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Bi-univalent properties for certain class of Bazilevič functions defined by convolution and with bounded boundary rotation

  • Mohamed K. AoufEmail author
  • Samar M. Madian
  • Adela O. Mostafa
Open Access
Original research
  • 84 Downloads

Abstract

In this paper, we obtain bi-univalent properties for certain class of Bazilevič functions defined by convolution and with bounded boundary rotation. We will find coefficient bounds for |a2| and |a3| for the class \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\mathsf {.}\)

Keywords

Bi-univalent Bazilevič functions Hadamard product Bounded boundary rotation 

2010 Mathematics Subject Classification

30C45 30C50 

Introduction

Let \(\mathcal {A}\) denote the class of analytic functions of the form:
$$ f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}\ \ (z\in\mathbb{U}:\mathbb{U} =\{z\in\mathbb{C}:\left\vert z\right\vert <1\}). $$
(1)
For \(h(z)\in \mathcal {A}\), given by \(h(z)=z+\sum \limits _{n=2}^{\infty }h_{n}z^{n},\) the Hadamard product (or convolution) of f(z) and h(z) is defined by:
$$ (f * h)(z)=z+\sum_{n=2}^{\infty} a_{n} h_{n} z^{n}=(h\times f)(z). $$
(2)

Definition 1

([1, 2], and [3] with p = 1). Let \(\mathcal {P}_{k}^{\lambda }(\rho)\, \left (0\leq \rho <1,\ k\geq 2 \text { and } \left \vert \lambda \right \vert <\frac {\pi }{2}\right)\) denote the class of functions \(p(z)=1+\sum \limits _{n=1}^{\infty }c_{n}z^{n},\) which are analytic in \(\mathbb {U}\) and satisfy the conditions:
$$(i)\ p(0)=1, $$
$$ (ii)\int\limits_{0}^{2\pi}\left\vert \frac{\mathfrak{R}\left\{ e^{i\lambda }p(z)\right\} -\rho\cos\lambda}{1-\rho}\right\vert \leq k\pi\cos \lambda\ \left(r<1,z=re^{i\theta}\in\mathbb{U}\right). $$
(3)

We note that:

(i) \(\mathcal {P}_{k}^{\lambda }(0)=\mathcal {P}_{k}^{\lambda }\ (\ k\geq 2\ \)and \(\left \vert \lambda \right \vert <\frac {\pi }{2})\ \) is the class of functions introduced by Robertson (see [4]), and he derived a variational formula for functions in this class.

(ii) \(\mathcal {P}_{k}^{0}(\rho)=\mathcal {P}_{k}(\rho)\ (0\leq \rho <1,\ k\geq 2)\ \)is the class of functions introduced by Padmanabhan and Parvatham [5] (see also Umarani and Aouf [6]).

(iii) \(\mathcal {P}_{k}^{0}(0)=\mathcal {P}_{k}(k\geq 2)\ \) is the class of functions having their real parts bounded in the mean on \(\mathbb {U}\), introduced by Robertson [4] and studied by Pinchuk [7].

(iv) \(\mathcal {P}_{2}^{0}(\rho)=\mathcal {P}\left (\rho \right)\ (0\leq \rho <1)\ \)is the class of functions with positive real part of order ρ, 0≤ρ<1.

(v) \(\mathcal {P}_{2}^{0}(0)=\mathcal {P}\) is the class of functions having positive real part for \(z\in \mathbb {U}\).

By the Koebe one-quarter theorem [8], we know that the image of \(\mathbb {U\ }\)under every univalent function \(f\in \mathcal {A}\) contains the disk with center in the origin and radius 1/4. Therefore, every univalent function f has an inverse f−1 satisfies:
$$ f^{-1}(f(z))=z\ (z\in\mathbb{U})\ \text{and}\ f(f^{-1}(w))=w\ (|w|< r_{0} (f),\ r_{0}(f)\geq1/4). $$
(4)
It is easy to see that the inverse function has the form:
$$ f^{-1}(w)=w-a_{2}w^{2}+\left(2a_{2}^{2}-a_{3}\right)w^{3}-\left(5a_{2}^{3}-5a_{2}a_{3} +a_{4}\right)w^{4}+....\ \ \ . $$
(5)

A function \(f\in \mathcal {A}\) is said to be bi-univalent in \(\mathbb {U}\) if both f and its inverse map g=f−1are univalent in \(\mathbb {U}\).

