期刊名称：Constantin Brancusi University's Annals, Letters and Social Sciences Serie
印刷版ISSN：1844-6051
出版年度：2008
期号：2
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-a)f(x)]=0, fєS. (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 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]. 1. Let f є S be the extremal function for which x is attained, which: Re[f( x )+( x -a)f′( x )]=0. Next we consider a variation of the function f given by Schiffer-Goluzin’s formula [1]: f*(x)=f(x)+ λV(x; ζ; ψ)+ 0(λ 2 ), | ζ |<1, λ>0, (2) ψ real number, where: f (x) f ( ) V(x; ; ) e e f (x) [ ] f (x) f ( ) f ( ) xf (x) f ( ) x f (x) f ( ) 2 e [ ] e [ ] x f ( ) 1 x f ( ) ⎧ ζ ζ ψ = − ⋅ ⋅ − ⎪ − ζ ζ ζ′ ⎨ ′ ′ ζ ζ ⎪ − ζ ⋅ + ζ − ζ ζ ζ − ζ ζ ζ ′ ′ ⎩ 2 iψ iψ 2 2 iψ -iψ 2 (3) Next we consider a variation x * of x: x * =x+λh+0(λ 2 ), x h 0 ∂ = ∂λ λ= * which satisfies the conditions: |x * |=x and Re[f * (x * )+(x * -a)f′ * (x * )]=0.
