In the article, our aim focuses on the certain quasi-subordination subclasses of analytic and bi-univalent functions associated with the Dziok-Srivastava operator. To state our results, at first we will recall some notations and basic properties for analytic and bi-univalent functions and Dziok-Srivastava operator.

Let $ \mathcal{A} $ be the class of normalized analytic function $ f(z) $ by

$ \begin{equation} f(z) = z+\sum\limits^{\infty}_{n = 2}a_{n}z^{n} \end{equation} $

(1.1)

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

Let the subclass $ \mathcal{S} $ of $ \mathcal{A} $ be the set of all univalent functions in $ \Delta $. According to the Koebe one quarter theorem [1], the inverse $ f^{-1} $ of every $ f\in\mathcal{S} $ satisfies

$ \begin{equation*} f^{-1}(f(z)) = z\; \; (z\in\Delta) \quad \text{and}\quad f(f^{-1}(w)) = w\; \; (w\in\Delta_{\rho}), \end{equation*} $

where $ \rho\geq \frac{1}{4} $ denotes the radius of the image $ f(\Delta) $ and $ \Delta_{\rho} = \{z\in\mathbb{C}: \mid z\mid<\rho\} $. It is recalled that

$ \begin{equation} f^{-1}(w) = w-a_{2}w^{2}+(2a^{2}_{2}-a_{3})w^{3}-(5a^{3}_{2}-5a_{2}a_{3}+a_{4})w^{4}+\cdots. \end{equation} $

(1.2)

If both the function $ f\in\mathcal{A} $ and its inverse $ f^{-1} $ are univalent in $ \Delta $, then it is bi-univalent. Denote by $ \Sigma $ the class of all bi-univalent functions $ f\in\mathcal{A} $ in $ \Delta $.

For given $ f, g\in\mathcal{A} $, define the Hadamard product or convolution $ f\ast g $ by

$ \begin{equation*} (f\ast g)(z) = z+\sum\limits^{\infty}_{n = 2}a_{n}b_{n}z^{n}\; \; (z\in\Delta), \end{equation*} $

where $ f(z) $ is given by (1.1) and $ g(z) = z+\sum\limits^{\infty}_{k = 2} b_{k}z^{k} $. Assume that the Gaussian hypergeometric function $ {}_{q}F_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s};z) $ is defined by

$ \begin{eqnarray*} {}_{q}F_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s};z)& = &\sum^{\infty}_{n = 0}\frac{\Pi^{q}_{k = 1}(\alpha_{k})_{n}}{\Pi^{s}_{j = 1}(\beta_{j})_{n}}\frac{z^{n}}{n!} \notag\\ & = &1+\sum^{\infty}_{n = 2}\frac{\Pi^{q}_{k = 1}(\alpha_{k})_{n-1}}{\Pi^{s}_{j = 1}(\beta_{j})_{n-1}}\frac{z^{n-1}}{(n-1)!}\; \; (z\in\Delta) \end{eqnarray*} $

for the complex parameters $ \alpha_{k} $ and $ \beta_{j} $ with $ \beta_{j}\neq0, -1, -2, -3,\ldots $ $ (k = 1,\cdots, q;j = 1, \cdots, s) $, where $ (\ell)_{n} $ denotes the Pochhammer symbol or shifted factorial by

$ (\ell)_{n} = \frac{\Gamma(\ell+n)}{\Gamma(\ell)} = \left\{\begin{array}{ll} 1, &\mbox{if}\; \; n = 0,\ell\in\mathbb{C}\setminus\{0\},\\ \ell(\ell+1)(\ell+2)\ldots (\ell+n-1), &\mbox{if}\; \; n\in\mathbb{N} = \{1,2,3,\ldots\}. \end{array}\right. $

Dziok and Srivastava [2, 3] ever introduced the convolution operator $ {}_{q}\mathcal{I}_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s}) $$ = {}_{q}\mathcal{I}_{s} $ later named by themselves as follows

$ \begin{eqnarray} {}_{q}\mathcal{I}_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s})f(z) & = &{z}_{q}F_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s};z)\ast f(z) \\ & = &z+\sum^{\infty}_{n = 2}p_{n}(q,s)a_{n}z^{n}\; \; (z\in\Delta), \end{eqnarray} $

(1.3)

where

$ \begin{equation} p_{n}(q,s) = \frac{\Pi^{q}_{k = 1}(\alpha_{k})_{n-1}}{\Pi^{s}_{j = 1}(\beta_{j})_{n-1}(n-1)!}. \end{equation} $

(1.4)

Note that

$ \begin{eqnarray*} &&z[{}_{q}\mathcal{I}_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s})f(z)]'\notag \\ & = &\alpha_{1}{}_{q}\mathcal{I}_{s}(\alpha_{1}+1,\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s})f(z)-(\alpha_{1}-1){}_{q}\mathcal{I}_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s})f(z) \notag \\ & = &\alpha_{2}{}_{q}\mathcal{I}_{s}(\alpha_{1},\alpha_{2}+1,\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s})f(z)-(\alpha_{2}-1){}_{q}\mathcal{I}_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s})f(z) \notag \\ &&\vdots\notag \\ & = &\alpha_{q}{}_{q}\mathcal{I}_{s}(\alpha_{1},\cdots,\alpha_{q}+1;\beta_{1},\cdots,\beta_{s})f(z)-(\alpha_{q}-1){}_{q}\mathcal{I}_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s})f(z)\; \; (z\in\Delta). \end{eqnarray*} $

Here we remind some reduced versions of Dziok-Srivastava operator $ {}_{q}\mathcal{I}_{s}(\alpha_{1},\cdots,\alpha_{q};\beta_{1},\cdots,\beta_{s}) $ for suitable parameters $ \alpha_{k}(k = 1,\cdots,q) $ and $ \beta_{j}(j = 1,\cdots,s) $; refer to the generalized Bernardi operator $ \mathcal{J}_{\eta} = {}_{2}\mathcal{I}_{1}(1,1+\eta;2+\eta)(\Re (\eta)>-1) $ [4]; Carlson-Shaffer operator $ \mathcal{L}(a,c) = {}_{2}\mathcal{I}_{1}(a,1;c) $ [5]; Choi-Saigo-Srivastava operator $ \mathcal{I}_{\lambda,\mu} = {}_{2}\mathcal{I}_{1}(\mu,1;\lambda+1)(\lambda>-1, \mu\geq0) $ [6]; Hohlov operator $ \mathcal{I}^{a,b}_{c} = {}_{2}\mathcal{I}_{1}(a,b;c) $ [7, 8]; Noor integral operator $ \mathcal{I}^{n} = {}_{2}\mathcal{I}_{1}(2,1;n+1) $ [9]; Owa-Srivastava fractional differential operator $ \Omega^{\lambda}_{z} = {}_{2}\mathcal{I}_{1}(2,1;2-\lambda)(0\leq\lambda<1) $ [10, 11]; Ruscheweyh derivative operator $ \mathcal{D}^{\delta} = {}_{2}\mathcal{I}_{1}(1+\delta,1;1) $ [12].

