) = \frac{[p+\eta]_{q}}{[n+\eta]_{q}}: = L_{n}. \end{equation}$

(1.4)

Here we remark that if $p = 1$ , it is exactly $q$ -Bernardi integral operator $\mathcal{J}^{\eta}_{q}$ ^[21]. Further, if $p = 1$ and $q\rightarrow1_{-}$ , obviously it is the classical Bernardi integral operator $\mathcal{J}_{\eta}$ ^[5]. In fact, Alexander ^[1] and Libera ^[18] integral operators are special versions of $\mathcal{J}_{\eta}$ for $\eta = 0$ and $\eta = 1$ , respectively.

For two analytic functions $f$ and $g$ , if there exists an analytic function $h$ satisfying $h(0) = 0$ and $\mid h(z)\mid < 1$ for $z\in\Delta$ so that $f(z) = g(h(z))$ , then $f$ is subordinate to $g$ , i.e., $f\prec g$ .

Let $\Lambda$ be the class of all analytic function $\phi$ via the form

$\begin{equation} \phi(z) = 1+\sum\limits^{\infty}_{n = 1}A_{n}z^{n}, \; \; (A_{1} \gt 0, z\in\Delta). \end{equation}$

(1.5)

It is well known that the $q$ -calculus ^[13,14], even the $(p, q)$ -calculus ^[6], is a generalization of the ordinary calculus without the limit symbol, and its related theory has been applied into mathematical, physical and engineering fields (see ^[11,15,25]). Since Ismail et al.^[12] firstly utilized the $q$ -derivative operator to investigate the $q$ -calculus of the class of starlike functions in disk, there had a great deal of work in this respect; for example, refer to Rehman et al. ^[24] for partial sums of generalized $q$ -Mittag-Leffler functions, Srivastava et al. ^[31] for Fekete-Szegö inequality for classes of $(p, q)$ -starlike and $(p, q)$ -convex functions and ^[27] for close-to-convexity of a certain family of $q$ -Mittag-Leffler functions, Seoudy and Aouf ^[26] for the coefficient estimates of $q$ -starlike and $q$ -convex functions and Uçar ^[33] for the coefficient inequality for $q$ -starlike functions. Besides, by involving some special functions and operators or increasing the complexity of function classes, many new subclasses of analytic functions associated with $q$ -calculus or $(p, q)$ -calculus were considered. Here we may refer to ^[9,23], Ahmad et al. ^[2] for convolution properties for a family of analytic functions involving $q$ -analogue of Ruscheweyh differential operator, Dweby and Darus ^[8] for subclass of harmonic univalent functions associated with $q$ -analogue of Dziok-Srivastava operator, Mahmmod and Sokól ^[20] for new subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator, Srivastava et al. ^[28] for coefficient inequalities for $q$ -starlike functions associated with the Janowski functions, ^[31] for Fekete-Szegö inequality for classes of $(p, q)$ -starlike and $(p, q)$ -convex functions using the $q$ -Bernardi integral operator and ^[32] for some results on the $q$ -analogues of the incomplete Fibonacci and Lucas polynomials. For the multivalent functions, Purohit ^[22] ever studied a new class of multivalently analytic functions associated with fractional $q$ -calculus operators, while Shi et al. ^[29] investigated the multivalent $q$ -starlike functions connected with circular domain. Moreover, Arif et al. ^[4] considered a $q$ -analogue of the Ruscheweyh type operator, and Srivastava et al. ^[30] dealt with basic and fractional q-calculus and associated Fekete-Szegö problems for $p$ -valently $q$ -starlike functions and $p$ -valently $q$ -convex functions of complex order using certain integral operators, and Khan et al. ^[17] for a new integral operator in $q$ -analog for multivalent functions. Stimulated by the previous results, in the paper we intend to introduce and investigate several new subclasses of $q$ -starlike and $q$ -convex type analytic and multivalent functions involving a generalized Bernardi integral operator, and establish the corresponding Fekete-Szegö type functional inequalities for these function classes. Besides, the corresponding bound estimates of the coefficients $a_{p+1}$ and $a_{p+2}$ are provided.

From now on we introduce some general subclasses of analytic and multivalent functions associated with the $q$ -derivative operator and the generalized Bernardi integral operator.

Definition 1.1. Let $f(z)\in\mathcal{A}_{p}$ and $\mu, \lambda\geq0$ . If the following subordination

$\begin{equation} (1-\lambda)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}+\frac{\lambda \mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}z^{p-1}}\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu-1}\prec\phi(z) \end{equation}$

(1.6)

is satisfied for $z\in\Delta$ , then we call that $f(z)$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ .

Definition 1.2. Let $f(z)\in\mathcal{A}_{p}$ and $0\leq\lambda\leq1$ . If the following subordination

$\begin{equation} (1-\lambda)\frac{z\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}+\frac{\lambda}{[p]_{q}} \left(1+\frac{qz\mathcal{D}_{q}[\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)](z)}{\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}\right)\prec\phi(z) \end{equation}$

(1.7)

is satisfied for $z\in\Delta$ , then we call that $f(z)$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ .

Definition 1.3. Let $f(z)\in\mathcal{A}_{p}$ and $\mu\geq0$ . If the following subordination

$\begin{equation} \left(\frac{z\mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}\right)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}\prec\phi(z) \end{equation}$

(1.8)

is satisfied for $z\in\Delta$ , then we call that $f(z)$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ .

