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Coefficient inequalities for a comprehensive class of bi-univalent functions related with bounded boundary variation

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Abstract

In the current work, we delineate a comprehensive class of bi-univalent functions connected with a bounded boundary variation to get the estimates of the first two Taylor–Maclaurin coefficients. In addition, certain special cases and some appealing interpretation of the results presented here are pointed out.

Introduction and definitions

Let \(\mathcal{A}\) be the class of functions f of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$
(1.1)

which are analytic in the open unit disc \(\Delta =\{z\in \mathbb{C}: \vert z \vert <1\}\) and normalized by the conditions \(f(0)=0\) and \(f^{\prime }(0)=1\). The Koebe one-quarter theorem [4] ensures that the image of Δ under every univalent function \(f\in \mathcal{A}\) contains the disc with the center in the origin and the radius \(1/4\). Thus, every univalent function \(f\in \mathcal{A}\) has an inverse \(f^{-1}:f(\Delta )\rightarrow \Delta \), satisfying \(f^{-1}(f(z))=z\), \(z\in \Delta \), and

$$ f \bigl(f^{-1}(w) \bigr)=w,\qquad \vert w \vert < r_{0}(f),\qquad r_{0}(f)\geq \frac{1}{4}. $$

In addition, it is straightforward to witness that the inverse function has the series expansion

$$ f^{-1}(w)=w-a_{2}w^{2}+ \bigl(2a_{2}^{2}-a_{3} \bigr)w^{3}- \bigl(5a _{2}^{3}-5a_{2}a_{3}+a_{4} \bigr)w^{4}+\cdots ,\quad w\in f(\Delta ). $$
(1.2)

A function \(f\in \mathcal{A}\) is said to be bi-univalent, if both f and \(f^{-1}\) are univalent in Δ, in the sense that \(f^{-1}\) has a univalent analytic continuation to Δ and we denote by Σ this class of bi-univalent functions. Actually, the study of the Taylor–Maclaurin coefficient inequalities for various classes of bi-univalent functions was recently revived by Srivastava et al. [16]. The huge flood of papers (for example) [1, 3, 5, 6, 9, 10, 12,13,14,15, 17,18,19] which emerged essentially from the pioneering work of Srivastava et al. [16]. One could refer [16], the above-mentioned work and the references therein for history, examples and different classes and its subclasses of bi-univalent functions. Recently, Çağlar et al. [2] found the upper bounds for the second Hankel determinant for certain subclasses of analytic and bi-univalent functions and Srivastava et al. [11] used the Faber polynomial expansions to address a new subclass of Σ and obtained bounds for their nth (\(n \geq 3\)) coefficients subject to a given gap series condition.

Definition 1.1

([7])

Let \(\mathcal{P}_{k}(\alpha )\), with \(k\geq 2\) and \(0\leq \alpha <1\), denote the class of univalent analytic functions P, normalized with \(P(0)=1\), and satisfying

$$ \int _{0}^{2\pi } \biggl\vert \frac{\operatorname{Re}P(z)-\alpha }{1-\alpha } \biggr\vert \,\operatorname{d}\theta \leq k\pi , $$

where \(z=re^{i\theta }\in \Delta \).

For \(\alpha =0\), we denote \(\mathcal{P}_{k}:=\mathcal{P}_{k}(0)\), hence the class \(\mathcal{P}_{k}\) corresponds to the class of functions p analytic in Δ, normalized with \(p(0)=1\), and having the expression

$$ p(z)= \int _{0}^{2\pi }\frac{1-ze^{it}}{1+ze^{it}}\,\operatorname{d}\mu (t), $$
(1.3)

where λ is a real-valued function with bounded variation, which ensures

$$ \int _{0}^{2\pi }d\mu (t)=2\pi \quad \text{and}\quad \int _{0}^{2\pi } \bigl\vert d\mu (t) \bigr\vert \leq k,\quad k\geq 2. $$
(1.4)

Obviously, \(\mathcal{P}:=\mathcal{P}_{2}\) is the celebrated class of Carathéodory functions, that is, the normalized functions with positive real part in the open unit disc Δ.