Let \(\sum \) denote the class of bi-univalent functions in \(\mathbb {U}\) in the form (1). For interesting examples about the class \(\sum \), see [9].

The object of this paper is to introduce new subclass of Bazilevič functions [10] for the class \(\sum \) with bounded boundary rotation and defined by using convolution as follows:

Definition 2

Let \(f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast },\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2\ {and}\ \left \vert \lambda \right \vert <\frac {\pi }{2},\ \)then \((f * h)(z)\in \sum \) is said to be in the class \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\ \)if it satisfies the following conditions:
$$ \left\{ (1-\alpha)\left(\frac{(f * h)(z)}{z}\right)^{\beta}+\alpha \frac{z(f * h)^{\prime}(z)}{(f * h)(z)}\left(\frac{(f * h)(z)} {z}\right)^{\beta}\right\} \in\mathcal{P}_{k}^{\lambda}(\rho)\ (z\in \mathbb{U)} $$
(6)
and
$$ {} \left\{ (1-\alpha)\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta} \,+\,\alpha\frac{w((f * h)^{-1}(w))^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\right\} \in\mathcal{P}_{k}^{\lambda}(\rho)\ (w\in\mathbb{U)}. $$
(7)

We note that by putting different values for h, α, β, k, λ, and ρ, in the above definition, we have:

(1) \(\mathcal {M}_{1,0,\rho,k,\beta }\left (f\times \frac {z}{1-z}\right)=R_{\sum } (\rho,k,\beta)\ (f\in \sum,\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2)\ \)(see [11], with γ=1);

(2) \(\mathcal {M}_{\alpha,0,\rho,k,1}(f * h)=\mathcal {L} _{\alpha,\rho,k}(f * h)\ \left (\ f,h\in \sum,\ \alpha \in \mathbb {C} ^{\ast },\ 0\leq \rho <1,\ k\geq 2\right)\ \)(see [12]);

(3) \(\mathcal {M}_{\eta,0,\rho,2,1}(f * h)=\mathcal {L}_{\eta,\rho }(f * h)\ \left (\ f,h\in \sum,\ \eta \geq 0,\ 0\leq \rho <1\right)\ \)(see [13] and [14]);

(4) \(\mathcal {M}_{\eta,0,\rho,2,1}\left (f\times \frac {z}{1-z}\right)=\mathcal {L}_{\eta,\rho }(f)(z)\ \left (\ f\in \sum,\ \eta \geq 0,\ 0\leq \rho <1\right)\ \)(see [15]);

(5) \(\mathcal {M}_{1,0,\rho,2,\beta }\left (f\times \frac {z}{1-z}\right)=\mathcal {L} _{\rho,\beta }(f)(z)\left (\ f\in \sum,\ \beta \geq 0,\ 0\leq \rho <1\right)\ \)(see [16]);

(6) \(\mathcal {M}_{1,0,\rho,2,1}\left (f\times \frac {z}{1-z}\right)=\mathcal {L}_{\rho }(f)(z)\left (\ f\in \sum,\ 0\leq \rho <1\right)\ \)(see [9]);

(7) \(\mathcal {M}_{\alpha,0,\rho,2,\beta }\left (f\times \frac {z}{1-z} \right)=\mathcal {NP}_{\sum }^{\beta,\alpha }(0,\rho)\ \left (f\in \sum,\ \beta,\alpha \geq 0,\ 0\leq \rho <1\right)\ \)(see [[17], with β=0]);

(8) \(\mathcal {M}_{1,0,\rho,2,\beta }\left (f\times \frac {z}{1-z}\right)=\mathcal {R}_{\sum }(\beta,\rho)\ \left (\ f\in \sum,\ \beta \geq 0,\ 0\leq \rho <1\ \right)\ \)(see [18]).

