Denote by $ \mathcal{A}_{m}(m\in\mathbb{N}) $ the class of multivalently analytic functions $ f $ which are expanded with the Taylor-Maclaurin's series

$ \begin{equation} f(z)=z^{m}+\sum\limits^{\infty}_{n=m+1}a_{n}z^{n} \end{equation} $

(1.1)

in the open unit disk $ \mathbb{U}=\{z\in\mathbb{C}: \mid z\mid < 1\} $, where $ \mathcal{A}_{1}=\mathcal{A} $ is univalent analytic function class, and by $ \Lambda $ the class of Schwarz functions $ \omega $ with $ \omega(0)=0 $ and $ \mid \omega(z)\mid < 1 $ for $ z\in\mathbb{U} $. For two analytic functions $ F $ and $ G $ in $ \mathbb{U} $, if there has a Schwarz function $ \omega\in\Lambda $ such that $ F(z)=G(\omega(z)) $, then $ F $ is subordinate to $ G $ in $ \mathbb{U} $, i.e. $ F\prec G $. In addition, if $ G $ is univalent in $ \mathbb{U} $, then the following equivalence ([1, 2])

$ \begin{equation*} F\prec G\Leftrightarrow F(0)=G(0)\; \text{and}\; F(\mathbb{U})\subset G(\mathbb{U}) \end{equation*} $

holds true. Besides, if $ \omega(z)=z $, then $ F $ is majorized by $ G $ in $ \mathbb{U} $, i.e. $ F\leq G $.

The theory of quantum calculus known as $ q $-calculus is equivalent to traditional infinitesimal calculus without the notion of limits. The $ q $-calculus, was started by Euler and Jacobi, found many interesting applications in various areas of mathematics, physics and engineering sciences. On a recent investigation done by Sahai and Yadav in the theory of special functions by [3], quantum calculus based on two parameters $ (p, q) $ was quoted. Indeed generalization of $ q $-calculus is the post quantum calculus, denoted $ (p, q) $-calculus. The $ (p, q) $-integer was introduced in order to give a generalization or to unify several forms of $ q $-oscillator algebras, well known in the earlier physics literature related to the representation theory of single parameter quantum algebras [4]. Throughout this article, we will use basic notations and definitions of the $ (p, q) $- calculus as follows: Let $ p > 0, q > 0 $. For any non-negative integer $ n $, the $ (p, q) $-integer number $ n $, denoted by $ [n]_{p, q} $, is defined as

$ \begin{equation} \left[ n \right]_{p, q} = \frac{{{p^n} - {q^n}}}{{p - q}}, \qquad [0]_{p, q} = 0. \end{equation} $

(1.2)

The twin-basic number is a natural generalization of the $ q $-number, that is $ {\left[n \right]_q} = \frac{{1- {q^n}}}{{1 - q}} $ $ (q \neq 1). $ Similarly, the $ (p, q) $-differential operator of a function $ f $, analytic in $ |z| < 1 $, is defined by

$ \begin{equation} {{\mathcal{D}_{p, q}}}f \left( z \right) = \frac{{f\left( pz \right) - f\left( {qz} \right)}}{{z\left( {p - q} \right)}}\quad (p \neq q, \ z \in \mathbb{U} =\{z \in \mathbb{C}:\ |z|<1\}). \end{equation} $

(1.3)

One can easily show that $ \mathcal{D}_{p, q} f(z)\rightarrow f'(z) $ as $ p \rightarrow 1^{-} $ and $ q \rightarrow 1^{-} $. Since we cannot obtain $ (p, q) $-integers just by replacing $ q $ by $ q/p $ in the definition of $ q $-integers, it is clear that $ q $-integers and $ (p, q) $-integers differs. However, the definition (1.2) reduces to quantum calculus for the case $ p=1 $. Thus, we can say that $ (p, q) $-calculus can be taken as a generalization of q-calculus. The $ (p, q) $-factorial is defined by $ [0]_{p, q}! = 1, \quad [n]_{p, q}! = \prod\limits_{k=1}^{n}[k]_{p, q}! \quad (n \geq 1). $ Note that for $ p \rightarrow 1^{-} $, the $ (p, q) $ - factorial reduces to the $ q $ - factorial. Also, clearly $ \lim_{p \rightarrow 1^-}\lim_{q \rightarrow 1^-}[n]_{p, q} = n, $ and $ \lim_{p \to\ 1^{-}}\lim_{q \to\ 1^{-}}[n]_{p, q}! = n!. $ For details on $ q $-calculus and $ (p, q) $-calculus, one can refer to [4, 5] and [3].

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

$ \begin{equation} \mathcal{D}_{p, q}f(z)=[m]_{p, q}z^{m-1}+\sum\limits^{\infty}_{n=m+1}[n]_{p, q}a_{n}z^{n-1}, \; \; (0<q\leq p\leq1), \end{equation} $

(1.4)

where the $ (p, q) $-derivative operator $ \mathcal{D}_{p, q}f(z) $ of $ f $ is defined by(1.3).

