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

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.

1 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 $|a_{3}|$.

2 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

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

3 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.