Logarithmic coefficients of starlike functions connected with k-Fibonacci numbers

Authors : Serap BULUT

Pages : 910-923

Doi:10.31801/cfsuasmas.808319

View : 17 | Download : 7

Publication Date : 2021-12-31

Article Type : Research Paper

Abstract :Let $\mathcal{A}$ denote the class of analytic functions in the open unit disc $\mathbb{U}$ normalized by $finsert ignore into journalissuearticles values(0);=f^{\prime }insert ignore into journalissuearticles values(0);-1=0,$ and let $\mathcal{S}$ be the class of all functions $f\in\mathcal{A}$ which are univalent in $\mathbb{U}$. For a function $f\in \mathcal{S}$, the logarithmic coefficients $\delta _{n}\,\leftinsert ignore into journalissuearticles values( n=1,2,3,\ldots \right); $ are defined by $\log \frac{finsert ignore into journalissuearticles values(z);}{z}=2\sum_{n=1}^{\infty }\delta _{n}z^{n}\qquad \leftinsert ignore into journalissuearticles values( z\in\mathbb{U}\right);.$ and it is known that $\left\vert \delta _{1}\right\vert \leq 1$ and $\left\vert \delta _{2}\right\vert \leq \frac{1}{2}\leftinsert ignore into journalissuearticles values( 1+2e^{-2}\right);=0,635\cdots .$ The problem of the best upper bounds for $\left\vert \delta_{n}\right\vert $ of univalent functions for $n\geq 3$ is still open. Let $\mathcal{SL}^{k}$ denote the class of functions $f\in \mathcal{A}$ such that $\frac{zf^{\prime }\leftinsert ignore into journalissuearticles values( z\right); }{finsert ignore into journalissuearticles values(z);}\prec \frac{1+\tau _{k}^{2}z^{2}}{1-k\tau _{k}z-\tau _{k}^{2}z^{2}},\quad \tau _{k}=\frac{k-\sqrt{k^{2}+4}}{2}\qquad \leftinsert ignore into journalissuearticles values( z\in \mathbb{U}\right);.$ In the present paper, we determine the sharp upper bound for $\left\vert\delta _{1}\right\vert ,\left\vert \delta _{2}\right\vert $ and $\left\vert\delta _{3}\right\vert $ for functions $f$ belong to the class $\mathcal{SL}^{k}$ which is a subclass of $\mathcal{S}$. Furthermore, a general formula is given for $\left\vert \delta _{n}\right\vert \,\leftinsert ignore into journalissuearticles values( n\in \mathbb{N}\right); $ as a conjecture.
Keywords : Analytic function, univalent function, shell like function, logarithmic coefficients, k Fibonacci number, subordination