On the radius of 𝛽-convexity of starlike functions of order 𝛼

Al-Amiri, Hassoon S.

doi:10.1090/S0002-9939-1973-0311883-1

On the radius of $\beta$-convexity of starlike functions of order $\alpha$
HTML articles powered by AMS MathViewer

by Hassoon S. Al-Amiri PDF

Proc. Amer. Math. Soc. 39 (1973), 101-109 Request permission

Abstract:

A function $f(z) = z + {a_2}{z^2} + \cdots$ is called $\beta$-convex if $f(z)f’(z)/z \ne 0$ in $D:|z| < 1$ and if \[ \operatorname {Re} \{ (1 - \beta )zf’(z)/f(z) + \beta (1 + zf''(z)/f’(z))\} > 0\] 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 $\alpha$.

References

Funktionentheorie

Theory of functions of a complex variable

12

16

Petru T. Mocanu, Une propriété de convexité généralisée dans la théorie de la représentation conforme, Mathematica (Cluj) 11(34) (1969), 127–133 (French). MR 273000

The order of starlikeness of certain univalent functions

18

The radius of $\alpha$-convexity of starlike functions

19

M. S. Robertson, Variational methods for functions with positive real part, Trans. Amer. Math. Soc. 102 (1962), 82–93. MR 133454, DOI 10.1090/S0002-9947-1962-0133454-X
Albert Schild, On starlike functions of order $\alpha$, Amer. J. Math. 87 (1965), 65–70. MR 174729, DOI 10.2307/2373224
V. A. Zmorovič, On bounds of convexity for starlike functions of order $\alpha$ in the circle $| z| <1$ and in the circular region $0<| z| <1$, Mat. Sb. (N.S.) 68 (110) (1965), 518–526 (Russian). MR 0197712