Definition 1.2

A function \(f\in \varSigma \) of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$

belongs to the class \(\mathcal{B}_{\varSigma }^{\lambda , \eta , \delta }(k; \alpha )\), \(\lambda \geq 0\), \(\delta \geq 1\), \(\eta \geq 0\), \(k\geq 2\) and \(0\leq \alpha <1\), if the subsequent conditions are fulfilled:

$$ ( 1-\delta ) \biggl(\frac{f(z)}{z} \biggr)^{\lambda }+ \delta f^{\prime }(z) \biggl(\frac{f(z)}{z} \biggr)^{\lambda -1}+ \nu \eta z f^{\prime \prime }(z) \in \mathcal{P}_{k}(\alpha ),\quad z \in \Delta , $$
(1.5)

and

$$ ( 1-\delta ) \biggl(\frac{g(w)}{w} \biggr)^{\lambda }+ \delta g^{\prime }(w) \biggl(\frac{g(w)}{w} \biggr)^{\lambda -1}+ \nu \eta w g^{\prime \prime }(w) \in \mathcal{P}_{k}(\alpha ),\quad w \in \Delta , $$
(1.6)

where the function \(g(w)=f^{-1}(w)\) is defined by (1.2) and \(\nu =\frac{2\delta +\lambda }{2\delta +1}\).

It is remarkable that the particular values of λ, δ, η, α and m direct the class \(\mathcal{B}_{\varSigma } ^{\lambda , \eta , \delta }(k; \alpha )\) to different subclasses, we exhibit the following subclasses:

(1) For \(\eta =0\), we obtain the class \(\mathcal{B}_{\varSigma }^{1, 0, \delta }(k; \alpha )\equiv \mathcal{N}_{\varSigma }^{\lambda , \delta }(k; \alpha )\). A function \(f\in \varSigma \) of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$

is said to be in \(\mathcal{N}_{\varSigma }^{\lambda , \delta }(k; \alpha )\), if

$$ ( 1-\delta ) \biggl(\frac{f(z)}{z} \biggr)^{\lambda }+ \delta f^{\prime }(z) \biggl(\frac{f(z)}{z} \biggr)^{\lambda -1}\in \mathcal{P}_{k}(\alpha ),\quad z\in \Delta , $$

and for \(g(w)=f^{-1}(w)\)

$$ ( 1-\delta ) \biggl(\frac{g(w)}{w} \biggr)^{\lambda }+ \delta g^{\prime }(w) \biggl(\frac{g(w)}{w} \biggr)^{\lambda -1}\in \mathcal{P}_{k}(\alpha ),\quad w\in \Delta , $$

holds.

Remark 1.3

For \(k=2\), the class \(\mathcal{N}_{\varSigma }^{\lambda , \delta }(2; \alpha )\equiv \mathcal{N}_{\varSigma }^{\lambda , \delta }(\alpha )\) was considered by Çağlar et al. [3].

(2) For \(\delta =1\) and \(\eta =0\), we observe the class \(\mathcal{B}_{ \varSigma }^{\lambda , 0, 1 }(k; \alpha )\equiv \mathcal{R}_{ \varSigma }^{\lambda }(k; \alpha )\). A function \(f\in \varSigma \) of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$

is said to be in \(\mathcal{R}_{\varSigma }^{\lambda }(k; \alpha )\), if

$$ f^{\prime }(z) \biggl(\frac{f(z)}{z} \biggr)^{\lambda -1}\in \mathcal{P}_{k}(\alpha ),\quad z\in \Delta , $$

and for \(g(w)=f^{-1}(w)\)

$$ g^{\prime }(w) \biggl(\frac{g(w)}{w} \biggr)^{\lambda -1}\in \mathcal{P}_{k}(\alpha ),\quad w\in \Delta , $$

holds.

Remark 1.4

For \(k=2\), the class \(\mathcal{R}_{\varSigma }^{\lambda }(2; \alpha ) \equiv \mathcal{R}_{\varSigma }^{\lambda }(\alpha )\) was considered in [8].

