| Summary: | Let S be the class of functions f(z)=z+a2z2…, f(0)=0, f′(0)=1 which are regular and univalent in theunit disk |z|<1.For 0≤x≤a≤1 we consider the equationRe [(x-a)f(x)]=0, fєS. and Re [(x3-a3)f(x)]=0. (1)Denote φ(x)=Re [(x-a)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 acompact class, there exists x .This problem was first proposed by Petru T. Mocanu in [2]. We will determine x by using thevariational method of Schiffer-Goluzin [1].
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