Remark 1.4. If we put

$\begin{equation*} \phi(z) = \left(\frac{1+z}{1-z}\right)^{\alpha}\; for\; 0 \lt \alpha\leq1 \end{equation*}$

$\begin{equation*} \phi(z) = \frac{1+(1-2\beta)z}{1-z}\; \; for\; 0\leq\beta \lt 1 \end{equation*}$

in Definition (1.1–1.3), then the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ ) reduces to $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \alpha)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \alpha)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \alpha)$ ) or $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \beta)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \beta)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \beta)$ ). Without the generalized Bernardi integral operator, the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ ) is the classical function class $\mathcal{LN}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}(\lambda; \phi)$ and $\mathcal{NS}(\mu; \phi)$ ) when $p = 1$ and $q\rightarrow1_{-}$ .

Let $\Omega$ be the class of functions $\omega(z)$ denoted by

$\begin{equation} \omega(z) = \sum\limits^{\infty}_{n = 1}E_{n}z^{n}, \; \; (z\in\Delta) \end{equation}$

(1.9)

via the inequality $\vert\omega(z)\vert < 1(z\in\Delta)$ . Now we recall some necessary Lemmas below.

Lemma 1.5 (^[16]). Let the function $\omega\in\Omega$ . Then

$\begin{equation*} \vert E_{2}-\tau E^{2}_{1}\vert\leq\max\{1, \vert\tau\vert\}, \; \; \; \; (\tau\in\mathbb{C}). \end{equation*}$

Specially, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Lemma 1.6 (^[7,10]). Let $\mathcal{P}$ be the class of all analytic functions $h(z)$ of the following form

$\begin{equation*} h(z) = 1+\sum\limits^{\infty}_{n = 1}c_{n}z^{n}, \; \; (z\in\Delta) \end{equation*}$

satisfying $\Re h(z) > 0$ and $h(0) = 1$ . Then there exist the sharp coefficient estimates $\mid c_{n}\mid\leq2(n\in\mathbb{N})$ . In Particular, the equality holds for all $n$ for the next function

$\begin{equation*} h(z) = \frac{1+z}{1-z} = 1+\sum\limits^{\infty}_{n = 1}2z^{n}. \end{equation*}$

Lemma 1.7 (^[3,19]). Let the function $\omega\in\Omega$ . Then

$\mid E_{2}-\kappa E^{2}_{1}\mid\leq\left\{\begin{array}{ll} -\kappa\; \; &\mathit{\mbox{if}}\; \; \kappa\leq-1, \\ 1\; \; &\mathit{\mbox{if}}\; \; -1\leq\kappa\leq1, \\ \kappa\; \; &\mathit{\mbox{if}}\; \; \kappa\geq1. \end{array}\right.$

For $\kappa < -1$ or $\kappa > 1$ , the inequality holds literally if and only if $\omega(z) = z$ or one of its rotations. If $< \kappa < 1$ , the inequality holds literally if and only if $\omega(z) = z^{2}$ or one of its rotations. In Particular, if $\kappa = -1$ , then the sharp result holds for the next function

$\begin{equation*} \omega(z) = \frac{z(z+\xi)}{1+\xi z}, \; \; \; \; (0\leq\xi\leq1) \end{equation*}$

or one of its rotations. If $\kappa = 1$ , then the sharp result holds for the next function

$\begin{equation*} \omega(z) = -\frac{z(z+\xi)}{1+\xi z}, \; \; \; \; (0\leq\xi\leq1) \end{equation*}$

or one of its rotations. If $-1 < \kappa < 1$ , then the upper bound is sharp as the followings

$\begin{equation*} \vert E_{2}-\kappa E^{2}_{1}\vert+(\kappa+1)\vert E_{1}\vert^{2}\leq1, \; \; \; \; (-1 \lt \kappa\leq0) \end{equation*}$

and

$\begin{equation*} \vert E_{2}-\kappa E^{2}_{1}\vert+(1-\kappa)\vert E_{1}\vert^{2}\leq1, \; \; \; \; (0 \lt \kappa \lt 1). \end{equation*}$

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

By (1.9) we give that

$\begin{eqnarray} \phi(\omega(z))& = &1+A_{1}E_{1}z+(A_{1}E_{2}+A_{2}E^{2}_{1})z^{2}\\ &+&(A_{1}E_{3}+2A_{2}E_{1}E_{2}+A_{3}E^{3}_{1})z^{3}+\ldots. \end{eqnarray}$

(2.1)

In the section, with Lemma 1.5 we study Fekete-Szegö functional problem for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ and provide the following theorem.

Theorem 2.1. Let $\delta\in\mathbb{C}$ . If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ , then

$\begin{equation*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert\leq\frac{A_{1}[p]_{q}}{\vert L_{2}\vert[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\max\left\{1;\left\vert\frac{A_{1}[p]_{q}\Theta}{2L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}, \end{equation*}$

where

$\begin{eqnarray*} \Theta& = &2\delta L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]\notag\\ &+&L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q} +2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]. \end{eqnarray*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Proof. Assume that $f(z)\in\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ . Then, from Definition 1.1, there exists an analytic function $\omega(z)\in\Omega$ such that

(2.2)

Part case

$\begin{eqnarray*} &&\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu-1}\\ & = &1+(\mu-1)L_{1}a_{p+1}z+\left[(\mu-1)L_{2}a_{p+2}+\frac{(\mu-1)(\mu-2)}{2}L^{2}_{1}a^{2}_{p+1}\right]z^{2}\notag\\ &+&\left[(\mu-1) L_{3}a_{p+3}+\frac{(\mu-1)(\mu-2)}{2}L_{1}L_{2}a_{p+1}a_{p+2}+\frac{(\mu-1)(\mu-2)(\mu-3)}{6}L^{3}_{1}a^{3}_{p+1}\right]z^{3}+\ldots, \end{eqnarray*}$

