New subclasses bi-univalent functions related to shell-like curves involving hypergeometric functions
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New subclasses bi-univalent functions related to shell-like curves involving hypergeometric functions G. Murugusundaramoorthy1
· H. Ö. Güney2
· K. Vijaya1
Received: 16 February 2020 / Accepted: 27 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In the present paper, new subclasses of bi-univalent functions of complex order associated with hypergeometric functions are introduced and coefficient estimates for functions in these classes are obtained. Several new (or known) consequences of the results are also pointed out. Keywords Hypergeometric function · Analytic function · Univalent function · Bi-univalent function · Bi-starlike function · Bi-convex function · Complex order · Coefficient bounds · Shell-like curve · Fibonacci numbers Mathematics Subject Classification 30C45
1 Introduction and definitions Let A denote the class of functions of the form f (z) = z +
∞
an z n
(1)
n=2
which are analytic in the open unit disc U = {z : |z| < 1} and normalized by the conditions f (0) = 0 and f (0) = 1. Further, let S denote the class of all functions in A which are univalent in U. Some of the important and well-investigated subclasses of the univalent function class S include (for example) the class S ∗ (α) of starlike functions of order α in U and the class K(α) of convex functions of order α (0 ≤ α < 1) in U.
B
H. Ö. Güney [email protected] G. Murugusundaramoorthy [email protected] K. Vijaya [email protected]
1
School of Advanced Sciences, Vellore Institute of Technology (Deemed to be University), Vellore 632014, India
2
Faculty of Science, Department of Mathematics, Dicle University, 21280 Diyarbakır, Turkey
123
G. Murugusundaramoorthy et al.
From Koebe one quarter theorem [11], it is well known that every function f ∈ S has an inverse f −1 , defined by f −1 ( f (z)) = z (z ∈ U) and f ( f −1 (w)) = w (|w| < r0 ( f ); r0 ( f ) ≥ 1/4) where
f −1 (w) = g(w) = w − a2 w 2 + 2a22 − a3 w 3 − 5a23 − 5a2 a3 + a4 w 4 + · · · .
(2)
A function f ∈ A is said to be bi-univalent in U if both f and f −1 are univalent in U. Let denote the class of bi-univalent functions in U given by (1). Earlier, Brannan and Taha [4] introduced certain subclasses of bi-univalent function class , namely bi-starlike ∗ (α) and bi-convex functions of order α denoted by K (α) functions of order α denoted by S corresponding to the function classes S ∗ (α) and K(α) respectively. Also, they determined non-sharp estimates on the first two Taylor–Maclaurin coefficients |a2 | and |a3 | (see also [38]). Many researchers have introduced and investigated several interesting subclasses of the bi-univalent function class and they have found non-sharp estimates on the first two Taylor–Maclaurin coefficients |a2 | and |a3 | (see [1–3,12,13,17,19,20,22,34–37,39–41]). An analytic function f is subordinate to an analytic function F in U, written as f ≺ F, provided there is an analytic function ω defined on U with ω(0) = 0 and |ω(z)| < 1 satisfying f (z) =
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