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α-Convexity of Schlicht functions

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Abstract

Upper and lower bounds are obtained for the radius ofα-convexity, Rα, of the schlicht within ¦z¦< 1 functions g(z), g(0)=0, and g′(0)=1, forα values ranging from 0 to 0.313.... The exact value of Rα is determined for 0.313... ≤α < 1. The results constitute the solution to a problem recently posed by the Roumanianmathematician P. T. Mocanu [1].

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Literature cited

  1. P. T. Mocanu, “Une propriété de convexité generalisée dans la théorie de la représentation conforme,” Mathematica (RSR),11, No. 1, 127–133 (1969).

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  2. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable [in Russian], Moscow (1966).

  3. G. Szegö and G. Polya, Problems and Theorems from Analysis [Russian translation], Vol. 2, Moscow (1956).

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Authors and Affiliations

  1. Tomsk State University, USSR

    V. V. Chernikov

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  1. V. V. Chernikov
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Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 227–232, February, 1972.

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Chernikov, V.V. α-Convexity of Schlicht functions. Mathematical Notes of the Academy of Sciences of the USSR 11, 141–144 (1972). https://doi.org/10.1007/BF01097933

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  • DOI: https://doi.org/10.1007/BF01097933

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