On the Fekete-Szego problem for alpha-quasi-convex functions

Abstract

Let $Q_\alpha(\alpha\ge 0)$ denote the class of normalized analytic alpha-quasi-convex functions $f$, defined in the unit disc, $D=\{z:|z|<1\}$, by the condition

$$\hbox{Re}\left[(1-\alpha){f'(z)\over g'(z)}+ \alpha {(zf'(z))'\over g'(z)}\right]>0,$$

Where $f(z)=z+\sum_{n=2}^\infty a_n z^n$ and where $g(z)=z+\sum_{n=2}^\infty b_nz^n$ is a convex univalent function in $D$. Sharp upper bounds are obtained for $|a_3-\mu a_2^2|$, when $\mu\ge 0$.