In 1967, Lewin [13] introduced the analytic and bi-univalent function and proved that $ \mid a_{2}\mid<1.51 $. Moreover, Brannan and Clunie [14] conjectured that $ \mid a_{2}\mid\leq\sqrt{2} $, and Netanyahu [15] obtained that $ \max\limits_{f\in\sum}\mid a_{2}\mid = \frac{4}{3} $. Later, Styer and Wright [17] showed that there exists function $ f(z) $ so that $ \mid a_{2}\mid>\frac{4}{3} $. However, so far the upper bound estimate $ \mid a_{2}\mid<1.485 $ of coefficient for functions in $ \sum $ by Tan [18] is best. Unfortunately, as for the coefficient estimate problem for every Taylor-Maclaurin coefficient $ \mid a_{n}\mid(n\in\mathbb{N}\setminus \{1,2\}) $ it is probably still an open problem. Based on the works of Brannan and Taha [19] and Srivastava et al. [20], many subclasses of analytic and bi-univalent functions class $ \sum $ were introduced and investigated, and the non-sharp estimates of first two Taylor-Maclaurin coefficients $ \mid a_{2}\mid $ and $ \mid a_{3}\mid $ were given; refer to Deniz [21], Frasin and Aouf [22], Hayami and Owa [23], Patil and Naik [24, 25], Srivastava et al. [26, 27], Tang et al. [28] and Xu et al. [29, 30] for more detailed information. Recently, Srivastava et al. [31, 32] gave some new subclasses of the function class $ \sum $ of analytic and bi-univalent functions to unify the works of Deniz [21], Frasin [33], Keerthi and Raja [34], Srivastava et al. [35], Murugusundaramoorthy et al.[36] and Xu et al. [29], etc. Besides, we also refer to Goyal et al. [37] for the subclasses of analytic and bi-univalent associated with quasi-subordination. Since Fekete-Szegö [38] studied the determination of the sharp upper bounds for the subclass of $ \mathcal{S} $, Fekete-Szegö functional problem was considered in many classes of functions; refer to Abdel-Gawad [39] for class of quasi-convex functions, Koepf [40] for class of close-to-convex functions, Orhan and Rǎducanu [16] for class of starlike functions, Magesh and Balaji [41] for class of convex and starlike functions, Orhan et al. [42] for the classes of bi-convex and bi-starlike type functions, Panigrahi and Raina [43] for class of quasi-subordination functions, Tang et al. [28] for classes of m-mold symmetric bi-univalent functions. In addition, Murugusundaramoorthy et al. [36, 44, 45] and Patil and Naik [46] ever introduced and investigated several new subclasses of the function class $ \sum $ of analytic and bi-univalent functions involving the hohlov operator. Moreover, Al-Hawary et al. [47] studied the Fekete-Szegö functional problem for the classes of analytic functions of complex order defined by the Dziok-Srivastava operator. Motivated by the statements above, in the article we are ready to introduce and investigate two new subclasses of the function class $ \sum $ of analytic and bi-univalent functions associated with the Dziok-Srivastava operator and quasi-subordination, and consider the corresponding bound estimates of the coefficient $ a_2 $ and $ a_3 $ as well as the corresponding Fekete-Szegö functional inequalities. Furthermore, the consequences and connections to some earlier known results would be pointed out.

For two analytic functions $ f $ and $ g $, if there exist two analytic functions $ \varphi $ and $ h $ with $ \mid\varphi(z)\mid\leq1 $, $ h(0) = 0 $ and $ \mid h(z)\mid<1 $ for $ z\in\Delta $ so that $ f(z) = \varphi(z)g(h(z)) $, then $ f $ is quasi-subordinate to $ g $, i.e., $ f\prec_{\text{quasi}}g $. Note that if $ \varphi\equiv1 $, then $ f $ is subordinate to $ g $ in $ \Delta $, i.e., $ f\prec g $. Further, if $ h(z) = z $, then $ f $ is majorized by $ g $ in $ \Delta $, i.e. $ f\leq g $. For the related work on quasi-subordination, refer to Robertson [48], and Frasin and Aouf [22]. Write

$ \begin{equation} \varphi(z) = B_{0}+B_{1}z+B_{2}z^{2}+B_{3}z^{3}+\ldots\quad(\mid\varphi(z)\mid\leq1, z\in\Delta). \end{equation} $

(1.5)

First we will introduce the following general subclasses of analytic and bi-univalent functions associated with the Dziok-Srivastava operator.

Definition 1.1 A function $ f(z)\in\sum $ given by (1.1), belongs to the class $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $ if the following quasi-subordinations are satisfied

$ \begin{equation} \left [\frac{z({}_{q}\mathcal{I}_{s}f)'(z)}{{}_{q}\mathcal{I}_{s}f(z)}\right]\left [\frac{({}_{q}\mathcal{I}_{s}f)(z)}{z}\right]^{\eta}-1\prec_{\text{quasi}}(\phi(z)-1) \end{equation} $

(1.6)

and

$ \begin{equation} \left [\frac{z({}_{q}\mathcal{I}_{s}g)'(w)}{{}_{q}\mathcal{I}_{s}g(w)}\right]\left [\frac{({}_{q}\mathcal{I}_{s}g)(w)}{w}\right]^{\eta}-1\prec_{\text{quasi}}(\phi(w)-1) \end{equation} $

(1.7)

for $ z, w\in\Delta $, where $ \eta\geq0 $ and the function $ g $ is the inverse of $ f $ given by (1.2).

Definition 1.2 A function $ f(z)\in\sum $ given by (1.1), belongs to the class $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}} $ $ (\tau,\mu,\lambda,\gamma;\phi) $ if the following quasi-subordinations are satisfied:

$ \begin{equation} \frac{1}{\tau} [\frac{z({}_{q}\mathcal{I}_{s}f)'(z)+\mu z^{2}({}_{q}\mathcal{I}_{s}f)''(z)}{(1-\lambda)z+\lambda(1-\gamma)({}_{q}\mathcal{I}_{s}f)(z)+\gamma z({}_{q}\mathcal{I}_{s}f)'(z)}-\frac{1}{[1+\gamma(1-\lambda)]} ]\prec_{\text{quasi}}(\phi(z)-1) \end{equation} $

(1.8)

and

$ \begin{equation} \frac{1}{\tau} [\frac{w({}_{q}\mathcal{I}_{s}g)'(z)+\mu w^{2}({}_{q}\mathcal{I}_{s}g)''(w)}{(1-\lambda)w+\lambda(1-\gamma)({}_{q}\mathcal{I}_{s}g)(w)+\gamma w({}_{q}\mathcal{I}_{s}g)'(w)}-\frac{1}{[1+\gamma(1-\lambda)]} ]\prec_{\text{quasi}}(\phi(w)-1) \end{equation} $

(1.9)

for $ z, w\in\Delta $, where $ \tau\in \mathbb{C}\setminus\{0\} $, $ 0\leq\mu\leq1 $, $ 0\leq\lambda\leq1 $, $ 0\leq\gamma\leq1 $ and the function $ g $ is the inverse of $ f $ given by (1.2).