Recall in [6] that a limacon function $ \vartheta: \mathbb{U}\rightarrow \mathbb{C} $ is defined by

$ \begin{equation} \vartheta(z)=1+\sqrt{2}z+\frac{1}{2}z^{2}, \end{equation} $

(1.5)

whose image is of the bean shape(see FIG. 1), that is to say, the interior of

$ \begin{equation} \{\zeta=x+yi\in\mathbb{C}:(4x^{2}+4y^{2}-8x-5)^{2}+8(4x^{2}+4y^{2}-12x-3)^{2}=0\}. \end{equation} $

(1.6)

With the development of quantum mechanics, the $ q $-calculus [7, 8] and $ (p, q) $-calculus [4], which are two generalizations of the ordinary calculus without the limit symbol, have been applied into many mathematical, physical and engineering fields (refer to [9] and [10]). Since the seminal paper of Ismail et al. [11], there has a great deal of work to generalize the analytic functions in $ q $-analysis and $ (p, q) $-analysis; referring to Srivastava's review paper [12] and [13-26]. For the multivalent analytic functions in $ q $-calculus sense, we also find some related advances in [25, 27-29]. In the article, we intend to probe this object by subordinating to a limacon [6] but not the Lemniscate of Bernoulli [30]. Specially, we introduce and study certain new subclass of analytic and multivalent Bazilevč functions related with $ (p, q) $-derivative operator and a limacon, and consider the corresponding estimates of the coefficients $ a_{m+1} $ and $ a_{m+2} $ as well as Fekete-Szegö functional inequalities. Meanwhile, the consequences and connections to all our results would also be pointed out.

Now, by making use of unified subordination technique by Ma-Mind[31] and $ (p, q) $-calculus sense, we define the following subclass of multivalent Bazilevič functions associated with a limacon function.

Definition 1.1 A function $ f\in\mathcal{A}_{m} $ is said to be in the class $ \mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) $ if the following subordination

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

(1.7)

holds for $ z\in\mathbb{U} $, where $ \delta > 0 $ and $ \lambda\in\mathbb{C} $.

Remark 1.2 In term of Definition 1.1, by choosing some special parameters, we get the following reduced versions:

● $ \mathcal{B}^{p, q}_{LC}(m, \lambda, 1)\equiv\mathcal{B}^{p, q}_{LC}(m, \lambda) $ and $ \mathcal{B}^{1, 1}_{LC}(m, \lambda)\equiv\mathcal{B}_{LC}(m, \lambda) $;

● $ \mathcal{B}^{p, q}_{LC}(m, 1, \delta)\equiv\mathcal{B}^{p, q}_{LC}(m, \delta) $ and $ \mathcal{B}^{1, 1}_{LC}(m, \delta)\equiv\mathcal{B}_{LC}(m, \delta) $;

● $ \mathcal{B}^{p, q}_{LC}(m, 1, 0)\equiv\mathcal{S}^{p, q}_{LC}(m) $ and $ \mathcal{S}^{1, 1}_{LC}(m)\equiv\mathcal{S}_{LC}(m) $;

● $ \lim_{p, q\rightarrow1_{-}}\mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) \equiv\mathcal{B}_{LC}(m, \lambda, \delta) $ and $ \mathcal{B}^{p, q}_{LC}(1, \lambda, \delta) \equiv\mathcal{B}^{p, q}_{LC}(\lambda, \delta). $

Remark 1.3 A function $ g $ belongs to the class $ \mathcal{B}_{LC}(1, 1, 0) $ if and only if there exists an analytic function $ q \prec \vartheta(z) $ such that

$ \begin{equation} \nonumber g(z)=z\exp\displaystyle{\int} ^{z}_{0}\frac{q(t)-1}{t}dt, \ \ \ z\in\mathbb{U}. \end{equation} $

The above intergral representation provides many examples of functions of the class $ \mathcal{B}_{LC}(1, 1, 0) $. Let $ q(t)=\vartheta(t^{n}) $, for $ n=1, 2, \cdot\cdot\cdot $, and $ t\in\mathbb{U} $. With with a simple calculation, we see that the function(see FIG. 2)

$ \begin{align*} \nonumber h(z)=&z\exp\int^{z}_{0}\frac{\sqrt{2}t^{n}+\frac{1}{2}t^{2n}}{t}dt \\=&z+\frac{\sqrt{2}}{n}z^{n+1}+\frac{1}{4n}z^{2n+1}+\cdot\cdot\cdot, \end{align*} $

is an extreme function for the class $ \mathcal{B}_{LC}(1, 1, 0) $.

