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:
$$ (ii)\int\limits_{0}^{2\pi}\left\vert \frac{\mathfrak{R}\left\{ e^{i\lambda }p(z)\right\} -\rho\cos\lambda}{1-\rho}\right\vert \leq k\pi\cos \lambda\ \left(r<1,z=re^{i\theta}\in\mathbb{U}\right). $$
(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.