Applications of $k$-Fibonacci numbers for the starlike analytic functions

Abstract

The $k-$ Fibonacci numbers $F_{k,n}\\:(k>0)$, defined recursively by $F_{k,0}=0$ , $F_{k,1}=1$ and $F_{k,n}=kF_{k,n}+F_{k,n-1}$ for $n\\geq1$ are used to define a new class $\\mathcal{S}\\mathcal{L}^k$. The purpose of this paper is to apply properties of $k$-Fibonacci numbers to consider the classical problem of estimation of the Fekete–Szegö problem for the class $\\mathcal{S}\\mathcal{L}^{k}$. An application for inverse functions is also given.