Also, we can obtain the following subclasses:

(i) \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }\left (f\times \frac {z} {1-z\ }\right)=\mathcal {\digamma }_{\alpha,\lambda,\rho,k,\beta }(f)\)
$$\begin{array}{*{20}l} & =\left\{ f\in\sum:(1-\alpha)\left(\frac{f(z)}{z}\right)^{\beta} +\alpha\frac{zf^{\prime}(z)}{f(z)}\left(\frac{f(z)}{z}\right)^{\beta} \in\mathcal{P}_{k}^{\lambda}(\rho)\right. \\ & \text{and }\left. (1-\alpha)\left(\frac{f^{-1}(w)}{w}\right)^{\beta }+\alpha\frac{w\left((f^{-1}(w)\right)^{\prime}}{f^{-1}(w)}\left(\frac{f^{-1}(w)} {w}\right) ^{\beta}\in\mathcal{P}_{k}^{\lambda}(\rho)\right\} ; \end{array} $$
(ii) \(\mathcal {M}_{\alpha,0,\rho,k,\beta }(f\ast h)=\mathcal {F}_{\alpha,\rho,k,\beta }(f * h)\)
$$\begin{array}{*{20}l} & =\left\{ f,h\in\sum:(1-\alpha)\left(\frac{(f * h)(z)}{z}\right)^{\beta}+\alpha\frac{z(f * h)^{\prime}(z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right)^{\beta}\in\mathcal{P}_{k}(\rho)\right. \\ & \text{and }\left. (1-\alpha)\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}+\alpha\frac{w((f * h)^{-1}(w))^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right) ^{\beta}\in\mathcal{P}_{k}(\rho)\right\} ; \end{array} $$
(iii) \(\mathcal {M}_{\alpha,0,\rho,2,\beta }(f\ast h)=\mathcal {F}_{\alpha,\rho,\beta }(f * h)\)
$$\begin{array}{*{20}l} & =\left\{ f,h\in\sum:\mathfrak{R}\left[ (1-\alpha)\left(\frac{(f * h)(z)} {z}\right)^{\beta}+\alpha\frac{z(f * h)^{\prime}(z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right)^{\beta}\right] >\rho\right. \\ & \text{and }\left. \mathfrak{R}\left[ (1-\alpha)\left(\frac{(f * h)^{-1}(w)} {w}\right)^{\beta}\,+\,\alpha\frac{w\left((f * h)^{-1}(w)\right)^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\right] >\rho\right\} ; \end{array} $$
(iv) \(\mathcal {M}_{\alpha,\lambda,0,k,\beta }(f * h)=\mathcal {M}_{\alpha,\lambda,k,\beta }(f * h)\)
$$\begin{array}{*{20}l} & =\left\{ f,h\in\sum:(1-\alpha)\left(\frac{(f * h)(z)}{z}\right)^{\beta}+\alpha\frac{z(f * h)^{\prime}(z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right)^{\beta}\in\mathcal{P}_{k}^{\lambda}\right. \\ & \text{and }\left. (1-\alpha)\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}+\alpha\frac{w\left((f * h)^{-1}(w)\right)^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\in\mathcal{P}_{k}^{\lambda }\right\} ; \end{array} $$
(v) \(\mathcal {M}_{\alpha,0,0,k,\beta }(f * h)=\mathcal {M}_{\alpha,k,\beta }(f * h)\)
$$\begin{array}{*{20}l} & =\left\{ f,h\in\sum:(1-\alpha)\left(\frac{(f * h)(z)}{z}\right)^{\beta}+\alpha\frac{z(f * h)^{\prime}(z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right)^{\beta}\in\mathcal{P}_{k}\right. \\ & \text{and }\left. (1-\alpha)\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}+\alpha\frac{w\left((f * h)^{-1}(w)\right)^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\in\mathcal{P}_{k}\right\} ; \end{array} $$
(vi) \(\mathcal {M}_{\alpha,0,0,2,\beta }(f * h)=\mathcal {M}_{\alpha,\beta }(f * h)\)
$$\begin{array}{*{20}l} & =\left\{ f,h\in\sum:(1-\alpha)\left(\frac{(f * h)(z)}{z}\right)^{\beta}+\alpha\frac{z(f * h)^{\prime}(z)}{(f\ast h)(z)}\left(\frac{(f * h)(z)}{z}\right)^{\beta}\in\mathcal{P}\right. \\ & \text{and }\left. (1-\alpha)\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}+\alpha\frac{w((f * h)^{-1}(w))^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\in\mathcal{P}\right\} ; \end{array} $$
(vii) \( \mathcal {M}_{1,\lambda,\rho,k,\beta }(f * h)=\mathbb {F}_{\lambda,\rho,k,\beta }(f * h)\)
$${\begin{aligned} &=\left\{ f,h\in\sum:\frac{z(f * h)^{\prime}(z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right) ^{\beta}\in\mathcal{P}_{k}^{\lambda} (\rho)\right. \text{and}\\ &\quad \left. \frac{w((f * h)^{-1}(w))^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\in \mathcal{P}_{k}^{\lambda}(\rho)\right\} ; \end{aligned}} $$
or
$$\begin{array}{*{20}l} & =\left\{ f\in\sum:\frac{e^{i\lambda}\left[ \frac{z(f * h)^{\prime} (z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right)^{\beta}\right] -\rho\cos\lambda-i\sin\lambda}{\left(1-\rho\right) \cos\lambda} \in\mathcal{P}_{k}\right. \\ & \text{and\ }\left. \frac{e^{i\lambda}\left[ \frac{z(f * h)^{\prime} (z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right)^{\beta}\right] -\rho\cos\lambda-i\sin\lambda}{\left(1-\rho\right) \cos\lambda} \in\mathcal{P}_{k}\right\} ; \end{array} $$
(viii) \(\mathcal {M}_{1,0,\rho,2,\beta }(f * h)=\mathbb {F}_{\rho,\beta }(f * h)\)
$$=\left\{ f,h\in\sum:\mathfrak{R}\left[ \frac{z(f * h)^{\prime}(z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right) ^{\beta}\right] >\rho \right.\\ \left. and\ \mathfrak{R}\left[ \frac{w((f * h)^{-1}(w))^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\right] >\rho\right\}. $$