$\begin{eqnarray*} &&\frac{\lambda \mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}z^{p-1}}\\ & = &\lambda+\frac{\lambda L_{1}a_{p+1}[p+1]_{q}}{[p]_{q}}z+\frac{\lambda L_{2}a_{p+2}[p+2]_{q}}{[p]_{q}}z^{2}\notag\\ &+&\frac{\lambda L_{3}a_{p+3}[p+3]_{q}}{[p]_{q}}z^{3}+\ldots, \end{eqnarray*}$

$\begin{eqnarray*} &&\frac{\lambda \mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}z^{p-1}} \left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu-1}\\ & = &\lambda+\lambda\left[\mu+\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\right]L_{1}a_{p+1}z\notag\\ &+&\lambda\left\{\left[\mu+\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\right]L_{2}a_{p+2}+(\mu-1)\left[\frac{\mu}{2}+\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\right]L^{2}_{1}a^{2}_{p+1}\right\}z^{2}+\ldots, \end{eqnarray*}$

$\begin{eqnarray*} &&(1-\lambda)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}\\ & = &(1-\lambda)+(1-\lambda)\mu L_{1}a_{p+1}z+(1-\lambda)\left[\mu L_{2}a_{p+2}+\frac{\mu(\mu-1)}{2}L^{2}_{1}a^{2}_{p+1}\right]z^{2}\notag\\ &+&(1-\lambda)\left[\mu L_{3}a_{p+3}+\frac{\mu(\mu-1)}{2}L_{1}L_{2}a_{p+1}a_{p+2}+\frac{\mu(\mu-1)(\mu-2)}{6}L^{3}_{1}a^{3}_{p+1}\right]z^{3}+\ldots. \end{eqnarray*}$

Since

$\begin{eqnarray*} &&(1-\lambda)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}+\frac{\lambda \mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}z^{p-1}} \left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu-1}\\ & = &1+\left[\mu+\lambda\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\right]L_{1}a_{p+1}z+\{\left[\mu+\lambda\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\right]L_{2}a_{p+2} \notag\\ &+&\left[\frac{(\mu-1)}{2}[\mu-2\lambda(\lambda\mu-\mu+1)]+\lambda(2\mu-\lambda\mu-1)\frac{[p+1]_{q}}{[p]_{q}}\right]L^{2}_{1}a^{2}_{p+1}\}z^{2}+\ldots, \end{eqnarray*}$

by (2.1) and (2.2) we see that

$\begin{equation*} A_{1}E_{1} = \left[\mu+\lambda\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\right]L_{1}a_{p+1}, \end{equation*}$

$\begin{eqnarray*} A_{1}E_{2}+A_{2}E^{2}_{1}& = &\left[\mu+\lambda\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\right]L_{2}a_{p+2}\notag\\ &+&\left[\frac{\mu-1}{2}[\mu-2\lambda(\lambda\mu-\mu+1)]+\lambda(2\mu-\lambda\mu-1)\frac{[p+1]_{q}}{[p]_{q}}\right]L^{2}_{1}a^{2}_{p+1}. \end{eqnarray*}$

Thereby

$\begin{equation} a_{p+1} = \frac{A_{1}E_{1}[p]_{q}}{L_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]} \end{equation}$

(2.3)

and

$\begin{eqnarray} a_{p+2}& = &\frac{(A_{1}E_{2}+A_{2}E^{2}_{1})[p]_{q}}{L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\\ &-&\frac{A^{2}_{1}E^{2}_{1}[p]^{2}_{q}\{(\mu-1)[\mu-2\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}\}}{2L_{2}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}. \end{eqnarray}$

(2.4)

Further, with (2.3) and (2.4) we obtain that

$\begin{equation*} a_{p+2}-\delta a^{2}_{p+1} = \frac{A_{1}[p]_{q}}{L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}[ E_{2}-\hbar E^{2}_{1}], \end{equation*}$

where

$\begin{eqnarray*} \hbar& = &\{2\delta L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]+L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}\notag\\ &+&2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]\}\frac{A_{1}[p]_{q}}{2L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}-\frac{A_{2}}{A_{1}}. \end{eqnarray*}$

Therefore, according to Lemma 1.5 we finish the proof of Theorem 2.1.

Corollary 2.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{A_{1}[p]_{q}}{\vert L_{2}\vert[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\\ &\times&\max\left\{1;\left\vert\frac{(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}}{2[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}. \end{eqnarray*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

When $\phi\in\mathcal{P}$ , combining (2.3) and (2.4) with Lemma 1.6 we instantly establish the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$ .

Corollary 2.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ , then

$\begin{equation*} \vert a_{p+1}\vert\leq\frac{2\vert E_{1}\vert[p]_{q}}{\vert L_{1}\vert[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]} \end{equation*}$

and

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq &\frac{2(\vert E_{2}\vert+\vert E_{1}\vert^{2})[p]_{q}}{\vert L_{2}\vert[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\notag\\ &+&\frac{2E^{2}_{1}[p]^{2}_{q}\vert(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}\vert}{\vert L_{2}\vert[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}. \end{eqnarray*}$

If we choose real $\delta$ and $\eta$ , then by Lemma 1.7 we derive the next result for Fekete-Szegö problem.

