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A class of double subordination-preserving integral operators. (English) Zbl 1108.30017

Summary: If H(U) denotes the space of analytic functions in the unit disk U, for the function h𝒜 and β, we define the integral operator I h;β :𝒦H(U)H(U) by

I β,γ (f)(z)=β 0 z f β (t)h -1 (t)h ' (t)dt 1/β ·

If “” stands for subordination, we determine simple sufficient conditions on g 1 , g 2 and β such that

zh ' (z) h(z) 1/β g 1 (z)zh ' (z) h(z) 1/β f(z)zh ' (z) h(z) 1/β g 2 (z)

implies

I h;β [g 1 ](z)I h;β [f](z)I h;β [g 2 ](z),

and say that I h;β is a double subordination-preserving integral operator, and we call such a kind of result a sandwich-type theorem.

Moreover, we prove that the above implication is sharp, in the sense that I h;β [g 1 ] is the largest function and I h;β [g 2 ] the smallest function so that the left-hand side, respectively the right-hand side of the above implication hold.

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