In order to obtain our main results, we have to recall here the following lemma.

Lemma 1

([3] with p = 1). If \(p(z)=1+\sum \limits _{n=1}^{\infty }c_{n}z^{n}\in \mathcal {P}_{k}^{\lambda }(\rho),\) then
$$ \left\vert c_{n}\right\vert \leq(1-\rho)\ k\ \cos\lambda. $$
(8)
The result is sharp. Equality is attained for the odd coefficients and even coefficients respectively for the functions:
$$p_{1}\left(z\right) =1+\left(1-\rho\right) \cos\lambda\ e^{-i\lambda} \left[ \left(\frac{k+2}{4}\right) \left(\frac{1-z}{1+z}\right) -\left(\frac{k-2}{4}\right) \left(\frac{1+z}{1-z}\right) -1\right], $$
$$p_{2}\left(z\right) =1+\left(1-\rho\right) \cos\lambda\ e^{-i\lambda} \left[ \left(\frac{k+2}{4}\right) \left(\frac{1-z^{2}}{1+z^{2}}\right) -\left(\frac{k-2}{4}\right) \left(\frac{1+z^{2}}{1-z^{2}}\right) -1\right]. $$

We note that for λ=0 in Lemma 1, we obtain the result obtained by Goswami et al. [19] [Lemma 2.1] for the class \(\mathcal {P}_{k}(\rho).\)

In this paper, we will obtain the coefficients bounds |a2| and |a3| for the class \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\), which defined in Definition 2.

Coefficient estimates for functions in the class \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\)

Theorem 1

Let \(f,h\in \sum,\ \alpha \in \mathbb {C} ^{\ast }\backslash \{-1,\frac {-1}{2}\},\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2,\ \left \vert \lambda \right \vert <\frac {\pi }{2},\) fh given by (2) and h2, h3≠0. If fh belongs to \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\), then:
$$ \left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{2k(1-\rho)\cos\lambda}{\left\vert 2\alpha+\beta\right\vert \left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}};\ \frac{k(1-\rho)\cos\lambda}{\left\vert \alpha+\beta\right\vert \left\vert h_{2}\right\vert }\right\} $$
(9)
and
$$ \left\vert a_{3}\right\vert \leq\frac{k(1-\rho)\cos\lambda}{\left\vert 2\alpha+\beta\right\vert \left\vert h_{3}\right\vert }+\frac{\left[ k(1-\rho)\cos\lambda\right]^{2}}{\left\vert \alpha+\beta\right\vert^{2}\left\vert h_{3}\right\vert }. $$
(10)

The result is sharp.