Lemma 1.3(see [1, 49]) Let $ \mathcal{P} $ be the class of all analytic functions $ q(z) $ of the following form

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

satisfying $ \Re q(z)>0 $ and $ q(0) = 1 $. 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 next function

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

Denote the functions $ s $ and $ t $ in $ \mathcal{P} $ by

$ \begin{equation} s(z) = \frac{1+u(z)}{1-u(z)} = 1+\sum\limits^{\infty}_{n = 1}c_{n}z^{n}\quad\text{and}\quad t(w) = \frac{1+v(w)}{1-v(w)} = 1+\sum\limits^{\infty}_{n = 1}d_{n}w^{n}\; \; (z,w\in\Delta). \end{equation} $

(2.1)

Equivalently, from (2.1) we know that

$ \begin{equation} u(z) = \frac{s(z)-1}{s(z)+1} = \frac{c_{1}}{2}z+\frac{1}{2}(c_{2}-\frac{c^{2}_{1}}{2})z^{2}+\ldots \; \; (z\in\Delta) \end{equation} $

(2.2)

and

$ \begin{equation} v(w) = \frac{t(w)-1}{t(w)+1} = \frac{d_{1}}{2}w+\frac{1}{2}(d_{2}-\frac{d^{2}_{1}}{2})w^{2}+\ldots\; \; (w\in\Delta). \end{equation} $

(2.3)

Given $ \phi\in\mathcal{P} $ with $ \phi'(0)>0 $, let $ \phi(\Delta) $ be symmetric with respect to the real axis. When the series expansion form of $ \phi $ is denoted by

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

(2.4)

by (2.2)–(2.3) and (2.4) it follows that

$ \begin{equation} \phi(u(z)) = 1+\frac{1}{2}E_{1}c_{1}z+[\frac{1}{2}E_{1}(c_{2}-\frac{c^{2}_{1}}{2})+\frac{1}{4}E_{2}c^{2}_{1}]z^{2}+\ldots\; \; (z\in\Delta) \end{equation} $

(2.5)

and

$ \begin{equation} \phi(v(w)) = 1+\frac{1}{2}E_{1}d_{1}w+[\frac{1}{2}E_{1}(d_{2}-\frac{d^{2}_{1}}{2})+\frac{1}{4}E_{2}d^{2}_{1}]w^{2}+\ldots\; \; (w\in\Delta). \end{equation} $

(2.6)

In the section we study the estimates for the class $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $. Now, we establish the next theorem.

Theorem 2.1 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $, then

$ \begin{equation} \mid a_{2}\mid\leq\min\{\frac{E_{1}}{(\eta+1)},\frac{2(\vert E_{2}-E_{1}\vert+ E_{1})}{\sqrt{(\eta+1)(\eta+2)}},\frac{E_{1}\sqrt{2E_{1}}}{\sqrt{\vert\Xi(\eta,B_{0},E_{1},E_{2})\vert}}\}\frac{\vert B_{0}\vert}{\vert p_{2}(q,s)\vert} \end{equation} $

(2.7)

and

$ \begin{equation} \mid a_{3}\mid\leq\frac{(\vert B_{0}+\vert B_{1}\vert) E_{1}}{(\eta+2)\vert p_{3}(q,s)\vert}+\min\{\frac{\vert B_{0}\vert E^{2}_{1}}{\eta+1}, \frac{2(\vert E_{2}-E_{1}\vert+E_{1})}{\eta+2}\}\frac{\vert B_{0}\vert}{(\eta+1)\vert p_{3}(q,s)\vert}, \end{equation} $

(2.8)

where

$ \begin{equation} \Xi(\eta,B_{0},E_{1},E_{2}) = (\eta+1)(\eta+2)B_{0}E^{2}_{1}+2(\eta+1)^{2}(E_{1}-E_{2}). \end{equation} $

(2.9)

Proof If $ f(z)\in\mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $, then by Definition 1.1 and Lemma 1.3, there exist two analytic functions $ u(z) $ and $ v(w)\in\mathcal{P} $ so that

$ \begin{equation} [\frac{z({}_{q}\mathcal{I}_{s}f)'(z)}{{}_{q}\mathcal{I}_{s}f(z)}] [\frac{({}_{q}\mathcal{I}_{s}f)(z)}{z}]^{\eta}-1 = \varphi(z)[\phi(u(z))-1] \end{equation} $

(2.10)

and

$ \begin{equation} [\frac{z({}_{q}\mathcal{I}_{s}g)'(w)}{{}_{q}\mathcal{I}_{s}g(w)}] [\frac{({}_{q}\mathcal{I}_{s}g)(w)}{w}]^{\eta}-1 = \varphi(w)[\phi(v(w))-1]. \end{equation} $

(2.11)

Expanding the left half parts of (2.11) and (2.12), we obtain that

$ \begin{eqnarray} &&[\frac{z({}_{q}\mathcal{I}_{s}f)'(z)}{{}_{q}\mathcal{I}_{s}f(z)}] [\frac{({}_{q}\mathcal{I}_{s}f)(z)}{z}]^{\eta}\\& = &1+(\eta+1)p_{2}(q,s)a_{2}z + [(\eta+2)p_{3}(q,s)a_{3}+\frac{(\eta-1)(\eta+2)}{2}p^{2}_{2}(q,s)a^{2}_{2}]z^{2}+\cdots\qquad \end{eqnarray} $

(2.12)

and

$ \begin{eqnarray} &&[\frac{z({}_{q}\mathcal{I}_{s}g)'(w)}{{}_{q}\mathcal{I}_{s}g(w)}] [\frac{({}_{q}\mathcal{I}_{s}g)(w)}{w}]^{\eta}\\ & = &1- (\eta+1)p_{2}(q,s)a_{2}w + [-(\eta+2)p_{3}(q,s)a_{3} +\frac{(\eta+2)(\eta+3)}{2}p^{2}_{2}(q,s)a^{2}_{2}]w^{2}+\cdots.\quad\qquad \end{eqnarray} $

(2.13)

In addition, we know that

$ \begin{equation} \varphi(z)[\phi(u(z))-1] = \frac{1}{2}B_{0}E_{1}c_{1}z+[\frac{1}{2}B_{1}E_{1}c_{1}+\frac{1}{2}B_{0}E_{1}(c_{2}-\frac{c^{2}_{1}}{2})+\frac{1}{4}B_{0}E_{2}c^{2}_{1}]z^{2}+\cdots \end{equation} $

(2.14)

and

$ \begin{equation} \varphi(w)[\phi(v(w))-1] = \frac{1}{2}B_{0}E_{1}d_{1}w+[\frac{1}{2}B_{1}E_{1}d_{1}+\frac{1}{2}B_{0}E_{1}(d_{2}-\frac{d^{2}_{1}}{2})+\frac{1}{4}B_{0}E_{2}d^{2}_{1}]w^{2}+\cdots. \end{equation} $

(2.15)

Therefore, from (2.10)–(2.15) we have that

$ \begin{equation} (\eta+1)p_{2}(q,s)a_{2} = \frac{1}{2}B_{0}E_{1}c_{1}, \end{equation} $

(2.16)

$ \begin{equation} (\eta+2)p_{3}(q,s)a_{3}+\frac{(\eta-1)(\eta+2)}{2}p^{2}_{2}(q,s)a^{2}_{2} = \frac{1}{2}B_{1}E_{1}c_{1}+\frac{1}{2}B_{0}E_{1}(c_{2}-\frac{c^{2}_{1}}{2})+\frac{1}{4}B_{0}E_{2}c^{2}_{1}, \end{equation} $

