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

Abstract

Let $\mathcal{A}$ denote the class of functions $f$ which are analytic in the open unit disk $\mathbb{U}$ and given by\[f(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}\qquad \left( z\in \mathbb{U}\right) .\]The coefficient functional $\phi _{\lambda }\left( f\right) =a_{3}-\lambda a_{2}^{2}$ on $f\in \mathcal{A}$ represents various geometric quantities. For example, $\phi _{1}\left( f\right) =a_{3}-a_{2}^{2}=S_{f}\left( 0\right) /6,$ where $S_{f}$ is the Schwarzian derivative. The problem of maximizing the absolute value of the functional $\phi _{\lambda }\left( f\right) $ is called the Fekete-Szegö problem.In a very recent paper, Shafiq \textit{et al}. [Symmetry 12:1043, 2020] defined a new subclass $\mathcal{SL}\left(k,q\right), (k>0, 0<q<1) $ consist of functions $f\in\mathcal{A}$ satisfying the following subordination:\[\frac{z\,D_{q}f\left( z\right) }{f(z)}\prec \frac{2\tilde{p}_{k}\left(z\right) }{\left( 1+q\right) +\left( 1-q\right) \tilde{p}_{k}\left( z\right)}\qquad \left( z\in \mathbb{U}\right) ,\]where\[\tilde{p}_{k}\left( z\right) =\frac{1+\tau _{k}^{2}z^{2}}{1-k\tau _{k}z-\tau_{k}^{2}z^{2}}, \qquad \tau _{k}=\frac{k-\sqrt{k^{2}+4}}{2},\]and investigated the Fekete-Szegö problem for functions belong to the class $\mathcal{SL}(k,q)$. This class is connected with $k$-Fibonacci numbers. The main purpose of this paper is to obtain sharp bounds on $\phi _{\lambda }\left( f\right)$ for functions $f$ belong to the class $\mathcal{SL}\left(k,q\right)$ when both $\lambda \in \mathbb{R}$ and $\lambda \in \mathbb{C}$, and to improve the result given in the above mentioned paper.