Proof 1 If \((f * h)\in \mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\), then from Definition 2, we have:
$$ (1-\alpha)\left(\frac{(f * h)(z)}{z}\right)^{\beta}+\alpha\frac{z(f * h)^{\prime}(z)}{(f * h)(z)}\left(\frac{(f * h)(z)}{z}\right)^{\beta }=p(z),\ p\in\mathcal{P}_{k}^{\lambda}(\rho) $$
(11)
and
$$ (1-\alpha)\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\,+\,\alpha \frac{w\left((f * h)^{-1}(w)\right)^{\prime}}{(f * h)^{-1}(w)}\left(\frac{(f * h)^{-1}(w)}{w}\right)^{\beta}\,=\,q(w),\ q\in\mathcal{P}_{k}^{\lambda} (\rho), $$
(12)
where p and q have Taylor expansions as follows:
$$ p(z)=1+p_{1}z+p_{2}z^{2}+p_{3}z^{3}+....,z\in\mathbb{U}, $$
(13)
$$ q(w)=1+q_{1}w+q_{2}w^{2}+q_{3}w^{3}+....,w\in\mathbb{U}. $$
(14)
By comparing the coefficients in (11) with (13) and coefficients in (12) with (14), we obtain:
$$ p_{1}=\left(\beta+\alpha\right) \ a_{2}h_{2}, $$
(15)
$$ p_{2}=\left(\beta+2\alpha\right) \ a_{3}h_{3}+\frac{\left(\beta +2\alpha\right) \left(\beta-1\right) }{2}\ a_{2}^{2}h_{2}^{2}, $$
(16)
$$ q_{1}=-\left(\beta+\alpha\right) \ a_{2}h_{2} $$
(17)
and
$$ q_{2}=\left(\beta+2\alpha\right) \ \left(2a_{2}^{2}h_{2}^{2}-a_{3}h_{3} \right)+\frac{\left(\beta+2\alpha\right) \left(\beta-1\right) }{2}\ a_{2}^{2}h_{2}^{2}. $$
(18)
Since \(p,q\in \mathcal {P}_{k}^{\lambda }(\rho)\ {and}\) by applying Lemma 1, we have:
$$ \left\vert p_{n}\right\vert \leq k(1-\rho)\cos\lambda\ (n\geq1) $$
(19)
and
$$ \left\vert q_{n}\right\vert \leq k(1-\rho)\cos\lambda\ (n\geq1). $$
(20)
From (16) and (18) and using inequalities (19) and (20), we obtain:
$$ \left\vert a_{2}\right\vert^{2}\leq\frac{1}{\left\vert 2\alpha+\beta \right\vert \left\vert \beta+1\right\vert }\frac{\left\vert p_{2}\right\vert +\left\vert q_{2}\right\vert }{\left\vert h_{2}\right\vert^{2}}\leq \frac{2k(1-\rho)\cos\lambda}{\left\vert 2\alpha+\beta\right\vert \left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}\ . $$
(21)
Also, from (15) and (19), we obtain:
$$ \left\vert a_{2}\right\vert \leq\frac{k(1-\rho)\cos\lambda}{\left\vert \alpha+\beta\right\vert \left\vert h_{2}\right\vert }\ . $$
(22)
Subtracting (18) from (16), we have:
$$ p_{2}-q_{2}=2\left(2\alpha+\beta\right) \ \left(a_{3}h_{3}-a_{2}^{2}h_{2}^{2}\right). $$
(23)
Also, we have:
$$ p_{1}^{2}+q_{1}^{2}=2\left(\alpha+\beta\right)^{2}a_{2}^{2}h_{2}^{2}. $$
(24)
After using (23), (24), (19), and (20), and some easily calculations, we obtain:
$$ \left\vert a_{3}\right\vert \leq\frac{k(1-\rho)\cos\lambda}{\left\vert 2\alpha+\beta\right\vert \left\vert h_{3}\right\vert }+\frac{\left[ k(1-\rho)\cos\lambda\right]^{2}}{\left\vert \alpha+\beta\right\vert^{2}\left\vert h_{3}\right\vert }, $$
(25)

which completes the proof of Theorem 1. The result is sharp in view of the fact that assertion (8) of Lemma 1 is sharp.