(2.17)

$ \begin{equation} -(\eta+1)p_{2}(q,s)a_{2} = \frac{1}{2}B_{0}E_{1}d_{1} \end{equation} $

(2.18)

and

$ \begin{equation} -(\eta+2)p_{3}(q,s)a_{3} +\frac{(\eta+2)(\eta+3)}{2}p^{2}_{2}(q,s)a^{2}_{2} = \frac{1}{2}B_{1}E_{1}d_{1}+\frac{1}{2}B_{0}E_{1}(d_{2}-\frac{d^{2}_{1}}{2})+\frac{1}{4}B_{0}E_{2}d^{2}_{1}. \end{equation} $

(2.19)

From (2.16) and (2.18), it infers that

$ \begin{equation} a_{2} = \frac{B_{0}E_{1}c_{1}}{2(\eta+1)p_{2}(q,s)} = -\frac{B_{0}E_{1}d_{1}}{2(\eta+1)p_{2}(q,s)}. \end{equation} $

(2.20)

Then, we show that

$ \begin{equation} c_{1} = -d_{1} \end{equation} $

(2.21)

and

$ \begin{equation} B^{2}_{0}E^{2}_{1}(c^{2}_{1}+d^{2}_{1}) = 8(\eta+1)^{2}p^{2}_{2}(q,s)a^{2}_{2}. \end{equation} $

(2.22)

By (2.17) and (2.19), we have that

$ \begin{equation} \frac{1}{4}B_{0}(E_{2}-E_{1})(c^{2}_{1}+d^{2}_{1})+\frac{1}{2}B_{0}E_{1}(c_{2}+d_{2}) = (\eta+1)(\eta+2)p^{2}_{2}(q,s)a^{2}_{2}. \end{equation} $

(2.23)

Therefore, by (2.22)–(2.23) we obtain that

$ \begin{equation} a^{2}_{2} = \frac{B^{2}_{0}E^{3}_{1}(c_{2}+d_{2})}{2(\eta+1)[(\eta+2)B_{0}E^{2}_{1}+2(\eta+1)(E_{1}-E_{2})]p^{2}_{2}(q,s)}. \end{equation} $

(2.24)

We follow from Lemma 1.3 and (2.22)–(2.24) that

$ \begin{equation*} \mid a_{2}\mid\leq\frac{\vert B_{0}\vert E_{1}}{(\eta+1)\vert p_{2}(q,s)\vert}, \end{equation*} $

$ \begin{equation*} \mid a_{2}\mid\leq\frac{2\vert B_{0}\vert(\vert E_{2}-E_{1}\vert+ E_{1})}{\sqrt{(\eta+1)(\eta+2)}\vert p_{2}(q,s)\vert} \end{equation*} $

and

$ \begin{equation*} \mid a_{2}\mid\leq\frac{\vert B_{0}\vert E_{1}\sqrt{2E_{1}}}{\sqrt{(\eta+1)\vert(\eta+2)B_{0}E^{2}_{1}+2(\eta+1)(E_{1}-E_{2})\vert}\vert p_{2}(q,s)\vert}, \end{equation*} $

then (2.7) holds. Similarly, from (2.17), (2.19) and (2.21), it also implies that

$ \begin{equation} B_{1}E_{1}(c_{1}-d_{1})+B_{0}E_{1}(c_{2}-d_{2}) = 4(\eta+2)p_{3}(q,s)a_{3}-4(\eta+2)p^{2}_{2}(q,s)a^{2}_{2}. \end{equation} $

(2.25)

Hence, from (2.22) and (2.25), we obtain that

$ \begin{equation*} a_{3} = \frac{B_{1}E_{1}(c_{1}-d_{1})+B_{0}E_{1}(c_{2}-d_{2})}{4(\eta+2)p_{3}(q,s)} +\frac{B^{2}_{0}E^{2}_{1}(c^{2}_{1}+d^{2}_{1})}{8(\eta+1)^{2}p_{3}(q,s)}. \end{equation*} $

Therefore, from Lemma 1.3 it shows that

$ \begin{equation*} \mid a_{3}\mid\leq\frac{(\vert B_{0}+\vert B_{1}\vert) E_{1}}{(\eta+2)\vert p_{3}(q,s)\vert}+\frac{B^{2}_{0}E^{2}_{1}}{(\eta+1)^{2}\vert p_{3}(q,s)\vert}. \end{equation*} $

On the other hand, by (2.23) and (2.25), we infer that

$ \begin{equation*} a_{3} = \frac{B_{1}E_{1}(c_{1}-d_{1})+B_{0}E_{1}(c_{2}-d_{2})}{4(\eta+2)p_{3}(q,s)} +\frac{B_{0}(E_{2}-E_{1})(c^{2}_{1}+d^{2}_{1})+2B_{0}E_{1}(c_{2}+d_{2})}{4(\eta+1)(\eta+2)p_{3}(q,s)}. \end{equation*} $

Thus, from Lemma 1.3, we see that

$ \begin{equation*} \mid a_{3}\mid\leq\frac{(\vert B_{0}+\vert B_{1}\vert) E_{1}}{(\eta+2)\vert p_{3}(q,s)\vert} +\frac{2\vert B_{0}\vert(\vert E_{2}-E_{1}\vert+E_{1})}{(\eta+1)(\eta+2)\vert p_{3}(q,s)\vert}. \end{equation*} $

Next, we consider Fekete-Szegö problems for the class $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $.

Theorem 2.2 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $ and $ \delta\in\mathbb{R} $, then

$ \mid a_{3}-\delta a^{2}_{2}\mid\leq\left\{\begin{array}{ll} \frac{\vert B_{0}\vert E_{1}}{2(\eta+2)\vert p_{3}(q,s)\vert},\; \; &\mbox{if}\; \; 2\vert B_{0}\vert E_{1}^{2}(\eta+2)\vert p^{2}_{2}(q,s)-\delta p_{3}(q,s)\vert\leq\mid\Xi\mid p^{2}_{2}(q,s),\\ \frac{2\vert B_{0}\vert^{2}E^{3}_{1}\vert p^{2}_{2}(q,s)-\delta p_{3}(q,s)\vert}{\vert\Xi p_{3}(q,s)\vert p^{2}_{2}(q,s)},\; \; &\mbox{if}\; \; 2\vert B_{0}\vert E_{1}^{2}(\eta+2)\vert p^{2}_{2}(q,s)-\delta p_{3}(q,s)\vert\geq\mid\Xi\mid p^{2}_{2}(q,s), \end{array}\right. $

where $ \Xi = \Xi(\eta,B_{0},E_{1},E_{2}) $ is the same as in Theorem 2.1.