Denote by $ \mathcal{P} $ the class of all analytic and univalent functions $ \ell(z) $ with

$ \begin{equation} \ell(z)=1+\sum\limits^{\infty}_{n=1}c_{n}z^{n}, \; \; (z\in\mathbb{D}) \end{equation} $

(1.8)

satisfying $ \Re\; [\ell(z)] > 0 $ and $ \ell(0)=1 $. To proceed our results, we have to be ready for some indispensable Lemmas below.

Lemma 1.4 [See [32, 33]] Let $ \ell(z)\in\mathcal{P} $. Then the sharp estimates

$ \mid c_{n}\mid\leq2\; (n\in\mathbb{N}) $

are true. In Particular, the equality holds for all $ n $ for the following function

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

Lemma 1.5 [See [31]] If $ \ell(z)\in\mathcal{P} $, then, for any complex $ \mu $,

$ \begin{align*} |c_2 - \mu c_1^2| \leq 2\max\left\lbrace 1, |2\mu - 1| \right\rbrace \end{align*} $

and the result is sharp for the functions

$ \begin{align*} \ell(z) = \dfrac{1+z}{1-z} \quad \mathit{\text{or}} \quad \ell(z) = \dfrac{1+z^2}{1-z^2}, \quad (z \in \mathbb{D}). \end{align*} $

Lemma 1.6 [See [31]] Assume that the function $ \ell(z)\in\mathcal{P} $ and $ \mu\in\mathbb{R} $. Then

$ \mid c_{2}-\mu c^{2}_{1}\mid\leq\left\{\begin{array}{ll} -4\mu+2\; \; &\mbox{if}\; \; \mu\leq0, \\ 2\; \; &\mbox{if}\; \; 0\leq\mu\leq1, \\ 4\mu-2\; \; &\mbox{if}\; \; \mu\geq1. \end{array}\right. $

For $ \mu < 0 $ or $ \mu > 1 $, the inequality holds literally if and only if $ \ell(z)=\frac{1+z}{1-z} $ or one of its rotations. If $ 0 < \mu < 1 $, the inequality holds literally if and only if $ \ell(z)=\frac{1+z^{2}}{1-z^{2}} $ or one of its rotations. In Particular, if $ \mu=0 $, then the sharp result holds for the following function

$ \begin{equation*} \ell(z)=\left(\frac{1}{2}+\frac{\varsigma}{2}\right)\frac{1+z}{1-z}+\left(\frac{1}{2}-\frac{\varsigma}{2}\right)\frac{1-z}{1+z}, \quad(0\leq\varsigma\leq1) \end{equation*} $

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

$ \begin{equation*} \frac{1}{\ell(z)}=\left(\frac{1}{2}+\frac{\varsigma}{2}\right)\frac{1+z}{1-z}+\left(\frac{1}{2}-\frac{\varsigma}{2}\right)\frac{1-z}{1+z}, \quad(0\leq\varsigma\leq1) \end{equation*} $

or one of its rotations. If $ 0 < \mu < 1 $, then the upper bound is sharp as the followings

$ \begin{equation*} \vert c_{2}-\mu c^{2}_{1}\vert+\mu\vert c_{1}\vert^{2}\leq2, \quad\left(0<\mu\leq\frac{1}{2}\right) \end{equation*} $

and

$ \begin{equation*} \vert c_{2}-\mu c^{2}_{1}\vert+(1-\mu)\vert c_{1}\vert^{2}\leq2, \quad\left(\frac{1}{2}<\mu<1\right). \end{equation*} $

Denote the function $ h\in\mathcal{P} $ by $ h(z)=\frac{1+u(z)}{1-u(z)}=1+\sum\limits^{\infty}_{n=1}c_{n}z^{n}. $ Then, from (1.8) we derive that

$ \begin{align} u(z) = \dfrac{h(z) - 1}{h(z) + 1} &= \dfrac{c_1}{2}z + \left( \dfrac{c_2}{2} - \dfrac{c_1^2}{4}\right) z^2 \\ &+ \left( \dfrac{c_3}{2} - \dfrac{c_2c_1}{2} + \dfrac{c_1^3}{8} \right)z^3 +\ldots, \; \; (z\in\mathbb{U}) \end{align} $

(2.1)

so that $ u(z)\in\Lambda $. By (1.5) and (2.1), we imply that

$ \begin{align} \vartheta(u(z))&=1+\frac{\sqrt{2}c_{1}}{2}z+\left[\frac{\sqrt{2}c_{2}}{2}-\frac{(2\sqrt{2}-1)c^{2}_{1}}{8}\right]z^{2}\\ &+\left[\frac{\sqrt{2}c_{3}}{2}-\frac{(2\sqrt{2}-1)c_{2}c_{1}}{4}+\frac{(\sqrt{2}-1)c^{3}_{1}}{8}\right]z^{3}\ldots, \; \; (z\in\mathbb{U}). \end{align} $

(2.2)

To begin with, we deal with the functional estimates for the class $ \mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) $ and establish the next theorems for the coefficient bounds and the corresponding Feteke-Szegö problems.