(3) For \(\lambda =0\); \(\delta =1\) and \(\eta =0\), we have \(\mathcal{B}_{ \varSigma }^{1, 0, 1 }(k; \alpha )\equiv \mathcal{S}_{\varSigma } ^{*}(k; \alpha )\). A function \(f\in \varSigma \) of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$

is said to be in \(\mathcal{S}_{\varSigma }^{*}(k; \alpha )\), if

$$ \frac{zf^{\prime }(z)}{f(z)}\in \mathcal{P}_{k}(\alpha ),\quad z\in \Delta , $$

and for \(g(w)=f^{-1}(w)\)

$$ \frac{wg^{\prime }(w)}{g(w)}\in \mathcal{P}_{k}(\alpha ),\quad w\in \Delta , $$

holds.

Remark 1.5

For \(k=2\), we attain the class \(\mathcal{S}_{\varSigma }^{*}(2; \alpha )\equiv \mathcal{S}_{\varSigma }^{*}(\alpha )\).

(4) For \(\lambda =1\), we have the class \(\mathcal{B}_{\varSigma }^{1, \eta , \delta }(k; \alpha )\equiv \mathcal{B}_{\varSigma }^{\eta , \delta }(k; \alpha )\). A function \(f\in \varSigma \) of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$

is said to be in \(\mathcal{B}_{\varSigma }^{\eta , \delta }(k; \alpha )\), if

$$ (1-\delta )\frac{f(z)}{z} + \delta f^{\prime }(z)+\eta zf^{\prime \prime }(z)\in \mathcal{P}_{k}(\alpha ),\quad z\in \Delta , $$

and for \(g(w)=f^{-1}(w)\)

$$ (1-\delta )\frac{g(w)}{w} + \delta g^{\prime }(w)+\eta wg^{\prime \prime }(w)\in \mathcal{P}_{k}(\alpha ),\quad w\in \Delta , $$

holds.

(5) For \(\delta =\lambda =1\), we obtain the class \(\mathcal{B}_{\varSigma } ^{1, \eta , 1 }(k; \alpha )\equiv \mathcal{F}_{\varSigma }(\eta , k;\alpha )\). A function \(f\in \varSigma \) of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$

is said to be in \(\mathcal{F}_{\varSigma }(\eta , k;\alpha )\), if

$$ f^{\prime }(z)+\eta zf^{\prime \prime }(z)\in \mathcal{P}_{k}( \alpha ),\quad z\in \Delta , $$

and for \(g(w)=f^{-1}(w)\)

$$ g^{\prime }(w)+\eta wg^{\prime \prime }(w)\in \mathcal{P}_{k}( \alpha ),\quad w\in \Delta , $$

holds.

(6) For \(\lambda =1\) and \(\eta =0\), we obtain the class \(\mathcal{B}_{ \varSigma }^{1, 0, \delta }(k; \alpha ),\equiv \mathcal{B}_{ \varSigma }(\delta , k;\alpha )\). A function \(f\in \varSigma \) of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$

is said to be in \(\mathcal{B}_{\varSigma }(\delta , m;\alpha )\), if

$$ (1-\delta )\frac{f(z)}{z}+\delta f^{\prime }(z)\in \mathcal{P}_{k}( \alpha ),\quad z\in \Delta , $$

and for \(g(w)=f^{-1}(w)\)

$$ (1-\delta )\frac{g(w)}{w}+\delta g^{\prime }(w)\in \mathcal{P}_{k}( \alpha ),\quad w\in \Delta , $$

holds.

Remark 1.6

For \(k=2\), the class \(\mathcal{B}_{\varSigma }(\delta , 2;\alpha )\equiv \mathcal{B}_{\varSigma }(\delta ;\alpha )\) was considered by Frasin and Aouf [5].

(7) For \(\delta =1\), \(\lambda =1\) and \(\eta = 0\), we have the class \(\mathcal{B}_{\varSigma }^{1, 0, 1 }(k; \alpha )\equiv \mathcal{P}_{\varSigma }(k;\alpha )\). A function \(f\in \varSigma \) of the form

$$ f(z) = z + \sum_{n=2}^{\infty } a_{n} z^{n} $$

is said to be in \(\mathcal{P}_{\varSigma }(m;\alpha )\), if

$$ f^{\prime }(z) \in \mathcal{P}_{k}(\alpha ),\quad z\in \Delta , $$

and for \(g(w)=f^{-1}(w)\)

$$ g^{\prime }(w) \in \mathcal{P}_{k}(\alpha ),\quad w\in \Delta , $$

holds.

