Distortion theorems for a special class of analytic functions

Shaffer, Dorothy Browne

doi:10.1090/S0002-9939-1973-0315113-6

Distortion theorems for a special class of analytic functions
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by Dorothy Browne Shaffer

Proc. Amer. Math. Soc. 39 (1973), 281-287

DOI: https://doi.org/10.1090/S0002-9939-1973-0315113-6

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Abstract:

Sharp bounds are derived for the derivative of analytic functions of class ${P_\alpha }$ defined by the condition $|p(z) - 1/2\alpha | \leqq 1/2\alpha ,0 \leqq \alpha \leqq 1,p(0) = 1$. These results are improved for the class of functions with missing terms. Application is made to the class of functions with derivative $\in {P_\alpha }$ and the radius of convexity is determined for this class.

References

Funktionentheorie

Theory of functions of a complex variable

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G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970/71), 159–177. MR 267103, DOI 10.4064/ap-23-2-159-177
T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104 (1962), 532–537. MR 140674, DOI 10.1090/S0002-9947-1962-0140674-7
Thomas H. MacGregor, A class of univalent functions, Proc. Amer. Math. Soc. 15 (1964), 311–317. MR 158985, DOI 10.1090/S0002-9939-1964-0158985-5
Dorothy Browne Shaffer, On bounds for the derivative of analytic functions, Proc. Amer. Math. Soc. 37 (1973), 517–520. MR 310256, DOI 10.1090/S0002-9939-1973-0310256-5