Logarithmic coefficients of starlike functions connected with k-Fibonacci numbers

Abstract

Let $mathcal{A}$ denote the class of analytic functions in the open unit disc $mathbb{U}$ normalized by $f(0)=f^{prime }(0)-1=0,$ and let $mathcal{S}$ be the class of all functions $finmathcal{A}$ which are univalent in $mathbb{U}$. For a function $fin mathcal{S}$, the logarithmic coefficients $delta _{n},left( n=1,2,3,ldots right) $ are defined by$log frac{f(z)}{z}=2sum_{n=1}^{infty }delta _{n}z^{n}qquad left( zinmathbb{U}right).$and it is known that $leftvert delta _{1}rightvert leq 1$ and $leftvert delta _{2}rightvert leq frac{1}{2}left( 1+2e^{-2}right)=0,635cdots .$ The problem of the best upper bounds for $leftvert delta_{n}rightvert $ of univalent functions for $ngeq 3$ is still open. Let $mathcal{SL}^{k}$ denote the class of functions $fin mathcal{A}$ such that$frac{zf^{prime }left( zright) }{f(z)}prec frac{1+tau _{k}^{2}z^{2}}{1-ktau _{k}z-tau _{k}^{2}z^{2}},quad tau _{k}=frac{k-sqrt{k^{2}+4}}{2}qquad left( zin mathbb{U}right).$In the present paper, we determine the sharp upper bound for $leftvertdelta _{1}rightvert ,leftvert delta _{2}rightvert $ and $leftvertdelta _{3}rightvert $ for functions $f$ belong to the class $mathcal{SL}^{k}$ which is a subclass of $mathcal{S}$. Furthermore, a general formula is given for $leftvert delta _{n}rightvert ,left( nin mathbb{N}right) $ as a conjecture.