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Bi-univalent properties for certain class of Bazilevič functions defined by convolution and with bounded boundary rotation
Journal of the Egyptian Mathematical Society volume 27, Article number: 11 (2019)
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 \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\mathsf {.}\)
Introduction
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.
Coefficient estimates for functions in the class \(\mathcal {M}_{\alpha,\lambda,\rho,k,\beta }(f * h)\)
Theorem 1
Let \(f,h\in \sum,\ \alpha \in \mathbb {C} ^{\ast }\backslash \{-1,\frac {-1}{2}\},\ \beta \geq 0,\ 0\leq \rho <1,\ k\geq 2,\ \left \vert \lambda \right \vert <\frac {\pi }{2},\) 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. ].
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Aouf, M.K., Madian, S.M. & Mostafa, A.O. Bi-univalent properties for certain class of Bazilevič functions defined by convolution and with bounded boundary rotation. J Egypt Math Soc 27, 11 (2019). https://doi.org/10.1186/s42787-019-0012-2
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DOI: https://doi.org/10.1186/s42787-019-0012-2