Theorem 2.1 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) $, then

$ \begin{equation} \vert a_{m+1}\vert\leq\frac{\sqrt{2}[m]_{p, q}}{\vert\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}\vert} \end{equation} $

(2.3)

and

$ \begin{eqnarray} \vert a_{m+2}\vert&\leq&\frac{(2\sqrt{2}-\frac{1}{2})[m]_{p, q}}{\vert\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}\vert}+\frac{2\vert\delta-1\vert[m]^{2}_{p, q}}{\vert\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}\vert^{2}}\\ &&\times\frac{\vert\lambda\vert\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta}{2}[m]_{p, q}}{\vert\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}\vert}. \end{eqnarray} $

(2.4)

Proof. Assume that $ f(z)\in\mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) $. Then, there exists a Schwarz function $ u(z)\in\Lambda $ so that

$ (1-\lambda)\left(\frac{f(z)}{z^{m}}\right)^{\delta}+\lambda\frac{z\mathcal{D}_{p, q}f(z)}{[m]_{p, q}f(z)}\left(\frac{f(z)}{z^{m}}\right)^{\delta}=\vartheta(u(z)). $

(2.5)

Since

$ \begin{eqnarray} &&(1-\lambda)\left(\frac{f(z)}{z^{m}}\right)^{\delta}+\lambda\frac{z\mathcal{D}_{p, q}f(z)}{[m]_{p, q}f(z)}\left(\frac{f(z)}{z^{m}}\right)^{\delta}\\ &=&1+\left[\lambda\left(\frac{[m+1]_{p, q}}{[m]_{p, q}}-1\right)+\delta\right]a_{m+1}z +\big\{\left[\lambda\left(\frac{[m+2]_{p, q}}{[m]_{p, q}}-1\right)+\delta\right]a_{m+2}\\ &&+\left[\lambda(\delta-1)\left(\frac{[m+1]_{p, q}}{[m]_{p, q}}-1\right)+\frac{\delta(\delta-1)}{2}\right]a^{2}_{m+1}\big\}z^{2}+\cdots \end{eqnarray} $

(2.6)

for $ f\in\mathcal{A}_{m}(m\in\mathbb{N}) $, combing (2.5) and (2.6) with (2.2) we get that

$ \begin{equation*} \frac{\sqrt{2}c_{1}}{2}=\left[\lambda\left(\frac{[m+1]_{p, q}}{[m]_{p, q}}-1\right)+\delta\right]a_{m+1} \end{equation*} $

and

$ \begin{eqnarray*} \frac{\sqrt{2}c_{2}}{2}-\frac{(2\sqrt{2}-1)c^{2}_{1}}{8}&=&\left[\lambda\left(\frac{[m+2]_{p, q}}{[m]_{p, q}}-1\right)+\delta\right]a_{m+2}\\ &&+\left[\lambda(\delta-1)\left(\frac{[m+1]_{p, q}}{[m]_{p, q}}-1\right)+\frac{\delta(\delta-1)}{2}\right]a^{2}_{m+1}, \end{eqnarray*} $

such that

$ \begin{equation} a_{m+1}=\frac{\sqrt{2}[m]_{p, q}c_{1}}{2[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]} \end{equation} $

(2.7)

and

$ \begin{eqnarray} a_{m+2}&=&\frac{[4\sqrt{2}c_{2}-(2\sqrt{2}-1)c^{2}_{1}][m]_{p, q}}{8[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}-\frac{[m]^{2}_{p, q}c^{2}_{1}}{2[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}\\ &&\times\frac{\lambda(\delta-1)\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta(\delta-1)}{2}[m]_{p, q}}{[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}. \end{eqnarray} $

(2.8)

In view of Lemma 1.4 we imply that Theorem 2.1 holds true.

By taking $ m=1 $ or $ \lambda=1 $ or $ \lambda=1, \delta=0 $ in Theorem 2.1, we deduce the corollaries below.