Proof From (2.25), it follows that

$ \begin{equation*} a_{3}-\frac{p^{2}_{2}(q,s)a^{2}_{2}}{p_{3}(q,s)} = \frac{B_{1}E_{1}(c_{1}-d_{1})+B_{0}E_{1}(c_{2}-d_{2})}{4(\eta+2)p_{3}(q,s)}. \end{equation*} $

By (2.24) we easily obtain that

$ \begin{eqnarray*} a_{3}-\delta a^{2}_{2}& = &\frac{B_{0}E_{1}[\Xi p^{2}_{2}(q,s)+2B_{0}E_{1}^{2}(\eta+2)(p^{2}_{2}(q,s)-\delta p_{3}(q,s))]c_{2}}{4(\eta+2)\Xi p_{3}(q,s)p^{2}_{2}(q,s)} \notag\\ &&+\frac{-B_{0}E_{1}[\Xi p^{2}_{2}(q,s)+2B_{0}E_{1}^{2}(\eta+2)(p^{2}_{2}(q,s)-\delta p_{3}(q,s))]d_{2}}{4(\eta+2)\Xi p_{3}(q,s)p^{2}_{2}(q,s)}. \end{eqnarray*} $

Hence, from Lemma 1.3, we imply that

$ \mid a_{3}-\delta a^{2}_{2}\mid\leq\left\{\begin{array}{ll} \frac{\vert B_{0}\vert E_{1}}{2(\eta+2)\vert p_{3}(q,s)\vert},\; \; &\mbox{if}\; \; 2\vert B_{0}\vert E_{1}^{2}(\eta+2)\vert p^{2}_{2}(q,s)-\delta p_{3}(q,s)\vert\leq\mid\Xi\mid p^{2}_{2}(q,s),\\ \frac{2\vert B_{0}\vert^{2}E^{3}_{1}\vert p^{2}_{2}(q,s)-\delta p_{3}(q,s)\vert}{\vert\Xi p_{3}(q,s)\vert p^{2}_{2}(q,s)},\; \; &\mbox{if}\; \; 2\vert B_{0}\vert E_{1}^{2}(\eta+2)\vert p^{2}_{2}(q,s)-\delta p_{3}(q,s)\vert\geq\mid\Xi\mid p^{2}_{2}(q,s). \end{array}\right. $

Corollary 2.3 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $ and $ \delta\in\mathbb{R} $, then

$ \vert a_{3}\vert\leq\left\{\begin{array}{ll} \frac{\vert B_{0}\vert E_{1}}{2(\eta+2)\vert p_{3}(q,s)\vert},\; \; &\mbox{if}\; \; 2\vert B_{0}\vert E_{1}^{2}(\eta+2)\leq\vert\Xi\vert,\\ \frac{2\vert B_{0}\vert^{2}E^{3}_{1}}{\vert\Xi p_{3}(q,s)\vert },\; \; &\mbox{if}\; \; 2\vert B_{0}\vert E_{1}^{2}(\eta+2)\geq\vert\Xi\vert, \end{array}\right. $

where $ \Xi = \Xi(\eta,B_{0},E_{1},E_{2}) $ is the same as in Theorem 2.1.

Remark 2.4 Without quasi-subordination (i.e., $ \varphi(z)\equiv1 $), if we choose some suitable parameters $ \alpha_{k} $ $ (k = 1,\cdots,q) $, $ \beta_{j} $ $ (j = 1,\cdots,s) $ and $ \eta $, we obtain the following reduced versions for $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $ in Theorem 2.1.

(Ⅰ) $ \mathcal{QH_{\sum}}^{a,b}_{c}(\alpha;\phi) = \mathcal{J}^{a,b;c}_{\sum}(\alpha,\phi) $, refer to Patil and Naik [46].

Remark 2.5 Without Dziok-Srivastava operator, we can collect the following reduced versions for $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) $ in Theorem 2.1.

(Ⅰ) $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\eta;\phi) = \mathcal{J}^{q}_{\eta}(\phi) $ $ (\eta\geq0) $, refer to Goyal et al. [37].

(Ⅱ) $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(1;\phi) = \mathcal{H}_{\sum}(\phi) $ for $ \varphi(z)\equiv1 $, refer to Ali et al. [50] and Tang et al. [51] for Corollary 2.2; $ \mathcal{QH_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(0;\phi) = \mathcal{S}^{*}_{\sum}(\phi) $ for $ \varphi(z)\equiv1 $, refer to Brannan and Taha. [19] and Tang et al. [51] for Corollary 2.4.

Now, we study the coefficients for the class $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\mu,\lambda,\gamma;\phi) $ and establish the next theorem.

Theorem 3.1 If $ f(z) $ given by (1.1) belongs to the class $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\mu,\lambda,\gamma;\phi) $, then

$ \begin{equation} \mid a_{2}\mid\leq\min\{\mathcal{F}_{1}, \mathcal{F}_{2}, \mathcal{F}_{3}\} \end{equation} $

(3.1)

for

$ \begin{equation*} \mathcal{F}_{1} = \frac{\mid B_{0}\tau\mid E_{1}(1+\gamma-\gamma\lambda)^{2}}{\vert p_{2}(q,s)\Psi(\mu,\lambda,\gamma)\vert},\quad \mathcal{F}_{2} = \sqrt{\frac{\mid B_{0}\mid(\mid E_{2}-E_{1}\mid+E_{1})\mid\tau\mid (1+\gamma-\gamma\lambda)^{3}}{\mid\Phi(\mu,\lambda,\gamma,p_{2}(q,s), p_{3}(q,s))\mid}} \end{equation*} $

and

$ \begin{equation*} \mathcal{F}_{3} = \frac{\mid B_{0}\tau\mid E^{3/2}_{1}(1+\gamma-\gamma\lambda)^{2}}{\sqrt{\mid B_{0}\tau E^{2}_{1}(1+\gamma-\gamma\lambda)\Phi(\mu,\lambda,\gamma,p_{2}(q,s), p_{3}(q,s))+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}(\mu,\lambda,\gamma)\mid}}, \end{equation*} $

and

$ \begin{equation*} \mid a_{3}\mid\leq\min\{\mathcal{G}_{1}, \mathcal{G}_{2}\} \end{equation*} $

for

$ \begin{equation*} \mathcal{G}_{1} = \frac{\mid\tau\mid(\vert B_{0}\vert+\vert B_{1}\vert) E_{1}(1+\gamma-\gamma\lambda)^{2}}{\vert p_{3}(q,s)\vert\Theta(\mu,\lambda,\gamma)}+\frac{\mid\tau\mid^{2}B_{0}^{2} E_{1}^{2}(1+\gamma-\gamma\lambda)^{4}}{p^{2}_{2}(q,s)\Psi^{2}(\mu,\lambda,\gamma)} \end{equation*} $

and

$ \begin{equation*} \mathcal{G}_{2} = \frac{\mid\tau\mid(\vert B_{0}\vert+\vert B_{1}\vert) E_{1}(1+\gamma-\gamma\lambda)^{2}}{\vert p_{3}(q,s)\vert\Theta(\mu,\lambda,\gamma)}+\frac{\mid\tau B_{0}\mid (\vert E_{2}-E_{1}\vert+E_{1})(1+\gamma-\gamma\lambda)^{3}}{\vert\Phi(\mu,\lambda,\gamma,p_{2}(q,s),p_{3}(q,s))\vert}, \end{equation*} $

where

$ \begin{equation} \Phi(\mu,\lambda,\gamma,p_{2}(q,s),p_{3}(q,s)) = (1+\gamma-\gamma\lambda)\Theta(\mu,\lambda,\gamma)p_{3}(q,s)-[\lambda(1-\gamma)+2\gamma]\Psi(\mu,\lambda,\gamma)p^{2}_{2}(q,s), \end{equation} $