Theorem 2.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

$\begin{equation*} \phi(z) = 1+\sum\limits^{\infty}_{n = 1}A_{n}z^{n}, \; \; (A_{1}, A_{2} \gt 0, z\in\Delta). \end{equation*}$

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ , then

$\vert a_{p+2}-\delta a^{2}_{p+1}\vert\leq\left\{\begin{array}{ll} \frac{[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\left\{A_{2}-\frac{A^{2}_{1}[p]_{q}\Theta}{2L^{2}_{1}\vert\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}\right\}, \; (\delta\leq\Upsilon_{1});\\ \frac{A_{1}[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}, \; (\Upsilon_{1}\leq\delta\leq\Upsilon_{2});\\ \frac{[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\left\{-A_{2}+\frac{A^{2}_{1}[p]_{q}\Theta}{2L^{2}_{1}\vert\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}}\right\}, \; (\delta\geq\Upsilon_{2}), \end{array}\right.$

where

$\begin{eqnarray*} \Upsilon_{1}& = &\frac{(A_{2}-A_{1})L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}} {A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\notag\\ &-&\frac{L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]} {2L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]} \end{eqnarray*}$

and

$\begin{eqnarray*} \Upsilon_{2}& = &\frac{(A_{2}+A_{1})L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}} {A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\notag\\ &-&\frac{L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]} {2L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}. \end{eqnarray*}$

Moreover, we take

$\begin{eqnarray*} \Upsilon_{3}& = &\frac{A_{2}L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}} {A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}\notag\\ &-&\frac{L^{2}_{1}[(\mu-1)[\mu-\lambda(\lambda\mu-\mu+1)][p]_{q}+2\lambda(2\mu-\lambda\mu-1)[p+1]_{q}]} {2L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}. \end{eqnarray*}$

Then, each of the following results is true:

(A) For $\delta\in[\Upsilon_{1}, \Upsilon_{3}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\left\{2(A_{1}-A_{2})L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}+A^{2}_{1}[p]_{q}\Theta\right\}\\ &\times&\frac{\vert a_{p+1}\vert^{2}}{2 A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]} \leq\frac{A_{1}[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}; \end{eqnarray*}$

(B) For $\delta\in[\Upsilon_{3}, \Upsilon_{2}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\left\{2(A_{1}+A_{2})L^{2}_{1}[\mu[p]_{q}+\lambda([p+1]_{q}-[p]_{q})]^{2}-A^{2}_{1}[p]_{q}\Theta\right\}\\ &\times&\frac{\vert a_{p+1}\vert^{2}}{2 A^{2}_{1}L_{2}[p]_{q}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]} \leq\frac{A_{1}[p]_{q}}{ L_{2}[\mu[p]_{q}+\lambda([p+2]_{q}-[p]_{q})]}, \end{eqnarray*}$

where

Remark 2.5. Fixing the parameter $p = 1$ in Theorems 2.1 and 2.4, we can state the new results for the univalent function classes $\mathcal{LN}^{\eta}_{1, q}(\mu, \lambda; \phi) = \mathcal{LN}^{\eta}_{q}(\mu, \lambda; \phi)$ . As Remark 1.4, we may consider $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \alpha)$ or $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \beta)$ to establish latest results. On the other hand, for the different parameters $\mu$ and $\lambda$ , we can deduce new results for $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ .

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

In the section we mainly consider Fekete-Szegö functional problem for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and establish the theorem as follows.

Theorem 3.1. Let $\delta\in\mathbb{C}$ . If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&\leq&\frac{A_{1}[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\notag\\ &\times&\max\left\{1;\left\vert\frac{ A_{1}[p]_{q}\Phi}{L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}, \end{eqnarray*}$

where

$\begin{equation*} \Phi = \delta L_{2}[p]_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]+L^{2}_{1}([p+1]_{q}-[p]_{q})[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q}]. \end{equation*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Proof. If $f\in\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , from Definition 1.2 there exists an analytic function $\omega(z)\in\Omega$ such that

(3.1)

Part case

$\begin{eqnarray*} &&\frac{z\mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}\\ & = &1+\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)L_{1}a_{p+1}z+\left[ \left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)L_{2}a_{p+2}-\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)L^{2}_{1}a^{2}_{p+1}\right]z^{2}+\ldots, \end{eqnarray*}$

$\begin{eqnarray*} &&\frac{1}{[p]_{q}} \left(1+\frac{qz\mathcal{D}_{q}[\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)](z)}{\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}\right)\\ & = &1+\frac{L_{1}[p+1]_{q}}{[p]_{q}}\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)a_{p+1}z\notag\\ &+&\left[\frac{L_{2}[p+2]_{q}}{[p]_{q}}\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)a_{p+2}-\frac{L^{2}_{1}[p+1]^{2}_{q}}{[p]^{2}_{q}}\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)a^{2}_{p+1}\right]z^{2}+\ldots. \end{eqnarray*}$

Since

$\begin{eqnarray*} &&(1-\lambda)\frac{z\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}+\frac{\lambda}{[p]_{q}} \left(1+\frac{qz\mathcal{D}_{q}[\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)](z)}{\mathcal{D}_{q}(\mathcal{J}^{\eta}_{p, q}f)(z)}\right)\\ & = &1+\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)+1\right]L_{1}a_{p+1}z\notag\\ &+&\left\{\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)+1\right]L_{2}a_{p+2}-\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+1]^{2}_{q}}{[p]^{2}_{q}}-1\right)+1\right]L^{2}_{1} a^{2}_{p+1}\right\}z^{2}+\ldots, \end{eqnarray*}$

by (2.1) and (3.1) we note that

$\begin{equation*} A_{1}E_{1} = \left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)+1\right]L_{1}a_{p+1} \end{equation*}$

and

$\begin{eqnarray*} A_{1}E_{2}+A_{2}E^{2}_{1}& = &\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+2]_{q}}{[p]_{q}}-1\right)+1\right]L_{2}a_{p+2}\\ &-&\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)\left[\lambda\left(\frac{[p+1]^{2}_{q}}{[p]^{2}_{q}}-1\right)+1\right]L^{2}_{1} a^{2}_{p+1}. \end{eqnarray*}$