Corollary 2.2 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{B}^{p, q}_{LC}(\lambda, \delta) $, then

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

and

$ \begin{eqnarray*} \vert a_{3}\vert\leq\frac{(2\sqrt{2}-\frac{1}{2})}{\vert\lambda([3]_{p, q}-1)+\delta\vert}+\frac{2\vert\delta-1\vert}{\vert\lambda([2]_{p, q}-1)+\delta\vert^{2}} \times\frac{\vert\lambda\vert\left([2]_{p, q}-1\right)+\frac{\delta}{2}}{\vert\lambda([3]_{p, q}-1)+\delta\vert}. \end{eqnarray*} $

Corollary 2.3 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{B}^{p, q}_{LC}(m, \delta) $, then

$ \begin{equation*} \vert a_{m+1}\vert\leq\frac{\sqrt{2}[m]_{p, q}}{([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}} \end{equation*} $

and

$ \begin{eqnarray*} \vert a_{m+2}\vert&\leq&\frac{(2\sqrt{2}-\frac{1}{2})[m]_{p, q}}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}+\frac{2\vert\delta-1\vert[m]^{2}_{p, q}}{[([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}\\ &&\times\frac{\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta}{2}[m]_{p, q}}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}. \end{eqnarray*} $

Corollary 2.4 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{S}^{p, q}_{LC}(m) $, then

$ \begin{equation*} \vert a_{m+1}\vert\leq\frac{\sqrt{2}[m]_{p, q}}{[m+1]_{p, q}-[m]_{p, q}} \end{equation*} $

and

$ \begin{eqnarray*} \vert a_{m+2}\vert&\leq&\frac{(2\sqrt{2}-\frac{1}{2})[m]_{p, q}}{[m+2]_{p, q}-[m]_{p, q}}+\frac{2[m]^{2}_{p, q}}{([m+1]_{p, q}-[m]_{p, q})([m+2]_{p, q}-[m]_{p, q})}. \end{eqnarray*} $

Theorem 2.5 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) $, then

$ \begin{equation} \vert a_{m+2}-\mu a^{2}_{m+1}\vert\leq\frac{\sqrt{2}[m]_{p, q}\max\left\lbrace 1, |2\hbar - 1| \right\rbrace}{\vert\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}\vert} \end{equation} $

(2.9)

holds for $ \mu\in\mathbb{C} $, where

$ \begin{eqnarray*} \hbar&=&\frac{(4-\sqrt{2})}{8}+\frac{\sqrt{2}\mu[m]_{p, q}[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}{2[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}\notag\\ &&+\frac{\sqrt{2}[m]_{p, q}}{2}\times\frac{\lambda(\delta-1)\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta(\delta-1)}{2}[m]_{p, q}}{[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}. \end{eqnarray*} $

Proof For $ \mu\in\mathbb{C} $, by using (2.7) and (2.8), we infer that

$ \begin{equation} a_{m+2}-\mu a^{2}_{m+1}=\frac{\sqrt{2}[m]_{p, q}}{2[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\left(c_{2}-\hbar c^{2}_{1}\right), \end{equation} $

(2.10)

where

$ \begin{eqnarray*} \hbar&=&\frac{(4-\sqrt{2})}{8}+\frac{\sqrt{2}\mu[m]_{p, q}[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}{2[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}\notag\\ &+&\frac{\sqrt{2}[m]_{p, q}}{2}\times\frac{\lambda(\delta-1)\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta(\delta-1)}{2}[m]_{p, q}}{[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}. \end{eqnarray*} $

Then, we apply Lemma 1.5 into the equality (2.10) and show that Theorem 2.5 holds true.

Similarly, by fixing $ \mu=0 $ or $ m=1 $, or $ \lambda=1 $, or $ \lambda=1 $, $ \delta=0 $ in Theorem 2.5, we obtain the next corollaries as follows.

Corollary 2.6 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) $, then

$ \begin{equation*} \vert a_{m+2}\vert\leq\frac{\sqrt{2}[m]_{p, q}\max\left\lbrace 1, |2\hbar - 1| \right\rbrace}{\vert\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}\vert}, \end{equation*} $

where

$ \begin{eqnarray*} \hbar=\frac{(4-\sqrt{2})}{8}+\frac{\sqrt{2}[m]_{p, q}}{2}\times\frac{\lambda(\delta-1)\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta(\delta-1)}{2}[m]_{p, q}}{[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}. \end{eqnarray*} $

Corollary 2.7 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{B}^{p, q}_{LC}(\lambda, \delta) $, then

$ \begin{equation*} \vert a_{3}-\mu a^{2}_{2}\vert\leq\frac{\sqrt{2}\max\left\lbrace 1, |2\hbar - 1| \right\rbrace}{\vert\lambda([3]_{p, q}-1)+\delta\vert} \end{equation*} $

holds for $ \mu\in\mathbb{C} $, where

$ \begin{eqnarray*} \hbar&=&\frac{(4-\sqrt{2})}{8}+\frac{\sqrt{2}\mu[\lambda([3]_{p, q}-1)+\delta]}{2[\lambda([2]_{p, q}-1)+\delta]^{2}}+\frac{\lambda(\delta-1)\left([2]_{p, q}-1\right)+\frac{\delta(\delta-1)}{2}}{\sqrt{2}[\lambda([2]_{p, q}-1)+\delta]^{2}}. \end{eqnarray*} $

Corollary 2.8 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{B}^{p, q}_{LC}(m, \delta) $, then

