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

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 \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\mathsf {.}\)

Let \(\mathcal {A}\) denote the class of analytic functions of the form:

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:

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:

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:

It is easy to see that the inverse function has the form:

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:

and

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) \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }\left (f\times \frac {z} {1-z\ }\right)=\mathcal {\digamma }_{\alpha,\lambda,\rho,k,\beta }(f)\)

(ii) \(\mathcal {M}_{\alpha,0,\rho,k,\beta }(f\ast h)=\mathcal {F}_{\alpha,\rho,k,\beta }(f * h)\)

(iii) \(\mathcal {M}_{\alpha,0,\rho,2,\beta }(f\ast h)=\mathcal {F}_{\alpha,\rho,\beta }(f * h)\)

(iv) \(\mathcal {M}_{\alpha,\lambda,0,k,\beta }(f * h)=\mathcal {M}_{\alpha,\lambda,k,\beta }(f * h)\)

(v) \(\mathcal {M}_{\alpha,0,0,k,\beta }(f * h)=\mathcal {M}_{\alpha,k,\beta }(f * h)\)

(vi) \(\mathcal {M}_{\alpha,0,0,2,\beta }(f * h)=\mathcal {M}_{\alpha,\beta }(f * h)\)

(vii) \( \mathcal {M}_{1,\lambda,\rho,k,\beta }(f * h)=\mathbb {F}_{\lambda,\rho,k,\beta }(f * h)\)

or

(viii) \(\mathcal {M}_{1,0,\rho,2,\beta }(f * h)=\mathbb {F}_{\rho,\beta }(f * h)\)

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

The result is sharp. Equality is attained for the odd coefficients and even coefficients respectively for the functions:

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.

Theorem 1

Let \(f,h\in \sum,\ \alpha \in \mathbb {C} ^{\ast }\backslash \{-1,\frac {-1}{2}\},\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2,\ \left \vert \lambda \right \vert <\frac {\pi }{2},\) f∗h given by (2) and h₂, h₃≠0. If f∗h belongs to \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\), then:

and

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:

and

where p and q have Taylor expansions as follows:

By comparing the coefficients in (11) with (13) and coefficients in (12) with (14), we obtain:

and

Since \(p,q\in \mathcal {P}_{k}^{\lambda }(\rho)\ {and}\) by applying Lemma 1, we have:

and

From (16) and (18) and using inequalities (19) and (20), we obtain:

Also, from (15) and (19), we obtain:

Subtracting (18) from (16), we have:

Also, we have:

After using (23), (24), (19), and (20), and some easily calculations, we obtain:

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:

and

The result is sharp.

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

Corollary 2

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2,\) f∗h given by (2) and h₂, h₃≠0. If \(f * h\in \mathcal {F}_{\alpha,\rho,k,\beta }(f * h)\), then:

and

The result is sharp.

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

Corollary 3

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \ \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ 0\leq \rho <1,\) f∗h given by (2) and h₂, h₃≠0. If \(f * h\in \mathcal {F}_{\alpha,\rho,\beta }(f * h)\), then:

and

The result is sharp.

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

Corollary 4

Let \(\ f,h\in \sum,\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2,\ \left \vert \lambda \right \vert <\frac {\pi }{2},\) f∗h given by (2) and h₂, h₃≠0. If \(f * h\in \mathbb {F} _{\lambda,\rho,k,\beta }(f * h)\), then:

and

The result is sharp.

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

Corollary 5

Let \(\ f,h\in \sum,\ \beta \geq 0,\ 0\leq \rho <1,\) f∗h given by (2) and h₂, h₃≠0. If \(f * h\in \mathbb {F}_{\rho,\beta }(f * h)\), then:

and

The result is sharp.

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

Corollary 6

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \ \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ \left \vert \lambda \right \vert <\frac {\pi }{2},\ k\geq 2,\) f∗h given by (2) and h₂, h₃≠0. If \(f * h\in \mathcal {M}_{\alpha,\lambda,k,\beta }(f * h)\), then:

and

The result is sharp.

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

Corollary 7

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C}^{\ast }\backslash \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\ k\geq 2,\) f∗h given by (2) and h₂, h₃≠0. If \(f * h\in \mathcal {M}_{\alpha,k,\beta }(f * h)\), then:

and

The result is sharp.

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

Corollary 8

Let \(\ f,h\in \sum,\ \alpha \in \mathbb {C} ^{\ast }\backslash \left \{-1,\frac {-1}{2}\right \},\ \beta \geq 0,\) f∗h given by (2) and h₂, h₃≠0. If \(f * h\in \mathcal {M}_{\alpha,\beta }(f * h)\), then:

and

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:

and

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. ].

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.

Affiliations

1. Faculty of Science, Department of Mathematics, Mansoura University, Mansoura, 35516, Egypt

   • Mohamed K. Aouf
   •  & Adela O. Mostafa

2. Basic Sciences Department, Higher Institute for Engineering and Technology, New Damietta, Egypt

   • Samar M. Madian

Contributions

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

Corresponding author

Correspondence to Mohamed K. Aouf.

Competing interests

The authors declare that they have no competing interests.

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