Then, it leads to

$\begin{equation} a_{p+1} = \frac{A_{1}E_{1}[p]^{2}_{q}}{L_{1}([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]} \end{equation}$

(3.2)

and

$\begin{eqnarray} a_{p+2}& = &\frac{[p]_{q}^{2}}{ L_{2}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\\ &\times&\left[(A_{1}E_{2}+A_{2}E^{2}_{1})+\frac{A^{2}_{1}E^{2}_{1}[p]_{q}\{\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q}\}}{ ([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}\right]. \end{eqnarray}$

(3.3)

Furthermore, in accordance with (3.2) and (3.3) we gain that

$\begin{equation*} a_{p+2}-\delta a^{2}_{p+1} = \frac{A_{1}[p]_{q}^{2}}{ L_{2}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}[ E_{2}-\varrho E^{2}_{1}], \end{equation*}$

where

$\begin{equation*} \varrho = \frac{A_{1}[p]_{q}\Phi}{L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}-\frac{A_{2}}{A_{1}}. \end{equation*}$

Thus, from Lemma 1.5 we give the Fekete-Szegö functional inequality in Theorem 3.1.

Corollary 3.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{A_{1}[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\\ &\times&\max\left\{1;\left\vert\frac{A_{1}[p]_{q}[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q}]}{([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}. \end{eqnarray*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

If $\phi\in\mathcal{P}$ , by (3.2) and (3.3) we take Lemma 1.6 to prove the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$ .

Corollary 3.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , then

$\begin{equation*} \vert a_{p+1}\vert\leq\frac{2\vert E_{1}\vert[p]^{2}_{q}}{\vert L_{1}\vert([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]} \end{equation*}$

and

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{2[p]_{q}^{2}}{ \vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\notag\\ &\times&\left[(\vert E_{2}\vert+\vert E_{1}\vert^{2})+\frac{2\vert E_{1}\vert^{2}[p]_{q}\{\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q}\}}{ ([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}\right]. \end{eqnarray*}$

On the other hand, if we take real $\delta$ and $\eta$ , then by Lemma 1.7 we give the next result for Fekete-Szegö problem.

Theorem 3.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

$\begin{equation*} \phi(z) = 1+\sum\limits^{\infty}_{n = 1}A_{n}z^{n}, \; \; (A_{1}, A_{2} \gt 0, z\in\Delta). \end{equation*}$

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ , then

$\begin{equation*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert\leq \end{equation*}$

$\left\{\begin{array}{ll} \frac{[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\left\{A_{2}-\frac{ A^{2}_{1}[p]_{q}\Phi}{L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}\right\}, \; (\delta\leq\Gamma_{1});\\ \frac{A_{1}[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}, \; (\Gamma_{1}\leq\delta\leq\Gamma_{2});\\ \frac{[p]_{q}^{2}}{\vert L_{2}\vert([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\left\{-A_{2}+\frac{ A^{2}_{1}[p]_{q}\Phi}{L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}}\right\}, \; (\delta\geq\Gamma_{2}), \end{array}\right.$

where

$\begin{eqnarray*} \Gamma_{1}& = &\{(A_{2}-A_{1})([p+1]_{q}-[p]_{q})\{\lambda([p+1]_{q}-[p]_{q})+[p]_{q}\}^{2}-A^{2}_{1}[p]_{q}[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q})]\} \notag\\ &\times&\frac{L^{2}_{1}([p+1]_{q}-[p]_{q})} {L_{2}A^{2}_{1}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]} \end{eqnarray*}$

and

$\begin{eqnarray*} \Gamma_{2}& = &\{(A_{2}+A_{1})([p+1]_{q}-[p]_{q})\{\lambda([p+1]_{q}-[p]_{q})+[p]_{q}\}^{2}-A^{2}_{1}[p]_{q}[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q})]\} \notag\\ &\times&\frac{L^{2}_{1}([p+1]_{q}-[p]_{q})} {L_{2}A^{2}_{1}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}. \end{eqnarray*}$

Moreover, we choose

$\begin{eqnarray*} \Gamma_{3}& = &\{A_{2}([p+1]_{q}-[p]_{q})\{\lambda([p+1]_{q}-[p]_{q})+[p]_{q}\}^{2}-A^{2}_{1}[p]_{q}[\lambda([p+1]^{2}_{q}-[p]^{2}_{q})+[p]^{2}_{q})]\} \notag\\ &\times&\frac{L^{2}_{1}([p+1]_{q}-[p]_{q})} {L_{2}A^{2}_{1}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}. \end{eqnarray*}$

Then, each of the following results is true:

(A) For $\delta\in[\Gamma_{1}, \Gamma_{3}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\frac{(A_{1}-A_{2})L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}+A^{2}_{1}[p]_{q}\Phi}{ A^{2}_{1}L_{2}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\vert a_{p+1}\vert^{2}\notag\\ &\leq&\frac{A_{1}[p]^{2}_{q}}{ L_{2}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}; \end{eqnarray*}$

