On the Fekete-Szegö problem for strongly \(\alpha\)-logarithmic quasiconvex functions (Q1768036)

Let \(S\) denote the class of normalized analytic functions, which are univalent in the open unit disk of the form \(f(z)= z+ \sum^\infty_{n=2} a_n z^n\), \(n= 1,2,\dots\). A classical result of M. Fekete and G. Szegő is the following sharp inequality for \(f\in S\) \[ |a_3-\mu a^2_2|\leq \begin{cases} 3- 4\mu\quad &\text{if }\mu\leq 0,\\ 1+ 2e^{-2\mu/(1- \mu)}\quad &\text{if }0\leq \mu\leq 1,\\ 4\mu- 3\quad &\text{if }\mu\geq 1.\end{cases} \] In this paper the authors introduce new subclasses of \(S\), called normalized strongly \(\alpha\)-logarithmic convex and quasiconvex functions of order \(\beta\) and obtain sharp Fekete-Szegő inequalities for functions belonging to this classes.