Remark 1.7

For \(k=2\), the class \(\mathcal{P}_{\varSigma }(2;\alpha )\equiv \mathcal{P}_{\varSigma }(\alpha )\) was introduced and studied by Srivastava et al. [16].

To prove the results discussed in this article, we need the following lemma.

Lemma 1.8

Let the function \(\varPhi (z)=1+\sum_{n=1}^{\infty }{h_{n}}{z^{n}} \), \(z\in \Delta \), such that \(\varPhi \in \mathcal{P}_{m}(\alpha )\). Then

$$ \vert h_{n} \vert \leq k(1-\alpha ),\quad n\geq 1. $$

In this study, we stumble on the estimates for the coefficients \(|a_{2}|\) and \(|a_{3}|\) for functions in the subclass \(\mathcal{B} _{\varSigma }^{\lambda , \eta , \delta }(k; \alpha )\). Also, we attain the upper bounds of the Fekete–Szegö inequality by means of the results of \(|a_{2}|\) and \(|a_{3}|\).

Main results

In the subsequent theorem, we find the coefficient estimates for functions in \(\mathcal{B}_{\varSigma }^{\lambda , \eta , \delta }(k; \alpha )\).

Theorem 2.1

Let \(f(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}\) be in the class \(\mathcal{B}_{\varSigma }^{\lambda , \eta , \delta }(k; \alpha )\). Then

$$\begin{aligned}& \vert a_{2} \vert \leq \min \biggl\{ \sqrt{\frac{2k(1-\alpha )}{(2 \delta + \lambda )(\lambda +1)+12\nu \eta }}; \frac{k(1-\alpha )}{ \delta +\lambda +2\nu \eta } \biggr\} , \\& \vert a_{3} \vert \leq \min \biggl\{ \frac{k(1-\alpha )}{2\delta +\lambda +6\nu \eta } + \frac{2k(1-\alpha )}{(2\delta +\lambda )( \lambda +1)+12\nu \eta }, \frac{k(1-\alpha )}{2\delta +\lambda +6 \nu \eta } + \frac{k^{2}(1-\alpha )^{2}}{ (\delta +\lambda +2 \nu \eta )^{2}} \biggr\} , \end{aligned}$$

and

$$\begin{aligned} \bigl\vert a_{3}-\mu a_{2}^{2} \bigr\vert &\leq \frac{k(1-\alpha )}{2\delta +\lambda +6\nu \eta }, \end{aligned}$$

where

$$ \mu = \frac{(2\delta +\lambda )(\lambda +3)+24\nu \eta }{2(2\delta + \lambda +6\nu \eta )}. $$

Proof

Since \(f\in \mathcal{B}_{\varSigma }^{\lambda , \eta , \delta }(k; \alpha )\), from Definition 1.2 we have

$$ ( 1-\delta ) \biggl(\frac{f(z)}{z} \biggr)^{\lambda }+ \delta f^{\prime }(z) \biggl(\frac{f(z)}{z} \biggr)^{\lambda -1}+ \nu \eta z f^{\prime \prime }(z) = p(z) $$
(2.1)

and

$$ ( 1-\delta ) \biggl(\frac{g(w)}{w} \biggr)^{\lambda }+ \delta g^{\prime }(w) \biggl(\frac{g(w)}{w} \biggr)^{\lambda -1}+ \nu \eta w g^{\prime \prime }(w)=q(w), $$
(2.2)

where \(p, q\in \mathcal{P}_{m}(\alpha )\) and \(g=f^{-1}\). Using the fact that the functions p and q have the following Taylor expansions:

$$\begin{aligned}& p(z)=1+p_{1}z+p_{2}z^{2}+p_{3}z^{3}+ \cdots ,\quad z\in \Delta , \end{aligned}$$
(2.3)
$$\begin{aligned}& q(w)=1+q_{1}w+q_{2}w^{2}+q_{3}w^{3}+ \cdots ,\quad w\in \Delta , \end{aligned}$$
(2.4)

