AN THE ECUATION Re [(x-a)f(x)]=0, fєS

• Main Author: Miodrag IOVANOV
• Format: Article
• Language: English
• Published: Academica Brancusi 2012-05-01
• Series: Fiabilitate şi Durabilitate
• Subjects: function; variation; finite number
• Online Access: http://www.utgjiu.ro/rev_mec/mecanica/pdf/2012-01.Supliment/68_Miodrag%20Iovanov.pdf

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].