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On the radius of $ \beta $-convexity of starlike functions of order $ \alpha $


Author: Hassoon S. Al-Amiri
Journal: Proc. Amer. Math. Soc. 39 (1973), 101-109
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9939-1973-0311883-1
MathSciNet review: 0311883
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Abstract: A function $ f(z) = z + {a_2}{z^2} + \cdots $ is called $ \beta $-convex if $ f(z)f'(z)/z \ne 0$ in $ D:\vert z\vert < 1$ and if

$\displaystyle \operatorname{Re} \{ (1 - \beta )zf'(z)/f(z) + \beta (1 + zf''(z)/f'(z))\} > 0$

for some $ \beta \geqq 0$ and all $ z$ in $ D$. Recently M. O. Reade and P. T. Mocanu have announced a sharp result about the radius of $ \beta $-convexity for starlike functions. The author generalizes this result to starlike functions of order $ \alpha $.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0311883-1
Keywords: Univalent functions, convex and starlike functions of order $ \alpha $, $ \beta $-convex functions, radius of convexity, radius of $ \beta $-convexity, extremal functions
Article copyright: © Copyright 1973 American Mathematical Society

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