## Fekete-Szegö problem for q-starlike functions in connected with k-Fibonacci numbers

Hacettepe Journal of Mathematics and Statistics, vol.51, no.6, pp.1661-1673, 2022 (SCI-Expanded)

• Publication Type: Article / Article
• Volume: 51 Issue: 6
• Publication Date: 2022
• Doi Number: 10.15672/hujms.1010314
• Journal Name: Hacettepe Journal of Mathematics and Statistics
• Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH
• Page Numbers: pp.1661-1673
• Keywords: analytic function, Fekete-Szegö problem, Fibonacci numbers, shell-like function, subordination, univalent function
• Kocaeli University Affiliated: Yes

#### Abstract

© 2022, Hacettepe University. All rights reserved.Let A denote the class of functions f which are analytic in the open unit disk U and given by ∞∑ f(z) = z + anzn (z ∈ U). n=2 The coefficient functional ϕλ (f) = a3 − λa22 on f ∈ A represents various geometric quantities. For example, ϕ1 (f) = a3 − a22=Sf (0) /6, where Sf is the Schwarzian derivative. The problem of maximizing the absolute value of the functional ϕλ (f) is called the FeketeSzegö problem. In a very recent paper, Shafiq et al. [Symmetry 12:1043, 2020] defined a new subclass SL (k, q), (k > 0, 0 < q < 1) consists of functions f ∈ A satisfying the following subordination: z Dq f (z) 2˜pk (z) ≺ (z ∈ U), f(z) (1 + q) + (1 − q) ˜pk (z) where 1 + τk2˜pk (z) =z2 1 − kτkz − τk2, τz2 k =k − √k2 + 4, 2 and investigated the Fekete-Szegö problem for functions belong to the class SL(k, q). This class is connected with k-Fibonacci numbers. The main purpose of this paper is to obtain sharp bounds on ϕλ (f) for functions f belong to the class SL (k, q) when both λ ∈ R and λ ∈ C, and to improve the result given in the above mentioned paper.