(B) For $\delta\in[\Gamma_{3}, \Gamma_{2}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\frac{(A_{1}+A_{2})L^{2}_{1}([p+1]_{q}-[p]_{q})^{2}[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]^{2}-A^{2}_{1}[p]_{q}\Phi}{ A^{2}_{1}L_{2}[p]^{2}_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}\vert a_{p+1}\vert^{2}\notag\\ &\leq&\frac{A_{1}[p]^{2}_{q}}{ L_{2}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]}, \end{eqnarray*}$

where

$\begin{equation*} \Phi = \delta L_{2}[p]_{q}([p+2]_{q}-[p]_{q})[\lambda([p+2]_{q}-[p]_{q})+[p]_{q}]+L^{2}_{1}[p+1]_{q}([p+1]_{q}-[p]_{q})[\lambda([p+1]_{q}-[p]_{q})+[p]_{q}]. \end{equation*}$

Remark 3.5. Similarly, by taking the parameter $p = 1$ in Theorems 3.1 and 3.4, we can obtain the new results for the univalent function classes $\mathcal{LM}^{\eta}_{1, q}(\lambda; \phi) = \mathcal{LM}^{\eta}_{q}(\lambda; \phi)$ . As Remark 1.4, we may consider $\mathcal{LM}^{\eta}_{p, q}(\lambda; \alpha)$ or $\mathcal{LM}^{\eta}_{p, q}(\lambda; \beta)$ to establish latest results. Clearly, for special parameter $\lambda$ , we can still imply new results for $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ .

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

In the section we investigate Fekete-Szegö functional problem for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ and obtain the corresponding theorem below.

Theorem 4.1. Let $\delta\in\mathbb{C}$ . If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&\leq&\frac{A_{1}[p]_{q}}{\vert L_{2}\vert(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\notag\\ &\times&\max\left\{1;\left\vert\frac{A_{1}[p]_{q}\Psi}{2L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}, \end{eqnarray*}$

where

$\begin{equation*} \Psi = 2\delta L_{2}[p]_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})+(\mu-1)[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]L^{2}_{1}. \end{equation*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Proof. Since $f\in\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , from Definition 1.3 there exists an analytic function $\omega(z)\in\Omega$ such that

(4.1)

Part case

$\begin{eqnarray*} &&\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu}\\ & = &1+\mu L_{1}a_{p+1}z+\left[\mu L_{2}a_{p+2}+\frac{\mu(\mu-1)}{2}L^{2}_{1}a^{2}_{p+1}\right]z^{2}\notag\\ &+&\left[\mu L_{3}a_{p+3}+\frac{\mu(\mu-1)}{2}L_{1}L_{2}a_{p+1}a_{p+2}+\frac{\mu(\mu-1)(\mu-2)}{6}L^{3}_{1}a^{3}_{p+1}\right]z^{3}+\ldots. \end{eqnarray*}$

Since

$\begin{equation*} \left(\frac{z\mathcal{D}_{q}\left(\mathcal{J}^{\eta}_{p, q}f\right)(z)}{[p]_{q}\mathcal{J}^{\eta}_{p, q}f(z)}\right)\left(\frac{\mathcal{J}^{\eta}_{p, q}f(z)}{z^{p}}\right)^{\mu} = 1+\left(\frac{[p+1]_{q}}{[p]_{q}}-1+\mu\right)L_{1}a_{p+1}z \end{equation*}$

$\begin{equation*} +\left\{\left(\frac{[p+2]_{q}}{[p]_{q}}-1+\mu\right)L_{2}a_{p+2}+(\mu-1)\left[\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)+\frac{\mu}{2} \right]L^{2}_{1}a^{2}_{p+1}\right\}z^{2}+\ldots, \end{equation*}$

from (2.1) and (4.1) we know that

$\begin{equation*} A_{1}E_{1} = \left(\frac{[p+1]_{q}}{[p]_{q}}-1+\mu\right)L_{1}a_{p+1} \end{equation*}$

and

$\begin{eqnarray*} A_{1}E_{2}+A_{2}E^{2}_{1}& = &\left(\frac{[p+2]_{q}}{[p]_{q}}-1+\mu\right)L_{2}a_{p+2}\\ &+&(\mu-1)\left[\left(\frac{[p+1]_{q}}{[p]_{q}}-1\right)+\frac{\mu}{2} \right]L^{2}_{1}a^{2}_{p+1}. \end{eqnarray*}$

Thus it deduces that

$\begin{equation} a_{p+1} = \frac{A_{1}E_{1}[p]_{q}}{L_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})} \end{equation}$

(4.2)

and

$\begin{eqnarray} a_{p+2}& = &\frac{(A_{1}E_{2}+A_{2}E^{2}_{1})[p]_{q}}{L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\\ &-&\frac{(\mu-1)A^{2}_{1}E^{2}_{1}[p]^{2}_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]}{2L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}. \end{eqnarray}$

(4.3)

Moreover, in the light of (4.2) and (4.3) we know that

$\begin{equation*} a_{p+2}-\delta a^{2}_{p+1} = \frac{A_{1}[p]_{q}}{L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}[ E_{2}-\aleph E^{2}_{1}], \end{equation*}$

where

$\begin{equation*} \aleph = \frac{2\delta L_{2}[p]_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})+(\mu-1)[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]L^{2}_{1}}{2L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}A_{1}[p]_{q}-\frac{A_{2}}{A_{1}}. \end{equation*}$

Hence, in view of Lemma 1.5 we get the Fekete-Szegö functional inequality in Theorem 4.1.

Corollary 4.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , then

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{A_{1}[p]_{q}}{\vert L_{2}\vert(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\\ &\times&\max\left\{1;\left\vert\frac{A_{1}(\mu-1)[p]_{q}\{\mu[p]_{q}+2([p+1]_{q}-[p]_{q})\}}{2(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}-\frac{A_{2}}{A_{1}}\right\vert\right\}. \end{eqnarray*}$

Moreover, the sharp result holds for the next function

$\begin{equation*} \omega(z) = z\; \; \; \; \mathit{\text{or}}\; \; \; \; \omega(z) = z^{2}, \; \; \; \; (z\in\Delta). \end{equation*}$

Once $\phi\in\mathcal{P}$ , together with (4.2) and (4.3) we apply Lemma 1.6 to prove the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$ .