Remark 1

For \(h(z)=\frac {z}{1-z\ },\ \beta =\alpha =1,\ k=2,\ \)and λ=0 in Theorem 1, we obtain the result obtained by Srivastava et al. [9] [Theorem 2].

Putting \(h(z)=\frac {z}{1-z\ }\ \)in Theorem 1, we obtain the following corollary.

Corollary 1

Let \(\ f\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2\ and\ \left \vert \lambda \right \vert <\frac {\pi }{2}.\ \)If \(f\in \mathcal {\digamma }_{\alpha,\lambda,\rho,k,\beta }(f)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{2k(1-\rho)\cos\lambda}{\left\vert 2\alpha+\beta\right\vert \left(\beta+1\right)} };\ \frac{k(1-\rho)\cos\lambda}{\left\vert \alpha+\beta\right\vert }\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{k(1-\rho)\cos\lambda}{\left\vert 2\alpha+\beta\right\vert }+\frac{\left[ k(1-\rho)\cos\lambda\right]^{2} }{\left\vert \alpha+\beta\right\vert^{2}}. $$
The result is sharp.

Putting λ=0 in Theorem 1, we obtain the following corollary.

Corollary 2

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2,\) fh given by (2) and h2, h3≠0. If \(f * h\in \mathcal {F}_{\alpha,\rho,k,\beta }(f * h)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{2k(1-\rho)}{\left\vert 2\alpha+\beta\right\vert \left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}};\ \frac{k(1-\rho)}{\left\vert \alpha+\beta\right\vert \left\vert h_{2}\right\vert }\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{k(1-\rho)}{\left\vert 2\alpha +\beta\right\vert \left\vert h_{3}\right\vert }+\frac{\left[ k(1-\rho)\right]^{2}}{\left\vert \alpha+\beta\right\vert^{2}\left\vert h_{3}\right\vert }. $$
The result is sharp.

Putting λ=0 and k=2 in Theorem 1, we obtain the following corollary.

Corollary 3

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \ \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ 0\leq \rho <1,\) fh given by (2) and h2, h3≠0. If \(f * h\in \mathcal {F}_{\alpha,\rho,\beta }(f * h)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{4(1-\rho)}{\left\vert 2\alpha+\beta\right\vert \left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}};\ \frac{2(1-\rho)}{\left\vert \alpha+\beta\right\vert \left\vert h_{2}\right\vert }\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{2(1-\rho)}{\left\vert 2\alpha +\beta\right\vert \left\vert h_{3}\right\vert }+\frac{\left[ 2(1-\rho)\right]^{2}}{\left\vert \alpha+\beta\right\vert^{2}\left\vert h_{3}\right\vert }. $$
The result is sharp.

Putting α=1 in Theorem 1, we obtain the following corollary.

Corollary 4

Let \(\ f,h\in \sum,\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2,\ \left \vert \lambda \right \vert <\frac {\pi }{2},\) fh given by (2) and h2, h3≠0. If \(f * h\in \mathbb {F} _{\lambda,\rho,k,\beta }(f * h)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{2k(1-\rho)\cos\lambda}{(2+\beta)\left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}};\ \frac{k(1-\rho)\cos\lambda}{(1+\beta)h_{2}}\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{k(1-\rho)\cos\lambda}{(2+\beta)\left\vert h_{3}\right\vert }+\frac{\left[ k(1-\rho)\cos\lambda\right]^{2}}{(1+\beta)^{2}\left\vert h_{3}\right\vert }. $$
The result is sharp.

Putting α=1, k=2, and λ=0 in Theorem 1, we obtain the following corollary.

Corollary 5

Let \(\ f,h\in \sum,\ \beta \geq 0,\ 0\leq \rho <1,\) fh given by (2) and h2, h3≠0. If \(f * h\in \mathbb {F}_{\rho,\beta }(f * h)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{4(1-\rho)}{(2+\beta)\left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}} ;\ \frac{2(1-\rho)}{(1+\beta)h_{2}}\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{2(1-\rho)}{(2+\beta)\left\vert h_{3}\right\vert }+\frac{\left[ 2(1-\rho)\right] ^{2}}{(1+\beta)^{2}\left\vert h_{3}\right\vert }. $$
The result is sharp.

