A general extremal problem for the class of close-to-convex functions

Abstract:

For $\beta \geqslant 0,{K_\beta }$ denotes the set of functions $f(z) = z + {a_2}{z^2} + \cdots$ defined on the unit disc U with the representation $f’(z) = a{p^\beta }(z)s(z)/z$, where $a \in C$, p is an analytic function with positive real part in U, and s is a normalized starlike function. If $0 \leqslant \beta \leqslant 1$, and $\zeta \in U$, let $F(u,v)$ be analytic in a neighborhood of $\{ (f(\zeta ),\zeta ):f \in {K_\beta }\}$. Then $\max \{ \operatorname {Re} F(f(\zeta ),\zeta ):f \in {K_\beta }\}$ occurs for a function of the form \[ f(z) = {(\beta + 1)^{ - 1}}{(x - y)^{ - 1}}[{(1 + xz)^{\beta + 1}}{(1 + yz)^{ - \beta - 1}} - 1],\] where $|x| = |y| = 1$ and $x \ne y$. If $0 < \beta < 1$ these are the only extremal functions. A consequence of this result is the determination of the value region $\{ f(\zeta )/\zeta :f \in {K_\beta }\}$ as $\{ {(\beta + 1)^{ - 1}}{(s - t)^{ - 1}}[{(1 + s)^{\beta + 1}}{(1 + t)^{ - \beta - 1}} - 1]:|s|,|t| \leqslant |\zeta |\}$.