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Integral operators that preserve the subordination. (English) Zbl 0898.30021

Let H(U) denote the space of all analytic functions in the open unit disc U of the complex plane. For fH(U), β,γ define

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

Let fg mean f is subordinate to g in U. The author investigates conditions on g, β, γ so that

z(f(z)/z) β z(g(z)/z) β zA β,γ (f)(z) z β zA β,γ (g)(z) z β ·

Similar results were earlier obtained by S. S. Miller, P. T. Mocanu and M. O. Reade.

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C25Covering theorems in conformal mapping theory
30C45Special classes of univalent and multivalent functions