Putting ρ=0 in Theorem 1, we obtain the following corollary.

Corollary 6

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \ \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ \left \vert \lambda \right \vert <\frac {\pi }{2},\ k\geq 2,\) fh given by (2) and h2, h3≠0. If \(f * h\in \mathcal {M}_{\alpha,\lambda,k,\beta }(f * h)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{2k\cos\lambda }{\left\vert 2\alpha+\beta\right\vert \left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}};\ \frac{k\cos\lambda}{\left\vert \alpha+\beta \right\vert \left\vert h_{2}\right\vert }\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{k\cos\lambda}{\left\vert 2\alpha +\beta\right\vert \left\vert h_{3}\right\vert }+\frac{\left[ k\cos \lambda\right]^{2}}{\left\vert \alpha+\beta\right\vert^{2}\left\vert h_{3}\right\vert }. $$
The result is sharp.

Putting ρ=λ=0 in Theorem 1, we obtain the following corollary.

Corollary 7

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ k\geq 2,\) fh given by (2) and h2, h3≠0. If \(f * h\in \mathcal {M}_{\alpha,k,\beta }(f * h)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{2k}{\left\vert 2\alpha+\beta\right\vert \left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}};\ \frac{k}{\left\vert \alpha+\beta\right\vert \left\vert h_{2} \right\vert }\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{k}{\left\vert 2\alpha+\beta\right\vert \left\vert h_{3}\right\vert }+\frac{k^{2}}{\left\vert \alpha+\beta\right\vert^{2}\left\vert h_{3}\right\vert }. $$
The result is sharp.

Putting ρ=λ=0 and k=2 in Theorem 1, we obtain the following corollary.

Corollary 8

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C} ^{\ast }\backslash \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\) fh given by (2) and h2, h3≠0. If \(f * h\in \mathcal {M}_{\alpha,\beta }(f * h)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{4}{\left\vert 2\alpha+\beta\right\vert \left(\beta+1\right) \left\vert h_{2}\right\vert^{2}}};\ \frac{2}{\left\vert \alpha+\beta\right\vert \left\vert h_{2} \right\vert }\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{2}{\left\vert 2\alpha+\beta\right\vert \left\vert h_{3}\right\vert} +\frac{4}{\left\vert \alpha+\beta\right\vert^{2}\left\vert h_{3}\right\vert }. $$
The result is sharp.

Putting λ=0, α=1 and \(h(z)=\frac {z}{1-z}\ \)in Theorem 1, we obtain the following corollary.

Corollary 9

Let \(\ f\in \sum,\ 0\leq \rho <1\ \) and β≥0. If \(f\in R_{\sum }(\rho,k,\beta)\), then:
$$\left\vert a_{2}\right\vert \leq\min\left\{ \sqrt[\ ]{\frac{2k(1-\rho)}{\left(2+\beta\right) \left(\beta+1\right) }};\ \frac{k(1-\rho)}{\left(1+\beta\right) }\right\} $$
and
$$\left\vert a_{3}\right\vert \leq\frac{k(1-\rho)}{\left(2+\beta\right) }+\frac{\left[ k(1-\rho)\right]^{2}}{\left(1+\beta\right)^{2}}. $$
The result is sharp.

Remark 2

The results in Corollary 9 correct the results obtained by Orhan et al. [11] [Theorem 2.11, with γ=1. ].

Notes

Acknowledgements

The authors are grateful to the referees for their valuable suggestions.

Funding

Higher Institute for Engineering and Technology, New Damietta, Egypt

Availability of data and materials

Not applicable.

Authors’ contributions

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

Competing interests

The authors declare that they have no competing interests.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Mohamed K. Aouf
    • 1
    Email author
  • Samar M. Madian
    • 2
  • Adela O. Mostafa
    • 1
  1. 1.Faculty of Science, Department of Mathematics, Mansoura UniversityMansouraEgypt
  2. 2.Basic Sciences Department, Higher Institute for Engineering and TechnologyNew DamiettaEgypt

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