Corollary 4.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , then

$\begin{equation*} \vert a_{p+1}\vert\leq\frac{2E_{1}[p]_{q}}{\vert L_{1}\vert(\mu[p]_{q}+[p+1]_{q}-[p]_{q})} \end{equation*}$

and

$\begin{eqnarray*} \vert a_{p+2}\vert&\leq&\frac{2(\vert E_{2}\vert+\vert E_{1}\vert^{2})[p]_{q}}{\vert L_{2}\vert(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\notag\\ &+&\frac{2\vert\mu-1\vert\vert E_{1}\vert^{2}[p]^{2}_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]}{\vert L_{2}\vert(\mu[p]_{q}+[p+2]_{q}-[p]_{q})(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}. \end{eqnarray*}$

Clearly, if we let $\delta$ and $\eta$ be real, then from Lemma 1.7 we also show the following result for Fekete-Szegö problem.

Theorem 4.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

$\begin{equation*} \phi(z) = 1+\sum\limits^{\infty}_{n = 1}A_{n}z^{n}, \; \; (A_{1}, A_{2} \gt 0, z\in\Delta). \end{equation*}$

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ , then

$\vert a_{p+2}-\delta a^{2}_{p+1}\vert\leq\left\{\begin{array}{ll} \frac{[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\left\{A_{2}-\frac{A^{2}_{1}[p]_{q}\Psi}{2L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}\right\}, \; (\delta\leq\Pi_{1});\\ \frac{A_{1}[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})};\\ \frac{[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}\left\{-A_{2}+\frac{A^{2}_{1}[p]_{q}\Psi}{2L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}}\right\}, \; (\delta\geq\Pi_{2}), \end{array}\right.$

where

$\begin{equation*} \Pi_{1} = \frac{2(A_{2}-A_{1})L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}-A^{2}_{1}(\mu-1)L^{2}_{1}[p]_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]} {2L_{2}A^{2}_{1}[p]^{2}_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})} \end{equation*}$

and

$\begin{equation*} \Pi_{2} = \frac{2(A_{2}+A_{1})L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}-A^{2}_{1}(\mu-1)L^{2}_{1}[p]_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]} {2L_{2}A^{2}_{1}[p]^{2}_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}. \end{equation*}$

Moreover, we put

$\begin{equation*} \Pi_{1} = \frac{2A_{2}L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}-A^{2}_{1}(\mu-1)L^{2}_{1}[p]_{q}[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]} {2L_{2}A^{2}_{1}[p]^{2}_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}. \end{equation*}$

Then, each of the following results is true:

(A) For $\delta\in[\Pi_{1}, \Pi_{3}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\frac{2(A_{1}-A_{2})L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}+A^{2}_{1}[p]_{q}\Psi}{ 2L_{2}A^{2}_{1}[p]_{q}[\mu[p]_{q}+([p+2]_{q}-[p]_{q})]}\vert a_{p+1}\vert^{2}\notag\\ &\leq&\frac{A_{1}[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}; \end{eqnarray*}$

(B) For $\delta\in[\Pi_{3}, \Pi_{2}]$ ,

$\begin{eqnarray*} \vert a_{p+2}-\delta a^{2}_{p+1}\vert&+&\frac{2(A_{1}+A_{2})L^{2}_{1}(\mu[p]_{q}+[p+1]_{q}-[p]_{q})^{2}-A^{2}_{1}[p]_{q}\Psi}{ 2L_{2}A^{2}_{1}[p]_{q}[\mu[p]_{q}+([p+2]_{q}-[p]_{q})]}\vert a_{p+1}\vert^{2}\notag\\ &\leq&\frac{A_{1}[p]_{q}}{ L_{2}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})}, \end{eqnarray*}$

where

$\begin{equation*} \Psi = 2\delta L_{2}[p]_{q}(\mu[p]_{q}+[p+2]_{q}-[p]_{q})+(\mu-1)[\mu[p]_{q}+2([p+1]_{q}-[p]_{q})]L^{2}_{1}. \end{equation*}$

Remark 4.5. Similarly, by choose the parameter $p = 1$ in Theorems 4.1 and 4.4, we can provide the new results for the univalent function classes $\mathcal{NS}^{\eta}_{1, q}(\mu; \phi) = \mathcal{NS}^{\eta}_{q}(\mu; \phi)$ . As Remark 1.4, we may consider $\mathcal{NS}^{\eta}_{p, q}(\mu; \alpha)$ or $\mathcal{NS}^{\eta}_{p, q}(\mu; \beta)$ to establish latest results. Besides, for the fixed parameter $\mu$ , we can still infer new results for $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ .

5. Conclusion

By involving a generalized Bernardi integral operator, several new subclasses of $q$ -starlike and $q$ -convex type analytic and multivalent functions are introduced to generalize the classical starlike and convex functions. Meanwhile, for these classes we may know integral operator and $q$ -derivative as well as multivalency how to change the coefficients of functions. In our main results, we establish the Fekete-Szegö type functional inequalities for these function classes. Further, the corresponding bound estimates of the coefficients $a_{p+1}$ and $a_{p+2}$ are interpreted. In fact, if we use the other integral operators or take $(p, q)$ -operator when certain function is univalent but not multivalent, we may get many similar results as in this article.

