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Zbl 0767.30014
Altintas, Osman; Owa, Shigeyoshi
Majorizations and quasi-subordinations for certain analytic functions.
(English)
[J] Proc. Japan Acad., Ser. A 68, No.7, 181-185 (1992). ISSN 0386-2194

This paper contains two theorems about functions analytic in the open unit disk $\Delta$. The first theorem yields a value of $r$ depending on $\alpha$ and $\beta$ such that the condition $\vert f(z)\vert\leq\vert g(z)\vert$ for $\vert z\vert<1$ implies $\vert f'(z)\vert\leq\vert g'(z)\vert$ for $\vert z\vert\leq r$. Here it is assumed that $f(z)=a\sb 1 z-\sum\sb{n=2}\sp \infty a\sb nz\sp n$ where $a\sb 1\ne 0$ and $a\sb n\geq 0$, $h(z)={{zg'(z)} \over {g(z)}}=1- \sum\sb{n=1}\sp \infty c\sb n z\sp n$ where $c\sb n\geq 0$ and $\text{Re}\{h(z)+\alpha zh'(z)\}>\beta$ for $\vert z\vert<1$ ($\text{Re }\alpha\geq 0$ and $0\leq\beta\leq 1$).\par The second theorem asserts that if $f(z)=z+\sum\sb{n=2}\sp \infty a\sb n z\sp n$ is quasi-subordinate to $g$ and $\text{Re}\bigl\{{\sp p\sqrt{{{g(z)} \over {s(z)}}}}>{1\over 2}\bigr\}$ for $\vert z\vert<1$ where $s$ is a normalized univalent function, then $\vert a\sb n\vert\leq {{(p+n)!} \over {(p+1)!(n-1)!}}$ for $n\geq 2$. Here $p\geq 1$, and the function $f(z)={z\over {(1-z)\sp{p+2}}}$ exhibits sharpness for the coefficient estimate.
[T.H.MacGregor (Albany)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
30C80 Maximum principle, etc. (one complex variable)
30C50 Coefficient problems for univalent and multivalent functions

Keywords: majorizations; quasi-subordinate

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