$ \begin{equation} \vert a_{m+2}-\mu a^{2}_{m+1}\vert\leq\frac{\sqrt{2}[m]_{p, q}\max\left\lbrace 1, |2\hbar - 1| \right\rbrace}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}} \end{equation} $

(2.11)

holds for $ \mu\in\mathbb{C} $, where

$ \begin{eqnarray*} \hbar&=&\frac{(4-\sqrt{2})}{8}+\frac{\sqrt{2}\mu[m]_{p, q}[([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}{2[([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}\notag\\ &&+\frac{\sqrt{2}(\delta-1)[m]_{p, q}}{2}\times\frac{\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta}{2}[m]_{p, q}}{[([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}. \end{eqnarray*} $

Corollary 2.9 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{S}^{p, q}_{LC}(m) $, then

$ \begin{equation} \vert a_{m+2}-\mu a^{2}_{m+1}\vert\leq\frac{\sqrt{2}[m]_{p, q}\max\left\lbrace 1, |2\hbar - 1| \right\rbrace}{[m+2]_{p, q}-[m]_{p, q}} \end{equation} $

(2.12)

holds for $ \mu\in\mathbb{C} $, where

$ \begin{eqnarray*} \hbar&=&\frac{(4-\sqrt{2})}{8}+\frac{\sqrt{2}[m]_{p, q}[\mu([m+2]_{p, q}-[m]_{p, q})-([m+1]_{p, q}-[m]_{p, q})]}{2([m+1]_{p, q}-[m]_{p, q})^{2}}. \end{eqnarray*} $

If we let $ \mu\in\mathbb{R} $ and $ \lambda\in\mathbb{R}_{+} $, then we are based on the proof of Theorem 2.5 and Lemma 1.6 to establish the Fekete-Szegö functional inequality for $ \mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) $.

Theorem 2.10 For $ \mu\in\mathbb{R} $ and $ \lambda\in\mathbb{R} $, if $ f(z)\in\mathcal{A}_{m} $ belongs to the class $ \mathcal{B}^{p, q}_{LC}(m, \lambda, \delta) $, then

$ \vert a_{m+2}-\mu a^{2}_{m+1}\vert\leq\left\{\begin{array}{ll} \frac{\sqrt{2}[m]_{p, q}(-2\hbar + 1)}{\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}, \; (\mu\leq\Xi_{1});\\ \frac{\sqrt{2}[m]_{p, q}}{\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}, \; (\Xi_{1}\leq\mu\leq\Xi_{2});\\ \frac{\sqrt{2}[m]_{p, q}(2\hbar - 1)}{\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}, \; (\mu\geq\Xi_{2}), \end{array}\right. $

where $ \hbar $ is the same as in Theorem 2.5, and

$ \begin{eqnarray*} \Xi_{1}&=&\frac{(-2\sqrt{2}+1)}{4}\frac{[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\\ &&-\frac{\lambda(\delta-1)\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta(\delta-1)}{2}[m]_{p, q}}{[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]} \end{eqnarray*} $

and

$ \begin{eqnarray*} \Xi_{2}&=&\frac{(2\sqrt{2}+1)}{4}\frac{[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\\ &&-\frac{\lambda(\delta-1)\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta(\delta-1)}{2}[m]_{p, q}}{[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}. \end{eqnarray*} $

In addition, we take

$ \begin{eqnarray*} \Xi_{3}&=&\frac{[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\\ &&-\frac{\lambda(\delta-1)\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta(\delta-1)}{2}[m]_{p, q}}{[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}. \end{eqnarray*} $

Then, each of the following inequalities holds:

(A) For $ \mu\in[\Xi_{1}, \Xi_{3}] $,

$ \begin{eqnarray*} &&\vert a_{m+2}-\mu a^{2}_{m+1}\vert+\frac{\sqrt{2}\hbar[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\vert a_{m+1}\vert^{2}\\ &\leq&\frac{\sqrt{2}[m]_{p, q}}{[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}; \end{eqnarray*} $

(B) For $ \mu\in[\Xi_{3}, \Xi_{2}] $,

$ \begin{eqnarray*} &&\vert a_{m+2}-\mu a^{2}_{m+1}\vert+\frac{\sqrt{2}(1-\hbar)[\lambda([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\vert a_{m+1}\vert^{2}\\ &\leq&\frac{\sqrt{2}[m]_{p, q}}{[\lambda([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}. \end{eqnarray*} $

Similarly, by letting $ m=1 $, or $ \lambda=1 $, or $ \lambda=1 $, $ \delta=0 $ in Theorem 2.10, we provide the following corollaries.