(3.2)

$ \begin{equation} \Psi(\mu,\lambda,\gamma) = 2-\gamma\lambda-\lambda+2\mu(1+\gamma-\gamma\lambda) \end{equation} $

(3.3)

and

$ \begin{equation} \Theta(\mu,\lambda,\gamma) = 3-2\gamma\lambda-\lambda+6\mu(1+\gamma-\gamma\lambda). \end{equation} $

(3.4)

Proof Here, we follow the method of Theorem 2.1. If $ f(z)\in\mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\mu,\lambda,\gamma;\phi) $, then by Definition 1.2 there exist two analytic functions $ u(z),v(z): \Delta\rightarrow\Delta $ with $ u(0) = 0 $ and $ v(0) = 0 $ such that

$ \begin{equation} \frac{1}{\tau} [\frac{z({}_{q}\mathcal{I}_{s}f)'(z)+\mu z^{2}({}_{q}\mathcal{I}_{s}f)''(z)}{(1-\lambda)z+\lambda(1-\gamma){}_{q}\mathcal{I}_{s}f(z)+\gamma z({}_{q}\mathcal{I}_{s}f)'(z)}-\frac{1}{[1+\gamma(1-\lambda)]}] = \varphi(z)[\phi(u(z))-1] \end{equation} $

(3.5)

and

$ \begin{equation} \frac{1}{\tau}[\frac{w ({}_{q}\mathcal{I}_{s}g)'(z)+\mu w^{2}({}_{q}\mathcal{I}_{s}g)''(w)}{(1-\lambda)w+\lambda(1-\gamma)({}_{q}\mathcal{I}_{s}g)(w)+\gamma w({}_{q}\mathcal{I}_{s}g)'(w)}-\frac{1}{[1+\gamma(1-\lambda)]}] = \varphi(w)[\phi(v(w))-1]. \end{equation} $

(3.6)

Expanding the left half parts of (3.5) and (3.6), we have that

$ \begin{eqnarray} &&\frac{1}{\tau} [\frac{z({}_{q}\mathcal{I}_{s}f)'(z)+\mu z^{2}({}_{q}\mathcal{I}_{s}f)''(z)}{(1-\lambda)z+\lambda(1-\gamma){}_{q}\mathcal{I}_{s}f(z)+\gamma z({}_{q}\mathcal{I}_{s}f)'(z)}-\frac{1}{[1+\gamma(1-\lambda)]}] \\ & = &\frac{\Psi(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{2}}p_{2}(q,s)a_{2}z \\&&+ [\frac{\Theta(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{2}}p_{3}(q,s)a_{3} -\frac{[\lambda(1-\gamma)+2\gamma]\Psi(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{3}}p^{2}_{2}(q,s)a^{2}_{2}]z^{2}+\cdots \end{eqnarray} $

(3.7)

and

$ \begin{eqnarray} &&\frac{1}{\tau}[\frac{w ({}_{q}\mathcal{I}_{s}g)'(z)+\mu w^{2}({}_{q}\mathcal{I}_{s}g)''(w)}{(1-\lambda)w+\lambda(1-\gamma)({}_{q}\mathcal{I}_{s}g)(w)+\gamma w({}_{q}\mathcal{I}_{s}g)'(w)}-\frac{1}{[1+\gamma(1-\lambda)]}] \\& = &-\frac{\Psi(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{2}}p_{2}(q,s)a_{2}w \\&& +\frac{1}{\tau} [\frac{\Theta(\mu,\lambda,\gamma)}{(1+\gamma-\gamma\lambda)^{2}}p_{3}(q,s)a_{3} -\frac{[\lambda(1-\gamma)+2\gamma]\Psi(\mu,\lambda,\gamma)}{(1+\gamma-\gamma\lambda)^{3}}p^{2}_{2}(q,s)a^{2}_{2}]w^{2}+\cdots. \end{eqnarray} $

(3.8)

Therefore, From (2.14)–(2.15) and (3.5)–(3.8), we get that

$ \begin{equation} \frac{\Psi(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{2}}p_{2}(q,s)a_{2} = \frac{1}{2}B_{0}E_{1}c_{1}, \end{equation} $

(3.9)

$ \begin{eqnarray} &&\frac{\Theta(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{2}}p_{3}(q,s)a_{3}- \frac{[\lambda(1-\gamma)+2\gamma]\Psi(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{3}}p^{2}_{2}(q,s)a^{2}_{2} \\ & = &\frac{1}{2}B_{1}E_{1}c_{1}+\frac{1}{2}B_{0}E_{1}(c_{2}-\frac{c^{2}_{1}}{2})+\frac{1}{4}B_{0}E_{2}c^{2}_{1}, \end{eqnarray} $

(3.10)

$ \begin{equation} -\frac{\Psi(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{2}}p_{2}(q,s)a_{2} = \frac{1}{2}B_{0}E_{1}d_{1} \end{equation} $

(3.11)

and

$ \begin{eqnarray} &&\frac{\Theta(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{2}}p_{3}(q,s)(2a^{2}_{2}-a_{3})- \frac{[\lambda(1-\gamma)+2\gamma]\Psi(\mu,\lambda,\gamma)}{\tau(1+\gamma-\gamma\lambda)^{3}}p^{2}_{2}(q,s)a^{2}_{2} \\ & = &\frac{1}{2}B_{1}E_{1}d_{1}+\frac{1}{2}B_{0}E_{1}(d_{2}-\frac{d^{2}_{1}}{2})+\frac{1}{4}B_{0}E_{2}d^{2}_{1}. \end{eqnarray} $

(3.12)

From (3.9) and (3.11), we know that

$ \begin{equation} a_{2} = \frac{B_{0}E_{1}c_{1}\tau(1+\gamma-\gamma\lambda)^{2}}{2p_{2}(q,s)\Psi(\mu,\lambda,\gamma)} = -\frac{B_{0}E_{1}d_{1}\tau(1+\gamma-\gamma\lambda)^{2}}{2p_{2}(q,s)\Psi(\mu,\lambda,\gamma)}. \end{equation} $

(3.13)

Then, it infers that

$ \begin{equation} c_{1} = -d_{1} \end{equation} $

(3.14)

and

$ \begin{equation} B_{0}^{2}E_{1}^{2}(c^{2}_{1}+d^{2}_{1}) = \frac{8p^{2}_{2}(q,s)\Psi(\mu,\lambda,\gamma)^{2}}{\tau^{2}(1+\gamma-\gamma\lambda)^{4}}a^{2}_{2}. \end{equation} $

(3.15)

By (3.10) and (3.12), we have that

$ \begin{equation} \frac{1}{4}B_{0}(c^{2}_{1}+d^{2}_{1})(E_{2}-E_{1})+\frac{1}{2}B_{0}E_{1}(c_{2}+d_{2}) = \frac{2a^{2}_{2}}{\tau(1+\gamma-\gamma\lambda)^{3}}\Phi(\mu,\lambda,\gamma,p_{2}(q,s),p_{3}(q,s)). \end{equation} $

(3.16)

