Certain classes of bi-univalent functions related to Shell-like curves connected with Fibonacci numbers
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Certain classes of bi-univalent functions related to Shell-like curves connected with Fibonacci numbers N. Magesh1
· C. Abirami2 · V. K. Balaji3
Received: 24 February 2020 / Accepted: 23 July 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In 2010, Srivastava et al. [38] revived the study of coefficient problems for bi-univalent functions. Due to the pioneering work of Srivastava et al. [38], there has been elicit curiosity to study the coefficient problems for various subclasses of bi-univalent functions. Motivated predominantly by Srivastava et al. [38], in this work, we consider certain classes of bi-univalent functions related with shell-like curves connected with Fibonacci numbers to obtain the estimates of second and third Taylor-Maclaurin coefficients and Fekete - Szegö inequalities. Further, special cases are also indicated. Some observations of the results presented here are also discussed. Keywords Univalent functions · Bi-univalent functions · Shell-like function · Convex shell-like function · Pseudo starlike function · Bazilevi´c function Mathematics Subject Classification Primary 30C45 · Secondary 30C50
1 Introduction and definitions Let R = (−∞, ∞) be the set of real numbers, and N := {1, 2, 3, · · · } = N0 \ {0}
B
N. Magesh [email protected] C. Abirami [email protected] V. K. Balaji [email protected]
1
Post-Graduate and Research Department of Mathematics, Government Arts College for Men, Krishnagiri, Tamilnadu 635 001, India
2
Faculty of Engineering and Technology, SRM University, Kattankulathur, Tamilnadu 603 203, India
3
Department of Mathematics, L.N. Govt College, Ponneri, Chennai, Tamilnadu, India
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be the set of positive integers. Let A denote the class of functions of the form f (z) = z +
∞
an z n
(1.1)
n=2
which are analytic in the open unit disk D = {z : z ∈ C and |z| < 1}, where C be the set of complex numbers. Further, by S , we shall denote the class of all functions in A which are univalent in D. Let P denote the class of functions of the form p(z) = 1 + p1 z + p2 z 2 + p3 z 3 + · · · ,
z∈D
which are analytic with { p(z)} > 0. Here p is called as Caratheodory functions [11]. It is well known that the following correspondence between the class P and the class of Schwarz functions w exists: p ∈ P if and only if p(z) = 1 + w(z)/1 − w(z). Let P (β), 0 ≤ β < 1, denote the class of analytic functions p in D with p(0) = 1 and { p(z)} > β. For analytic functions f and g in D, f is said to be subordinate to g if there exists an analytic function w such that w(0) = 0,
|w(z)| < 1 and
f (z) = g(w(z)),
z ∈ D,
denoted by f ≺ g,
z∈D
or, conventionally, by f (z) ≺ g(z),
z ∈ D.
In particular, when g is univalent in D, f ≺g
(z ∈ D) ⇔ f (0) = g(0) and
f (D) ⊂ g(D).
Some of the important and well-investigated subclasses of S include (for example) the class S ∗ (α) of starlike functions of order α (0 α < 1) in D and the class K(α) of convex functions of order α (0 α
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