Optimal synthesis of a planar mechanism using Matlab: Example of slider crank in a rowing motion

Med Amine LARIBI; Med Amine LARIBI

doi:10.3934/steme.2023001

STEM Education

2023, Volume 3, Issue 1: 1-14. doi: 10.3934/steme.2023001

Previous Article Next Article

Research article Special Issues

Optimal synthesis of a planar mechanism using Matlab: Example of slider crank in a rowing motion

Med Amine LARIBI ^,

Department of GMSC, Pprime Institute CNRS, ISAE-ENSMA, University of Poitiers, UPR 3346, France; med.amine.laribi@univ-poitiers.fr

Academic Editor: Mingfeng Wang

Received: 09 December 2022 Revised: 26 February 2023 Accepted: 07 March 2023 Published: 10 March 2023

This paper illustrates the conducted effort to introduce the methodology of optimal synthesis of mechanisms through an engineering problem under Matlab. This project is part of a first course on poly-articulated mechanisms and robotics at the graduate level. The course combines both the theoretical and implementation under Matlab aspects to achieve its goals. The example of the rowing machine is considered to introduce the crank-slider mechanism and the kinematic analysis using complex numbers. The mechanism is associated with a spring to ensure the device simulating the resistance in the rowing bench simulator. The optimal synthesis problem is formulated for the desired effort to perform and solved using the Sequential Quadratic Programming method. All developments are implemented under Matlab.

Keywords:

Citation: Med Amine LARIBI. Optimal synthesis of a planar mechanism using Matlab: Example of slider crank in a rowing motion[J]. STEM Education, 2023, 3(1): 1-14. doi: 10.3934/steme.2023001

Related Papers:

[1]	Huo Tang, Shahid Khan, Saqib Hussain, Nasir Khan . Hankel and Toeplitz determinant for a subclass of multivalent $ q $-starlike functions of order $ \alpha $. AIMS Mathematics, 2021, 6(6): 5421-5439. doi: 10.3934/math.2021320
[2]	Khadeejah Rasheed Alhindi, Khalid M. K. Alshammari, Huda Ali Aldweby . Classes of analytic functions involving the q-Ruschweyh operator and q-Bernardi operator. AIMS Mathematics, 2024, 9(11): 33301-33313. doi: 10.3934/math.20241589
[3]	Muhammad Sabil Ur Rehman, Qazi Zahoor Ahmad, H. M. Srivastava, Nazar Khan, Maslina Darus, Bilal Khan . Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions. AIMS Mathematics, 2021, 6(2): 1110-1125. doi: 10.3934/math.2021067
[4]	İbrahim Aktaş . On some geometric properties and Hardy class of q-Bessel functions. AIMS Mathematics, 2020, 5(4): 3156-3168. doi: 10.3934/math.2020203
[5]	Chunhong Li, Dandan Yang, Chuanzhi Bai . Some Opial type inequalities in (p, q)-calculus. AIMS Mathematics, 2020, 5(6): 5893-5902. doi: 10.3934/math.2020377
[6]	Saqib Hussain, Shahid Khan, Muhammad Asad Zaighum, Maslina Darus . Certain subclass of analytic functions related with conic domains and associated with Salagean q-differential operator. AIMS Mathematics, 2017, 2(4): 622-634. doi: 10.3934/Math.2017.4.622
[7]	Muhammad Amer Latif, Mehmet Kunt, Sever Silvestru Dragomir, İmdat İşcan . Post-quantum trapezoid type inequalities. AIMS Mathematics, 2020, 5(4): 4011-4026. doi: 10.3934/math.2020258
[8]	Yue Wang, Ghulam Farid, Babar Khan Bangash, Weiwei Wang . Generalized inequalities for integral operators via several kinds of convex functions. AIMS Mathematics, 2020, 5(5): 4624-4643. doi: 10.3934/math.2020297
[9]	Mohammad Faisal Khan, Ahmad A. Abubaker, Suha B. Al-Shaikh, Khaled Matarneh . Some new applications of the quantum-difference operator on subclasses of multivalent $ q $-starlike and $ q $-convex functions associated with the Cardioid domain. AIMS Mathematics, 2023, 8(9): 21246-21269. doi: 10.3934/math.20231083
[10]	Ala Amourah, B. A. Frasin, G. Murugusundaramoorthy, Tariq Al-Hawary . Bi-Bazilevič functions of order $ \vartheta +i\delta $ associated with $ (p, q)- $ Lucas polynomials. AIMS Mathematics, 2021, 6(5): 4296-4305. doi: 10.3934/math.2021254