Corollary 2.11 For $ \mu\in\mathbb{R} $ and $ \lambda\in\mathbb{R}_{+} $, if $ f(z)\in\mathcal{A} $ belongs to the class $ \mathcal{B}^{p, q}_{LC}(\lambda, \delta) $, then

$ \vert a_{3}-\mu a^{2}_{2}\vert\leq\left\{\begin{array}{ll} \frac{\sqrt{2}(-2\hbar + 1)}{\lambda([3]_{p, q}-1)+\delta}, \; (\mu\leq\Pi_{1});\\ \frac{\sqrt{2}}{\lambda([3]_{p, q}-1)+\delta}, \; (\Pi_{1}\leq\mu\leq\Pi_{2});\\ \frac{\sqrt{2}(2\hbar - 1)}{\lambda([3]_{p, q}-1)+\delta}, \; (\mu\geq\Pi_{2}), \end{array}\right. $

where $ \hbar $ is the same as in Corollary 2.7, and

$ \begin{eqnarray*} \Pi_{1}&=&\frac{(-2\sqrt{2}+1)}{4}\frac{[\lambda([2]_{p, q}-1)+\delta]^{2}}{[\lambda([3]_{p, q}-1)+\delta]}-\frac{\lambda(\delta-1)\left([2]_{p, q}-1\right)+\frac{\delta(\delta-1)}{2}}{\lambda([3]_{p, q}-1)+\delta} \end{eqnarray*} $

and

$ \begin{eqnarray*} \Pi_{2}&=&\frac{(2\sqrt{2}+1)}{4}\frac{[\lambda([2]_{p, q}-1)+\delta]^{2}}{[\lambda([3]_{p, q}-1)+\delta]}-\frac{\lambda(\delta-1)\left([2]_{p, q}-1\right)+\frac{\delta(\delta-1)}{2}}{\lambda([3]_{p, q}-1)+\delta}. \end{eqnarray*} $

In addition, we take

$ \begin{eqnarray*} \Pi_{3}&=&\frac{[\lambda([2]_{p, q}-1)+\delta]^{2}}{\lambda([3]_{p, q}-1)+\delta}-\frac{\lambda(\delta-1)\left([2]_{p, q}-1\right)+\frac{\delta(\delta-1)}{2}}{\lambda([3]_{p, q}-1)+\delta}. \end{eqnarray*} $

Then, each of the following inequalities holds:

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

$ \begin{eqnarray*} \vert a_{3}-\mu a^{2}_{2}\vert+\frac{\sqrt{2}\hbar[\lambda([2]_{p, q}-1)+\delta]^{2}}{\lambda([3]_{p, q}-1)+\delta}\vert a_{2}\vert^{2} \leq\frac{\sqrt{2}}{\lambda([3]_{p, q}-1)+\delta}; \end{eqnarray*} $

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

$ \begin{eqnarray*} \vert a_{3}-\mu a^{2}_{2}\vert+\frac{\sqrt{2}(1-\hbar)[\lambda([2]_{p, q}-1)+\delta]^{2}}{\lambda([3]_{p, q}-1)+\delta}\vert a_{2}\vert^{2}\leq\frac{\sqrt{2}}{\lambda([3]_{p, q}-1)+\delta}. \end{eqnarray*} $

Corollary 2.12 For $ \mu\in\mathbb{R} $, if $ f(z)\in\mathcal{A}_{m} $ belongs to the class $ \mathcal{B}^{p, q}_{LC}(m, \delta) $, then

$ \vert a_{m+2}-\mu a^{2}_{m+1}\vert\leq\left\{\begin{array}{ll} \frac{\sqrt{2}[m]_{p, q}(-2\hbar + 1)}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}, \; (\mu\leq\Sigma_{1});\\ \frac{\sqrt{2}[m]_{p, q}}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}, \; (\Sigma_{1}\leq\mu\leq\Sigma_{2});\\ \frac{\sqrt{2}[m]_{p, q}(2\hbar - 1)}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}, \; (\mu\geq\Sigma_{2}), \end{array}\right. $

where $ \hbar $ is the same as in Corollary 2.8, and

$ \begin{eqnarray*} \Sigma_{1}&=&\frac{(-2\sqrt{2}+1)}{4}\frac{[([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\\ &&-\frac{(\delta-1)[\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta}{2}[m]_{p, q}]}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}} \end{eqnarray*} $

and

$ \begin{eqnarray*} \Sigma_{2}&=&\frac{(2\sqrt{2}+1)}{4}\frac{[([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\\ &&-\frac{(\delta-1)[\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta}{2}[m]_{p, q}]}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}. \end{eqnarray*} $

In addition, we take

$ \begin{eqnarray*} \Sigma_{3}&=&\frac{[([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\\ &&-\frac{(\delta-1)[\left([m+1]_{p, q}-[m]_{p, q}\right)+\frac{\delta(\delta-1)}{2}[m]_{p, q}]}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}. \end{eqnarray*} $

Then, each of the following inequalities holds:

