On the Fekete-Szegő problem for close-to-convex functions

Abstract:

Let $S$ be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result \[ {\max _{f \in S}}{\text {|}}{a_3} - \lambda a_2^2| = 1 + 2{e^{ - 2\lambda /1 - \lambda )}}\] for $\lambda \in [0,1]$. We consider the corresponding problem for the family $C$ of close-to-convex functions and get \[ \max \limits _{f \in C} |{a_3} - \lambda a_2^2 = \left \{ {\begin {array}{*{20}{c}} {3 - 4\lambda } & {{\text {if}}\lambda \in [0,1/3],} \\ {1/3 + 4/(9\lambda )} & {{\text {if}}\lambda \in {\text {[1/3,2/3],}}} \\ 1 & {{\text {if}}\lambda \in {\text {[2/3,1]}}{\text {.}}} \\ \end {array} } \right .\] As an application it is shown that $||{a_3}| - |{a_2}|| \leq 1$ for close-to-convex functions, in contrast to the result in $S$ \[ \max \limits _{f \in s} {\text {||}}{a_3}| - |{a_2}|| = 1.029....\]