Therefore, by (3.15)–(3.16) we know that

$ \begin{equation} a^{2}_{2} = \frac{\frac{1}{4}B^{2}_{0}E^{3}_{1}\tau^{2}(1+\gamma-\gamma\lambda)^{4}(c_{2}+d_{2})}{B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi(\mu,\lambda,\gamma,p_{2}(q,s), p_{3}(q,s))+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}(\mu,\lambda,\gamma)}. \end{equation} $

(3.17)

Therefore, from (3.15)–(3.17) and Lemma 1.3, we obtain that

$ \begin{eqnarray*} \mid a_{2}\mid&\leq&\frac{\mid B_{0}\tau\mid E_{1}(1+\gamma-\gamma\lambda)^{2}}{\vert p_{2}(q,s)\vert\Psi(\mu,\lambda,\gamma)}, \\ \mid a_{2}\mid&\leq&\sqrt{\frac{\mid B_{0}\mid(\mid E_{2}-E_{1}\mid+E_{1})\mid\tau\mid (1+\gamma-\gamma\lambda)^{3}}{\mid\Phi(\mu,\lambda,\gamma,p_{2}(q,s), p_{3}(q,s))\mid}} \end{eqnarray*} $

and

$ \begin{equation*} \mid a_{2}\mid\leq\frac{\mid B_{0}\tau\mid E^{3/2}_{1}(1+\gamma-\gamma\lambda)^{2}}{\sqrt{\mid B_{0}\tau E^{2}_{1}(1+\gamma-\gamma\lambda)\Phi(\mu,\lambda,\gamma,p_{2}(q,s), p_{3}(q,s))+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}(\mu,\lambda,\gamma)\mid}}. \end{equation*} $

Similarly, from (3.10) and (3.12), it implies that

$ \begin{equation} \frac{1}{2}B_{1}E_{1}(c_{1}-d_{1})+\frac{1}{2}B_{0}E_{1}(c_{2}-d_{2}) = \frac{2p_{3}(q,s)\Theta(\mu,\lambda,\gamma)(a_{3}-a^{2}_{2})}{\tau(1+\gamma-\gamma\lambda)^{2}}. \end{equation} $

(3.18)

Hence, by (3.15) and (3.18), it follows that

$ \begin{equation*} a_{3} = \frac{\tau [B_{1}E_{1}(c_{1}-d_{1})+B_{0}E_{1}(c_{2}-d_{2})](1+\gamma-\gamma\lambda)^{2}}{4p_{3}(q,s)\Theta(\mu,\lambda,\gamma)}+\frac{\tau^{2} B_{0}^{2}E_{1}^{2}(1+\gamma-\gamma\lambda)^{4}(c^{2}_{1}+d^{2}_{1})}{8p^{2}_{2}(q,s)\Psi^{2}(\mu,\lambda,\gamma)}. \end{equation*} $

So, we obtain from Lemma 1.3 that

$ \begin{equation*} \mid a_{3}\mid\leq\frac{\mid\tau\mid(\vert B_{0}\vert+\vert B_{1}\vert) E_{1}(1+\gamma-\gamma\lambda)^{2}}{\mid p_{3}(q,s)\Theta(\mu,\lambda,\gamma)\mid}+\frac{\mid\tau\mid^{2}B_{0}^{2} E_{1}^{2}(1+\gamma-\gamma\lambda)^{4}}{p^{2}_{2}(q,s)\Psi^{2}(\mu,\lambda,\gamma)}. \end{equation*} $

On the other hand, by (3.16) and (3.18), we infer that

$ \begin{eqnarray*} a_{3}& = &\frac{\tau [B_{1}E_{1}(c_{1}-d_{1})+B_{0}E_{1}(c_{2}-d_{2})](1+\gamma-\gamma\lambda)^{2}}{4p_{3}(q,s)\Theta(\mu,\lambda,\gamma)} \notag\\ &&+\frac{\tau B_{0}[(E_{2}-E_{1})(c^{2}_{1}+d^{2}_{1})+2E_{1}(c_{2}+d_{2})](1+\gamma-\gamma\lambda)^{3}}{8\Phi(\mu,\lambda,\gamma,p_{2}(q,s),p_{3}(q,s))}. \end{eqnarray*} $

Thus, from Lemma 1.3 we see that

$ \begin{equation*} \mid a_{3}\mid\leq\frac{\mid\tau\mid(\vert B_{0}\vert+\vert B_{1}\vert) E_{1}(1+\gamma-\gamma\lambda)^{2}}{\mid p_{3}(q,s)\mid\Theta(\mu,\lambda,\gamma)}+\frac{\mid\tau B_{0}\mid (\vert E_{2}-E_{1}\vert+E_{1})(1+\gamma-\gamma\lambda)^{3}}{\vert\Phi(\mu,\lambda,\gamma,p_{2}(q,s),p_{3}(q,s))\vert}. \end{equation*} $

Next, we consider Fekete-Szegö problems for the class $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\mu,\lambda,\gamma;\phi) $.

Theorem 3.2 Let $ f(z) $ given by (1.1) belong to the class $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\mu,\lambda,\gamma;\phi) $ and $ \delta\in\mathbb{R} $. Then

$ \begin{equation*} \mid a_{3}-\delta a^{2}_{2}\mid\leq\frac{\vert B_{1}\tau\vert E_{1}(1+\gamma-\gamma\lambda)^{2}}{\vert p_{3}(q,s)\vert\Theta}+\frac{\vert1-\delta\vert B^{2}_{0}E^{3}_{1}\vert\tau\vert(1+\gamma-\gamma\lambda)^{4}}{\vert B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}\vert}, \end{equation*} $

if

$ \begin{equation*} \vert B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}\vert\leq\vert(1-\delta)B_{0}\tau p_{3}(q,s)\vert\Theta E_{1}^{2}(1+\gamma-\gamma\lambda)^{2}, \end{equation*} $

or

$ \begin{equation*} \mid a_{3}-\delta a^{2}_{2}\mid\leq\frac{(\vert B_{0}\vert+\vert B_{1}\vert)\vert\tau\vert E_{1}(1+\gamma-\gamma\lambda)^{2}}{\vert p_{3}(q,s)\vert\Theta}, \end{equation*} $

if

$ \begin{equation*} \vert B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}\vert\geq\vert(1-\delta)B_{0}\tau p_{3}(q,s)\vert\Theta E_{1}^{2}(1+\gamma-\gamma\lambda)^{2}, \end{equation*} $

where $ \Phi = \Phi(\mu,\lambda,\gamma,p_{2}(q,s),p_{3}(q,s)) $, $ \Theta = \Theta(\mu,\lambda,\gamma) $ and $ \Psi = \Psi(\mu,\lambda,\gamma) $ are the same as in Theorem 3.1.