(A) For $ \mu\in[\Sigma_{1}, \Sigma_{3}] $,

$ \begin{eqnarray*} &&\vert a_{m+2}-\mu a^{2}_{m+1}\vert+\frac{\sqrt{2}\hbar[([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\vert a_{m+1}\vert^{2}\\ &\leq&\frac{\sqrt{2}[m]_{p, q}}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}; \end{eqnarray*} $

(B) For $ \mu\in[\Sigma_{3}, \Sigma_{2}] $,

$ \begin{eqnarray*} &&\vert a_{m+2}-\mu a^{2}_{m+1}\vert+\frac{\sqrt{2}(1-\hbar)[([m+1]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]^{2}}{[m]_{p, q}[([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}]}\vert a_{m+1}\vert^{2}\\ &\leq&\frac{\sqrt{2}[m]_{p, q}}{([m+2]_{p, q}-[m]_{p, q})+\delta[m]_{p, q}}. \end{eqnarray*} $

Corollary 2.13 For $ \mu\in\mathbb{R} $, if $ f(z)\in\mathcal{A}_{m} $ belongs to the class $ \mathcal{S}^{p, q}_{LC}(m) $, then

$ \vert a_{m+2}-\mu a^{2}_{m+1}\vert\leq\left\{\begin{array}{ll} \frac{\sqrt{2}[m]_{p, q}(-2\hbar + 1)}{[m+2]_{p, q}-[m]_{p, q}}, \; (\mu\leq\Upsilon_{1});\\ \frac{\sqrt{2}[m]_{p, q}}{[m+2]_{p, q}-[m]_{p, q}}, \; (\Upsilon_{1}\leq\mu\leq\Upsilon_{2});\\ \frac{\sqrt{2}[m]_{p, q}(2\hbar -1)}{[m+2]_{p, q}-[m]_{p, q}}, \; (\mu\geq\Upsilon_{2}), \end{array}\right. $

where $ \hbar $ is the same as in Corollary 2.9, and

$ \begin{eqnarray*} \Upsilon_{1}=\frac{(-2\sqrt{2}+1)}{4}\frac{([m+1]_{p, q}-[m]_{p, q})^{2}}{[m]_{p, q}([m+2]_{p, q}-[m]_{p, q})}+\frac{[m+1]_{p, q}-[m]_{p, q}}{[m+2]_{p, q}-[m]_{p, q}} \end{eqnarray*} $

and

$ \begin{eqnarray*} \Upsilon_{2}=\frac{(2\sqrt{2}+1)}{4}\frac{([m+1]_{p, q}-[m]_{p, q})^{2}}{[m]_{p, q}([m+2]_{p, q}-[m]_{p, q})}+\frac{[m+1]_{p, q}-[m]_{p, q}}{[m+2]_{p, q}-[m]_{p, q}}. \end{eqnarray*} $

In addition, we take

$ \begin{eqnarray*} \Upsilon_{3}&=&\frac{([m+1]_{p, q}-[m]_{p, q})^{2}}{[m]_{p, q}([m+2]_{p, q}-[m]_{p, q})}+\frac{[m+1]_{p, q}-[m]_{p, q}}{[m+2]_{p, q}-[m]_{p, q}}. \end{eqnarray*} $

Then, each of the following inequalities holds:

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

$ \begin{eqnarray*} \vert a_{m+2}-\mu a^{2}_{m+1}\vert+\frac{\sqrt{2}\hbar([m+1]_{p, q}-[m]_{p, q})^{2}}{[m]_{p, q}([m+2]_{p, q}-[m]_{p, q})}\vert a_{m+1}\vert^{2}\leq\frac{\sqrt{2}[m]_{p, q}}{[m+2]_{p, q}-[m]_{p, q}}; \end{eqnarray*} $

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

$ \begin{eqnarray*} \vert a_{m+2}-\mu a^{2}_{m+1}\vert+\frac{\sqrt{2}(1-\hbar)([m+1]_{p, q}-[m]_{p, q})^{2}}{[m]_{p, q}([m+2]_{p, q}-[m]_{p, q})}\vert a_{m+1}\vert^{2}\leq\frac{\sqrt{2}[m]_{p, q}}{[m+2]_{p, q}-[m]_{p, q}}. \end{eqnarray*} $

Our objective is to generalize some classical interesting results in geometric function theory from the ordinary analysis to $ q $-analysis or $ (p, q) $-analysis. Here, by using the $ (p, q) $-derivative operator and Bazilevič function, certain new subclass of analytic and multivalent functions can be defined to improve the classical starlike functions and even $ q $-starlike functions. In our main results, for this class we obtain the corresponding bound estimates of the coefficients $ a_{m+1} $ and $ a_{m+2} $ and the Fekete-Szegö functional inequalities. Of course, we may choose another special functions to replace Bazilevič function and the limacon function, which are leaved out the interested readers to realize.