Majorization-subordination theorems for locally univalent functions. III
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- by Douglas Michael Campbell
- Trans. Amer. Math. Soc. 198 (1974), 297-306
- DOI: https://doi.org/10.1090/S0002-9947-1974-0349987-5
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Abstract:
A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to ${\mathfrak {n}_\alpha }$, the linear invariant family of locally univalent functions of finite order $\alpha$. If $f(z)$ is subordinate to $F(z)$ in the open unit disc, $f’(0) \geqslant 0$, and $F(z)$ is in ${\mathfrak {n}_\alpha },1.65 \leqslant \alpha < \infty$, then $f’(z)$ is majorized by $F’(z)$ in $|z| \leqslant (\alpha + 1) - {({\alpha ^2} + 2\alpha )^{1/2}}$. The result is sharp.References
- Douglas M. Campbell, Majorization-subordination theorems for locally univalent functions, Bull. Amer. Math. Soc. 78 (1972), 535–538. MR 299769, DOI 10.1090/S0002-9904-1972-12987-2
- Douglas Michael Campbell, Majorization-subordination theorems for locally univalent functions. II, Canadian J. Math. 25 (1973), 420–425. MR 315120, DOI 10.4153/CJM-1973-042-6
- 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
- Christian Pommerenke, Linear-invariante Familien analytischer Funktionen I, Math. Ann. 155 (1964), no. 2, 108–154 (German). For a review of this paper see [MR0168751]. MR 1513275, DOI 10.1007/BF01344077
- Tao-Shing Shah, On the radius of superiority in subordination, Sci. Record (N.S.) 1 (1957), 329–333. MR 100095
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 198 (1974), 297-306
- MSC: Primary 30A42
- DOI: https://doi.org/10.1090/S0002-9947-1974-0349987-5
- MathSciNet review: 0349987