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Certain properties of the class of univalent functions defined by Ruscheweyh derivative. (English) Zbl 1093.30012

Let A denote the class of analytic functions in U={z:|z|<1} and TA denote the class of functions of the form

f(z)=z- n=2 a n z n ,a n 0,n=1,2,·

Let us put

D λ f(z):=f(z)*z (1-z) 1+λ =z- n=2 a n B n (λ)z n ,

where

B n (λ)=Γ(n+λ) (n-1)!Γ(1+λ)

and let

D(α,β,λ):fT:Rez(D λ f(z)) ' D λ f(z)>αz(D λ f(z)) ' D λ f(z)-1+βzU,

where α0, 0β<1, λ>-1. In this paper some special properties of the classes D(α,β,λ) are investigated. It is shown that D(α,β,λ) are the convex sets. It is proved that if fD(α,β,λ) then for δ>0 the function

G δ (z)=(1-δ)f(z)+δ 0 z f(t) tdt

also belongs to D(α,β,λ). Some estimations for the functional integral of order δ, δ<0, defined by

D z δ f(z):=1 Γ(-δ) 0 z f(t) (z-t) 1+δ dt

where fD(α,β,λ) are given.

MSC:
30C45Special classes of univalent and multivalent functions