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Zbl 0772.30012
Nunokawa, Mamoru; Hoshino, Shinichi
One criterion for multivalent functions. (English)
[J] Proc. Japan Acad., Ser. A 67, No.2, 35-37 (1991). ISSN 0386-2194

It is known [cf. the first author, Proc. Japan Acad., Ser. A 65, No. 10, 326-328 (1989; Zbl 0705.30018)] that: If $f(z)=z\sp p+a\sb{p+1} z\sp{p+1}+\cdots+p\ge 2$ is analytic in $E=\{z: \vert z\vert<1\}$ and $$\text{Re }f\sp{(p)}(z)>-{\log(4/e)\over\log(e/2)} p!,\quad z\in E,$$ then $f$ is $p$-valent in $E$.\par In this note the following result is proved: Theorem. If $f(z)=z\sp p+a\sb{p+1} z\sp{p+1}+\cdots+p\ge 3$, is analytic in $E$ and $$\text{Re } f\sp{(p)}(z)>- {1-4(\log(4/e))(\log(e/2))\over 4(\log(4/e))(\log(e/2))} p!,\quad z\in E,$$ then $f$ is $p$-valent in $E$.
[J.Stankiewicz (Rzeszów)]

MSC 2000:: *30C45 Special classes of univalent and multivalent functions

Keywords: $p$-valent

Citations: Zbl 0705.30018

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