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A class of superordination-preserving integral operators. (English) Zbl 1019.30023

Let H(U) denote the class of analytic functions in the unit disk U and let the integral operator A β,γ (f)(z):KH(U), KH(U) be defined by

A β,γ (f)(z)=β+γ)/ z γ 0 z f β (t) t γ-1 d t 1/β ,β,γ·

If f,FH(U) and F is univalent in U we say that f is subordinate to F or F is superordinate to f, written f(z)F(z), if f(0)=F(0) and f(U)F(U). In a recent paper S. S. Miller and P. T. Mocanu have determined conditions on φ such that

h(z)φp (z) , z p ' (z) , z 2 p '' (z) ; zimpliesq(z)p(z),

for all functions p that satisfy the above superordination. In this paper the author determines sufficient conditions on g,β and γ such that the following differential superordination holds:

z[g(z)/z β zf ( z ) / z β implieszA β,γ (g) (z) / z β zA β,γ (f) (z) / z β ·

The function z[A β,γ (g)(z)/z] β is the largest function so that the right-hand side holds, for all functions f satisfying the left-hand side differential super-ordination. The particular case g(z)=ze λz is considered.

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C45Special classes of univalent and multivalent functions