and equating the coefficients in (2.1) and (2.2), from (1.2) we obtain

$$\begin{aligned}& (\delta +\lambda + 2\nu \eta )a_{2} = p_{1}, \end{aligned}$$
(2.5)
$$\begin{aligned}& (2\delta +\lambda ) \biggl[ \biggl(\frac{\lambda -1}{2} \biggr)a_{2} ^{2}+ \biggl(1+\frac{6\eta }{2\delta +1} \biggr)a_{3} \biggr] = p_{2}, \end{aligned}$$
(2.6)
$$\begin{aligned}& - (\delta +\lambda +2\nu \eta )a_{2} = q_{1}, \end{aligned}$$
(2.7)
$$\begin{aligned}& (2\delta +\lambda ) \biggl[ \biggl(\frac{\lambda +3}{2}+\frac{12\eta }{2 \delta +1} \biggr)a_{2}^{2}- \biggl(1+\frac{6\eta }{2\delta +1} \biggr)a _{3} \biggr] = q_{2}. \end{aligned}$$
(2.8)

In view of the fact that \(p, q\in \mathcal{P}_{m}(\alpha )\) and Lemma 1.8, the following inequalities hold:

$$\begin{aligned} \vert p_{k} \vert \leq k(1-\alpha ), \qquad \vert q_{k} \vert \leq k(1-\alpha ), \quad k\geq 1. \end{aligned}$$
(2.9)

It follows from (2.6) and (2.8), additionally, by means of the inequalities (2.9), that

$$ \vert a_{2} \vert \leq \sqrt{ \frac{2k(1-\alpha )}{(2\delta + \lambda )(\lambda +1)+12\nu \eta }}. $$
(2.10)

From (2.5) and (2.7), we have

$$ p_{1}=-q_{1} $$

and

$$\begin{aligned}& a_{2}^{2} = \frac{p_{1}^{2}}{ (\delta +\lambda +2\nu \eta ) ^{2}}, \end{aligned}$$
(2.11)

which, by applying (2.9), shows

$$\begin{aligned}& \vert a_{2} \vert \leq \frac{k(1-\alpha )}{\delta +\lambda +2\nu \eta }. \end{aligned}$$

Next, combining the above inequality with (2.10), the first inequality of the conclusion is proved.

On the other hand, by subtracting (2.8) from (2.6), we have

$$\begin{aligned} a_{3} = \frac{p_{2}-q_{2}}{2 (2\delta +\lambda +6\nu \eta )} + a _{2}^{2}. \end{aligned}$$
(2.12)

By using (2.10) in (2.12), we show

$$\begin{aligned}& \vert a_{3} \vert \leq \frac{k(1-\alpha )}{2\delta +\lambda +6\nu \eta } + \frac{2k(1- \alpha )}{(2\delta +\lambda )(\lambda +1)+12\nu \eta } \end{aligned}$$

and using (2.11) in (2.12), we get

$$\begin{aligned}& \vert a_{3} \vert \leq \frac{k(1-\alpha )}{2\delta +\lambda +6\nu \eta } + \frac{k ^{2}(1-\alpha )^{2}}{ (\delta +\lambda +2\nu \eta )^{2}}. \end{aligned}$$

From (2.8), we have

$$\begin{aligned} \frac{(2\delta +\lambda )(\lambda +3)+24\nu \eta }{2(2\delta +\lambda +6\nu \eta )} a_{2}^{2} - a_{3} = \frac{q_{2}}{2\delta +\lambda +6 \nu \eta }. \end{aligned}$$

Furthermore, using (2.9), we finally deduce

$$\begin{aligned}& \bigl\vert a_{3}-\mu a_{2}^{2} \bigr\vert \leq \frac{ \vert q_{2} \vert }{2 \delta +\lambda +6\nu \eta } \leq \frac{k(1-\alpha )}{2\delta + \lambda +6\nu \eta }, \end{aligned}$$

where

$$ \mu = \frac{(2\delta +\lambda )(\lambda +3)+24\nu \eta }{2(2\delta + \lambda +6\nu \eta )}, $$

which completes our proof. □

Remark 2.2

For \(k=2\), the results obtained in Theorem 2.1 improves the results of Yousef et al. [20, Theorem 4.1].

