期刊名称：Constantin Brancusi University's Annals, Letters and Social Sciences Serie
印刷版ISSN：1844-6051
出版年度：2011
期号：3
出版社：Academica Brancusi Publisher
摘要：Let S be the class of functions f(z)=z+a2z 2…, 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 [(x 2 -a 2 )f(x)]=0, fєS. (1) Denote φ(x)=Re [(x 2 -a 2 )f(x)]. Because φ(0)=0 and φ( a)=0 it follows that there is x0є(0,a) such that: φ′( x0)=0 and y0є(-a,0) such that: φ′( y0)=0 The aim of this paper is to find max{x| φ′( x)=0} and min{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. If y is min{y| φ′(y)=0}, then for y< y the equation φ′( y)=0 does not have real roots. Since S is a compact class, there exists x and y . This problem was first proposed by Petru T. Mocanu in [2]. We will determine x and y by using the variational method of Schiffer-Goluzin [1].
