AN EXTREMAL REGION FOR UNIVALENT FUNCTIONS

Abstract

Let S be the class of functions f(z)=z+22a z + …,f(0)=0 , f′(0)=1 which are regular andunivalent in the unit disk |z|0 we consider the equation:Re [( 2 2 x a ) f x ] 0 , f S , x 1,1 .Denote φ(x)= Re [( 2 2 x a ) f x ].Because φ(0)=0 and φ( a)=0 it follows that there is y (-a,0) such that φ′( y)=0 and z (0, a) suchthat (z) 0.The aim of this paper is to find min{y| φ′( y)=0}and max{z| φ′( z)=0}[3].If y and z is min{y| φ′(y)=0}, respectively max{z| φ′(z)=0} then for x z the equation φ′( x)=0 doesnot have real roots. Since S is a compact class, there exists .This problem was first proposed by Petru T. Mocanu .We will determine by using the variational method ofSchiffer-Goluzin .

Keywords

• Let f є S be the extremal function for which z is attained
• which