An extremal problem for univalent functions

Let S be the class of functions f(z)=z+a2z2 ,… f(0)=0, f′(0)=1 which are regular and univalent in the unit disk |z|<1. For 0≤x≤a≤1 we consider the equation Re [(x3-a3)f(x)]=0, fєS. (1) Denote φ(x)=Re [(x3-a3)f(x)]. Because φ(0)=0 and φ(a)=0 it follows that there is x0є(0,a) such that: φ′( x0)=0. The aim of this paper is to find max{x| φ′(x)=0}. If x is max{x| φ′(x)=0}, then for x> x the equation φ′( x)=0 does not have real roots. Since S is a compact class, there exists x . This problem was first proposed by Petru T. Mocanu in [2]. We will determine: x by using the variational method of Schiffer-Goluzin [1].