AN EXTREMAL PROBLEM FOR UNIVALENT FUNCTIONS
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 [(x3 -a3 )f(x)]=0, fєS. (1) Denote φ(x)=Re [(x3 -a3 )f(x)]. Because φ(0)=0 and φ(a)=0 it follows that there is x0є(0,a) suc...
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| Format: | Article |
| Language: | English |
| Published: |
Academica Brâncuşi
2010-12-01
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| Series: | Analele Universităţii Constantin Brâncuşi din Târgu Jiu : Seria Economie |
| Subjects: | |
| Online Access: | http://www.utgjiu.ro/revista/ec/pdf/2010-04.I/19_MIODRAG_IOVANOV.pdf |
| Summary: | 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 [(x3
-a3
)f(x)]=0, fєS. (1)
Denote φ(x)=Re [(x3
-a3
)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]. |
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| ISSN: | 1844-7007 1844-7007 |