Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 0754.30009
Owa, Shigeyoshi; Nishimoto, Katsuyuki; Lee, Sang Keun; Cho, Nak Eun
A note on certain fractional operator.
(English)
[J] Bull. Calcutta Math. Soc. 83, No.2, 87-90 (1991). ISSN 0008-0659

Let $S$ denote the class of functions $f(z)=z+\sum\sp \infty\sb{n=2} a\sb n z\sp n$ which are analytic and univalent in the unit disk $U=\{z\mid\vert z\vert<1\}$. Let $n$ be a non-negative integer, and let $\alpha$ be such that $0\le \alpha <1$. For $f(z)\in S$, the authors consider $D\sp{n+\alpha}\sb z f(z)$, the fractional derivative of order $n+\alpha$, and conjecture that $$\vert D\sp{n+\alpha}\sb z f(z)\vert\le{\Gamma(n+1+\alpha)(n+\alpha+\vert z\vert)\over (1-\vert z\vert)\sp{n+2+\alpha}}$$ for $z\in U$, with equality when $f(z)$ is the Koebe function $k(z)={z\over (1-z)\sp 2}$ $(z\in U)$. If true, this will generalize an earlier result of Landau (1926), namely: $$\vert f\sp{(n)}(z)\vert\le{n!(n+\vert z\vert)\over (1-\vert z\vert)\sp{n+2}}$$ for $z\in U$, $n=1,2,3,\dots$, and $f(z)\in S$.
[M.T.McGregor (Swansea)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
26A24 Differentiation of functions of one real variable

Keywords: analytic; univalent; fractional derivative; Koebe function

Cited in: Zbl 0863.30013

Highlights
Master Server