Proof From (3.18), it follows that

$ \begin{equation*} a_{3}-a^{2}_{2} = \frac{B_{1}E_{1}\tau(c_{1}-d_{1})(1+\gamma-\gamma\lambda)^{2}}{4p_{3}(q,s)\Theta(\mu,\lambda,\gamma)}+\frac{B_{0}E_{1}\tau(c_{2}-d_{2})(1+\gamma-\gamma\lambda)^{2}}{4p_{3}(q,s)\Theta(\mu,\lambda,\gamma)}. \end{equation*} $

By (3.17) we easily obtain that

$ a_{3}-\delta a^{2}_{2} = \frac{B_{1}E_{1}\tau(c_{1}-d_{1})(1+\gamma-\gamma\lambda)^{2}}{4p_{3}(q,s)\Theta(\mu,\lambda,\gamma)} $

$ \begin{equation*} +\frac{B_{0}E_{1}\tau(1+\gamma-\gamma\lambda)^{2}[B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}+(1-\delta)B_{0}E_{1}^{2}\tau p_{3}(q,s)\Theta(1+\gamma-\gamma\lambda)^{2} ]c_{2}}{4p_{3}(q,s)\Theta(\mu,\lambda,\gamma)[B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}]} \end{equation*} $

$ \begin{equation*} +\frac{B_{0}E_{1}\tau(1+\gamma-\gamma\lambda)^{2}[B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}-(1-\delta)B_{0}E_{1}^{2}\tau p_{3}(q,s)\Theta(1+\gamma-\gamma\lambda)^{2} ]d_{2}}{4p_{3}(q,s)\Theta(\mu,\lambda,\gamma)[B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}]} \end{equation*} $

Hence, from Lemma 1.3, we imply that

$ \begin{equation*} \mid a_{3}-\delta a^{2}_{2}\mid\leq\frac{\vert B_{1}\tau\vert E_{1}(1+\gamma-\gamma\lambda)^{2}}{\vert p_{3}(q,s)\vert\Theta}+\frac{\vert1-\delta\vert B^{2}_{0}E^{3}_{1}\vert\tau\vert(1+\gamma-\gamma\lambda)^{4}}{\vert B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}\vert}, \end{equation*} $

when

$ \begin{equation*} \vert B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}\vert\leq\vert(1-\delta) B_{0}\tau p_{3}(q,s)\vert\Theta E_{1}^{2}(1+\gamma-\gamma\lambda)^{2}, \end{equation*} $

or

$ \begin{equation*} \mid a_{3}-\delta a^{2}_{2}\mid\leq\frac{(\vert B_{0}\vert+\vert B_{1}\vert)\vert\tau\vert E_{1}(1+\gamma-\gamma\lambda)^{2}}{\vert p_{3}(q,s)\vert\Theta}, \end{equation*} $

when

$ \begin{equation*} \vert B_{0}E^{2}_{1}\tau(1+\gamma-\gamma\lambda)\Phi+(E_{1}-E_{2})p^{2}_{2}(q,s)\Psi^{2}\vert\geq\vert(1-\delta)B_{0}\tau p_{3}(q,s)\vert\Theta E_{1}^{2}(1+\gamma-\gamma\lambda)^{2}. \end{equation*} $

Remark 3.3 In fact, from Theorems 3.1 and 3.2, we may consider the coefficient bound estimates and Fekete-Szegö problem for the classes $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\gamma,1,\gamma;\phi) $, $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}} $ $ (\tau,1,0,1;\phi) $, $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(1,\mu,0,0;\phi) $, $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(1,0,1,\gamma;\phi) $ and $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(1,0,\lambda,0;\phi) $, etc. Similarly, if we choose some suitable parameters $ \alpha_{k}(k = 1,\cdots,q) $, $ \beta_{j}(j = 1,\cdots,s) $, $ \tau, \mu, \lambda $ and $ \gamma $ without quasi-subordination (i.e., $ \varphi(z)\equiv1 $), we provide the following reduced versions for $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\mu,\lambda,\gamma;\phi) $ in Theorem 3.1.

(Ⅰ) $ \mathcal{QS_{\sum}}^{a,1}_{ a}(1,0,1,0;\phi) = \mathcal{S}_{\sum}(\phi) $, refer to Ma and Minda [52];

(Ⅱ) $ \mathcal{QS_{\sum}}^{a,1}_{ a}(1,0,\lambda,0;\phi) = \mathcal{G}^{\phi,\phi}_{\sum}(\gamma) $, refer to Magesh and Yamini [53];

(Ⅲ) $ \mathcal{QS_{\sum}}^{a,1}_{ a}(\tau,\mu,0,0;\phi) = \sum(\tau,\mu,\phi) $, refer to Srivastava and Bansal [35];

(Ⅳ) $ \mathcal{QS_{\sum}}^{a,1}_{ a}(\tau,\gamma,1,\gamma;\phi) = \mathcal{S}_{\sum}(\gamma,\tau;\phi) $, refer to Deniz [21];

(Ⅴ) $ \mathcal{QS_{\sum}}^{a,b}_{c}(\tau,\gamma,1,\gamma;\phi) \equiv\mathcal{S}^{a,b,c}_{\sum}(\tau,\gamma;\phi) $, $ \mathcal{QS_{\sum}}^{a,b}_{c}(\tau,\mu,0,0;\phi) \equiv\mathcal{K}^{a,b,c}_{\sum}(\tau,\mu;\phi) $ and $ \mathcal{QS_{\sum}}^{a,b}_{ c}(1,0,1,0;\phi) = \mathcal{J}_{\sum}(0,\phi) $, refer to refer to Patil and Naik [46];

(Ⅵ) $ \mathcal{QS_{\sum}}^{a,1}_{ a}(\tau,1,0,1;\phi) = \mathcal{S}_{\sum}(\tau,1,0,1;\phi) $ and $ \mathcal{QS_{\sum}}^{a,1}_{ a}(\tau,\gamma,\lambda,\gamma;\phi) = \mathcal{S}_{\sum}(\tau,\gamma,\lambda,\gamma;A,B) $, refer to srivastava et al. for Corollary 1 and Example 10 in [32], respectively. Here, the function $ \phi $ in the second equality is defined by

$ \begin{equation} \phi(z) = \frac{1+Az}{1+Bz}\; \; \; \; (-1\leq A<B\leq1). \end{equation} $

(3.19)

Remark 3.4 Let $ \varphi(z)\equiv1 $ and $ \phi(z) = \frac{1+(1-2\beta)z}{1-z} $ for $ 0\leq\beta<1 $. If we take some suitable parameters $ \tau, \mu, \lambda $ and $ \gamma $, we also have the following reduced versions for $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\mu,\lambda,\gamma;\phi) $ in Theorems 3.1 and 3.2. For example, $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,0,1,0;\beta) = {}_{q}\mathcal{S}_{s}(\tau,\beta) $, $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,1,1,1;\beta) = {}_{q}\mathcal{C}_{s}(\tau,\beta) $, refer to Al-Hawary et al. [47] for the classes of analytic and univalent (but not bi-univalent) functions.

Remark 3.5 In addition, if we only choose some suitable parameters $ \tau, \mu, \lambda $ and $ \gamma $, we give some reduced versions for $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\mu,\lambda,\gamma;\phi) $ without Dziok-Srivastava operator in Theorem 3.1. For example, $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(\tau,\gamma,0,0;\phi) = \mathcal{K}^{q}_{\gamma,\tau}(\phi) $$ (1\geq\gamma\geq0) $, $ \mathcal{QS_{\sum}}^{\alpha_{1},\cdots,\alpha_{q}}_{ \beta_{1},\cdots,\beta_{s}}(1,\alpha,1,0;\phi) = \mathcal{H}^{q}_{\alpha}(\phi) $$ (\alpha\geq0) $, refer to Goyal et al. [37].