Sharpness of certain Campbell and Pommerenke estimates |
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Authors: | J Godula V V Starkov |
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Institution: | (1) Institute of Mathematics, M. Curie-Sklodowska University, Lublin;(2) Petrozavodsk State University, USSR |
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Abstract: | The paper is concerned with the sharpness of some well-known estimates in universal linear-invariant families
of regular functions. It is shown that the estimate of | arg ƒ′(z)|,z ∈ Δ = {z: |z| < 1} obtained by Pommerenke in 1964 is sharp; the extremal function is found. A lower estimate for the Schwarzian derivative
in
is obtained. For
, a sharp estimate of order of the function ƒr(z)=ƒ(rz)/r withr ∈ (0, 1) is found; this estimate is applied to solve other problems.
Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 665–672, May, 1998. |
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Keywords: | conformal map linear-invariant family of regular functions locally univalent function convex function Pommerenke rotation theorem |
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