Acknowledgments

We thank the referees for their careful readings and using comments so that this manuscript is greatly improved. This work is supported by Institution of Higher Education Scientific Research Project in Ningxia of the People's Republic of China under Grant NGY2017011, Natural Science Foundation of Ningxia of the People's Republic of China under Grant 2020AAC03066, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People's Republic of China under Grant 2018MS01026, the Natural Science Foundation of the People's Republic of China under Grant 11561001, 42064004 and 11762016.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Govorov G (2000) Geodynamics of Small-Kuril paleoarc system after geochronological and petrochemical data. Dokl Earth Sci 372: 521–524. In Russian.
[2]	Govorov G (2002) Phanerozoic Magmatic Belts and Origin of the Okhotsk Sea Geoblock Structure. Dalnauka Vladivostok: 1–97. In Russian.
[3]	Parfenov L, Popeko V, Popeko L (1983) The main structural and material complexes of Shikotan Island and their geological nature (Lesser Kurile Islands). Russ Geol Geophys 10: 24–34. In Russian.
[4]	Melankholina EN (1978) Gabbroids and parallel dikes in the structure of the island of Shikotan (Lesser Kuril Islands). Geotectonics. 3: 128–136.
[5]	Gorshkov G, Markhinin E, Rodionova R, et al. (1964) Description of the volcanoes of the Kuril Islands. Geol USSR 31: 581–604. In Russian.
[6]	Sasa Y (1932) On the geological structure of Shikotan Island (Lesser Kuril Ridge). Geol J 39: 465. In Russian.
[7]	Goliokko B (1992) The structure and geological development of the southern part of the Kuril island arc in the Late Cretaceous-Miocene in connection with the subduction of the Pacific plate. Dissertation, Shirshov Institute of Oceanology of Russian Academy of Sciences. In Russian.
[8]	Bogatikov O, Tsvetkov A (1988) Magmatic Evolution of Island Arcs, Nauka, Moscow, 1–247. In Russian.
[9]	Frolova T, Burikova I, Guschin A, et al. (1985) The Origin of the Volcanic Series of Island Arcs, Nedra, Moscow, 275. In Russian.
[10]	Frolova T, Burikova I, Frolov V, et al. (1977) Peculiarities of volcanism of the Lesser Kuril Arc. Bulletin MOIP Branch of the Geology 4: 38–50. In Russian.
[11]	Govorov G, Tsvetkov A (1985) Basaltoid magmatism of the Lesser Kuril Arc, In: Volcanic and Volcanic-Sedimentary Rocks of the Far East, Far Eastern Branch of the USSR Academy of Sciences, Vladivostok. In Russian.
[12]	Krasilov V, Blohina N, Kundyshev A, et al. (1986) New data on the stratigraphy and geological history of the Lesser Kuril Arc. Dokl Earth Sci 291: 177–180. In Russian.
[13]	Terekhov E, Mozherovsky A (2007) Bottom relief of the central part of the Sea of Okhotsk (View from space). Dep. VINITI 327-В2007. Available from: https://elibrary.ru/item.asp?id = 18282357. In Russian.
[14]	Markevich VS, Mozherovsky AV, Terekhov EP (2012) Palynological characteristics of the sediments of the Malokuril'skaya formation (Maastrichtian-Danian), Shikotan Island. Stratigr Geol Correl 20: 466–477. https://doi.org/10.1134/S0869593812040041 doi: 10.1134/S0869593812040041
[15]	Mozherovsky AV, Terekhov EP (2016) Authigenic minerals of Meso-Cenozoic volcanic-sedimentary rocks of marginal seas bottom of the North-Western Pacific. Stand Glob J Geol Explor Res 3: 105–114.
[16]	Palechek TN, Terekhov EP, Mozherovskii AV (2008) Campanian-Maastrichtian radiolarians from the Malokuril'skaya formation, the Shikotan Island. Stratigr Geol Correl 16: 650–663. https://doi.org/10.1134/S0869593808060051 doi: 10.1134/S0869593808060051
[17]	Tsvetkov AA, Govorov GI, Tsvetkova MV, et al. (1986) The Evolution of Magmatism in the Lesser Kuril Ridge of the Kuril Island-Arc System. Int Geol Rev 28: 180–196. https://doi.org/10.1080/00206818609466259 doi: 10.1080/00206818609466259

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Some features of geological structure of the Shikotan Island (Lesser Kuril Arc)—A view from "Space"

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Abstract

1. Introduction

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

5. Conclusion

Acknowledgments

Conflict of interest

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Abstract

1. Introduction

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

5. Conclusion

Acknowledgments

Conflict of interest

References

AIMS Geosciences

Some features of geological structure of the Shikotan Island (Lesser Kuril Arc)—A view from "Space"

Related Papers:

Abstract

1. Introduction

2. Functional estimates for the class LNηp,q(μ,λ;ϕ) \mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)

3. Functional estimates for the class LMηp,q(λ;ϕ) \mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)

4. Functional estimates for the class NSηp,q(μ;ϕ) \mathcal{NS}^{\eta}_{p, q}(\mu; \phi)

5. Conclusion

Acknowledgments

Conflict of interest

References

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Catalog

Abstract

1. Introduction

2. Functional estimates for the class LNηp,q(μ,λ;ϕ) \mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)

3. Functional estimates for the class LMηp,q(λ;ϕ) \mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)

4. Functional estimates for the class NSηp,q(μ;ϕ) \mathcal{NS}^{\eta}_{p, q}(\mu; \phi)

5. Conclusion

Acknowledgments

Conflict of interest

References

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

2. Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

3. Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

4. Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$