Corollary 2.3

Let \(f(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}\) be in the class \(\mathcal{B}_{\varSigma }^{\lambda , \eta , \delta }(\alpha )\). Then

$$\begin{aligned}& \vert a_{2} \vert \leq \min \biggl\{ \sqrt{\frac{4(1-\alpha )}{(2 \delta + \lambda )(\lambda +1)+12\nu \eta }}; \frac{2(1-\alpha )}{ \delta +\lambda +2\nu \eta } \biggr\} , \\& \vert a_{3} \vert \leq \min \biggl\{ \frac{2(1-\alpha )}{2\delta +\lambda +6\nu \eta } + \frac{4(1-\alpha )}{(2\delta +\lambda )( \lambda +1)+12\nu \eta }, \frac{2(1-\alpha )}{2\delta +\lambda +6 \nu \eta } + \frac{4(1-\alpha )^{2}}{ (\delta +\lambda +2\nu \eta )^{2}} \biggr\} , \end{aligned}$$

and

$$\begin{aligned} \biggl\vert a_{3}-\frac{(2\delta +\lambda )(\lambda +3)+24\nu \eta }{2(2 \delta +\lambda +6\nu \eta )} a_{2}^{2} \biggr\vert &\leq \frac{2(1- \alpha )}{2\delta +\lambda +6\nu \eta }. \end{aligned}$$

Corollary 2.4

Let \(f(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}\) be in the class \(\mathcal{N}_{\varSigma }^{\lambda , \delta }(k; \alpha )\). Then

$$\begin{aligned}& \vert a_{2} \vert \leq \min \biggl\{ \sqrt{\frac{2k(1-\alpha )}{(2 \delta + \lambda )(\lambda +1)}}; \frac{k(1-\alpha )}{\delta + \lambda } \biggr\} , \\& \vert a_{3} \vert \leq \min \biggl\{ \frac{k(1-\alpha )}{2\delta +\lambda } + \frac{2k(1-\alpha )}{(2\delta +\lambda )(\lambda +1)}, \frac{k(1-\alpha )}{2\delta +\lambda } + \frac{k^{2}(1-\alpha )^{2}}{ (\delta +\lambda )^{2}} \biggr\} , \end{aligned}$$

and

$$\begin{aligned} \biggl\vert a_{3}-\frac{\lambda +3}{2} a_{2}^{2} \biggr\vert &\leq \frac{k(1- \alpha )}{2\delta +\lambda }. \end{aligned}$$

Concluding remarks and observations

In this paper, we investigate the estimates of second and third Taylor–Maclaurin coefficients for a comprehensive class \(\mathcal{B} _{\varSigma }^{\lambda , \eta , \delta }(k; \alpha )\) of bi-univalent functions. Also, the corresponding coefficient estimates for functions in the subclasses \(\mathcal{R}_{\varSigma }^{\lambda }(k; \alpha )\), \(\mathcal{S}_{\varSigma }^{*}(k; \alpha )\), \(\mathcal{B} _{\varSigma }^{\eta , \delta }(k; \alpha )\), \(\mathcal{F}_{\varSigma }(\eta , k;\alpha )\), \(\mathcal{B}_{\varSigma }(\delta , k;\alpha )\) and \(\mathcal{P}_{\varSigma }(k;\alpha )\) as mentioned above can be derived easily and so we omit the details. Also, some interesting remarks on the results presented here are given.

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Acknowledgements

The authors wish to express their sincere appreciation to the editor and referees for valuable comments and suggestions.

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Funding

This work was supported by the Natural Science Foundation of the People’s Republic of China (Grant No. 11561001), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant No. NJYT-18-A14), the Natural Science Foundation of Inner Mongolia of the People’s Republic of China (Grant No. 2018MS01026), the Higher School Foundation of Inner Mongolia of the People’s Republic of China (Grant Nos. NJZY17300, NJZY17301 and NJZY18217) and the Natural Science Foundation of Chifeng of Inner Mongolia.

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All authors equally worked on the results and they read and approved the final manuscript.

Correspondence to Huo Tang.

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MSC

  • 30C45
  • 30C50

Keywords

  • Univalent functions
  • Bi-univalent functions
  • Bounded boundary rotation
  • Fekete–Szegö inequality