Comprehensive subclass of m -fold symmetric bi-univalent functions defined by subordination
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Comprehensive subclass of m-fold symmetric bi-univalent functions defined by subordination Serap Bulut1 · Safa Salehian2
· Ahmad Motamednezhad3
Received: 7 February 2020 / Accepted: 24 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In the present paper, we define and investigate the subclass Gm (λ, τ, γ , θ, ψ) of m-fold symmetric bi-univalent functions in the open unit disk U associated with subordination. Moreover, we find estimates on general coefficient |amk+1 | (k ≥ 2) for functions belong to this subclass Gm (λ, τ, γ , θ, ψ). The results presented in this paper would generalize some related works of several earlier authors. Keywords Bi-univalent functions · Coefficient estimates · Hadamard product · m-fold symmetric bi-univalent functions · Subordination Mathematics Subject Classification 30C45 · 30C50
1 Introduction Let A be the class of normalized analytic functions f on the unit disk U = {z ∈ C : |z| < 1}, in the form ∞ an z n . (1) f (z) = z + n=2
We denote by S , the class of functions f ∈ A which are univalent in U. The Koebe OneQuarter Theorem [7] showed that the range of every function of class S contains the disk
B
Safa Salehian [email protected] Serap Bulut [email protected] Ahmad Motamednezhad [email protected]
1
Faculty of Aviation and Space Sciences, Kocaeli University, Arslanbey Campus, 41285 Kartepe, Kocaeli, Turkey
2
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran
3
Faculty of Mathematical Sciences, Shahrood University of Technology, P.O.Box 316-36155, Shahrood, Iran
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S. Bulut et al.
{w ∈ C : |w| < 41 }. So every function f ∈ S has an inverse f −1 , which is defined by f −1 ( f (z)) = z and f(f
−1
(w)) = w
(z ∈ U)
1 . |w| < r0 ( f ), r0 ( f ) ≥ 4
In fact, the inverse function f −1 is given by f −1 (w) = w − a2 w 2 + (2a22 − a3 )w 3 − (5a23 − 5a2 a3 + a4 )w 4 + · · · .
(2)
f −1
are univalent in U. We A function f ∈ A is said to be bi-univalent in U, if both f and denote by σ , the class of bi-univalent functions in U given by (1). Lewin [14] investigated the class σ of bi-univalent functions and showed that |a2 | < 1.51, for the Taylor-Maclaurin coefficient |a2 | of functions belong to σ . Subsequently, Brannan et √ al. [2] conjectured that |a2 | ≤ 2. For a brief history and interesting examples of functions in the class σ , one may refer to a pioneering paper by Srivastava et al. [19]. In fact, this widely-cited work by Srivastava et al. [19] actually revived the study of analytic and bi-univalent functions in recent years. It has led to a flood of papers on the subject by (for example) Srivastava et al. [6,22–25], and others [1,3,4,10,12,27]. However, finding upper bounds of the Taylor–Maclaurin coefficients |an |(n ∈ N−{2, 3}) for each f ∈ σ is coefficient estimate problem and still an open problem. For each function f ∈ S , function (3) h(z) = m f (z m ) is univalent and maps unit disk U into a region with m-fold symmetry. A fu
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