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]	Meriam, J.L. and Kraige, L.G., Engineering mechanics, Volume 2 Dynamics, Wiley, 1997.
[2]	Thibaut L., Ceuppens S., De Loof H., De Meester J., Goovaerts L., Struyf A., et al. (2018) Integrated STEM education: A systematic review of instructional practices in secondary education. European Journal of STEM Education 3: 2.
[3]	Ghani U., Zhai X., Ahmad R. (2021) Mathematics skills and STEM multidisciplinary literacy: Role of learning capacity. STEM Education 1: 104.
[4]	Koleza, E. and Skordoulis, C.D., Innovating STEM Education: Increased Engagement and Best Practices. Champaign, IL: Common Ground Research Networks. https://doi.org/10.18848/978-1-86335-251-2/CGP
[5]	Erdman, A.G. and Sandor, G.N., Mechanism Design: Analysis and Synthesis, 4th ed.; Pearson: London, UK, 2001.
[6]	Cabrera J., Simon A., Prado M. (2002) Optimal synthesis of mechanisms with genetic algorithms. Mech Mach Theory 37: 1165-1177.
[7]	Laribi M., Mlika A., Romdhane L., Zeghloul S. (2004) A combined genetic algorithm-fuzzy logic method GA-FL in mechanisms synthesis. Mech Mach Theory 39: 717-735.
[8]	McCarthy, J.M. and Joskowicz, L., Kinematic synthesis, Cambridge University Press, 2009.
[9]	Hernández A., Muñoyerro A., Urízar M., Amezua E. (2021) Comprehensive approach for the dimensional synthesis of a four-bar linkage based on path assessment and reformulating the error function. Mech Mach Theory 156: 104126.
[10]	Constans, E. and Dyer, K.B., Introduction to Mechanism Design with Computer Applications, CRC Press Taylor & Francis Group, 2019. https://doi.org/10.1201/b22145
[11]	Nocedal, J. and Wright, S., Numerical Optimization, Springer, 2006.

This article has been cited by:

1.	Pinhong Long, Jinlin Liu, Murugusundaramoorthy Gangadharan, Wenshuai Wang, Certain subclass of analytic functions based on $ q $-derivative operator associated with the generalized Pascal snail and its applications, 2022, 7, 2473-6988, 13423, 10.3934/math.2022742
2.	H. M. Srivastava, Sarem H. Hadi, Maslina Darus, Some subclasses of p-valent $$\gamma $$-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator, 2023, 117, 1578-7303, 10.1007/s13398-022-01378-3
3.	Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi, Applications of fuzzy differential subordination theory on analytic $ p $ -valent functions connected with $ \mathfrak{q} $-calculus operator, 2024, 9, 2473-6988, 21239, 10.3934/math.20241031
4.	Ekram E. Ali, Hari M. Srivastava, Abdel Moneim Y. Lashin, Abeer M. Albalahi, Applications of Some Subclasses of Meromorphic Functions Associated with the q-Derivatives of the q-Binomials, 2023, 11, 2227-7390, 2496, 10.3390/math11112496

Author's biography Dr. Med Amine LARIBI is an Associate Professor in the Fundamental and Applied Sciences Faculty of the University of Poitiers (UP), where he teaches robotics and mechanic. He has a Mechanical Engineer Degree (specialization on Mechanical Design) from École Nationale d'Ingénieurs de Monsatir (E.N.I.M.) in 2001. M.S. in Mechanical Design, 2002. He received his Ph.D. in Mechanics from University of Poitiers in 2005 and National Habilitation in Mechanics from University of Poitiers in 2018. He servers as Associate Editor in several reputed internationals journals including ASME Journal of Medical Devices, Robotic Intelligence and Automation (Emerald Publishing), Mechanical Sciences (Copernicus Publications), MDPI Robotics and MDPI Machines. He is a member of international scientific committee of RAAD, MESROB and ROMANSY. His research interests, at the Department of G.M.S.C. of PPRIME Institute, include aspects on robots design, mechanism synthesis, cable driven robots, parallel robots, haptic interfaces, collaborative robots with more than 53 published peer reviewed journal papers, 9 edited books, 10 edited journal special issue, 6 PhD students. He is leading several national and international research projects in the fields of medical robotics and biomimetics

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)