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

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 \(z\in \mathbb {U}\).

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:

(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:

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

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.