Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k -Fibonacci numbers involving
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Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function B. A. Frasin1
· S. R. Swamy2
· J. Nirmala3
Received: 19 May 2020 / Accepted: 7 October 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract The aim of the present paper is to introduce some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function φ(s) = 1+e2 −s , s ≥ 0 in the open unit disc D. We investigate the j upper bounds on initial coefficients for functions of the form gφ (z) = z + ∞ j=2 φ(s)d j z , in these newly introduced special families and also discuss the Fekete–Szegö problem. Some interesting consequences of the results established here are also indicated. Keywords Holomorphic function · Bi-univalent function · Fekete–Szegö inequality · Fibonacci numbers · Modified sigmoid function Mathematics Subject Classification Primary 11B39 · 30C45 · 33C45; Secondary 30C50 · 33C05
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B. A. Frasin [email protected] S. R. Swamy [email protected] J. Nirmala [email protected]
1
Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
2
Department of Computer Science and Engineering, RV College of Engineering, Bangalore, Karnataka 560 059, India
3
Department of Mathematics, Maharani’s Science College for Women, Bangalore, Karnataka 560 001, India
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B. A. Frasin et al.
1 Introduction and definitions Let C be the set of complex numbers, R be the set of real numbers and N be the set of natural numbers. Let A be the family of normalized functions that have the form g(z) = z + d2 z 2 + d3 z 3 + · · · = z +
∞
djz j,
(1.1)
j=2
which are holomorphic in D = {z ∈ C : |z| < 1} and let S be the collection of all members of A that are univalent in D. It is well- known (see [5]) that every function g ∈ S has an inverse g −1 satisfying z = g −1 (g(z)), z ∈ D and ω = g(g −1 (ω)), |ω| < r0 (g), r0 (g) ≥ 1/4, where g −1 = f (ω) = ω − d2 ω2 + (2d22 − d3 )ω3 − (5d23 − 5d2 d3 + d4 )ω4 + · · · .
(1.2)
A member g of A is said to be bi-univalent in D if both g and g −1 are univalent in D. We denote the family of bi-univalent functions that have the form (1.1), by . One can see the works of [4,11,12,18] and [26] for detailed investigations of the family and various subfamilies of the family . We recall the principle of subordination between two holomorphic functions g(z) and f (z) in D. It is known that g(z) is subordinate to f (z), written as g(z) ≺ f (z), z ∈ D, if there is a ψ(z) holomorphic in D, with ψ(0) = 0 and |ψ(z)| < 1, z ∈ D, such that g(z) = f (ψ(z)), z ∈ D. In particular, if f is univalent in D g(z) ≺ f (z) ⇐⇒ g(0) = f (0) and g(D) ⊂ f (D). ∞ In 2007, Falcón and Plaza [9] have examined the k-Fibonacci number sequence Fk, j j=0 , k ∈ R+ , given by Fk, j+1 = k Fk, j + Fk, j−1 ,
Fk,0 = 0,
Fk,1 = 1,
(1.3)
where j ∈ N and j
Fk, j
(k − tk ) j
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