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Zbl 0583.30016
Owa, Shigeyoshi; Srivastava, H.M.
Some applications of the generalized Libera integral operator.
(English)
[J] Proc. Japan Acad., Ser. A 62, 125-128 (1986). ISSN 0386-2194

For a function f(z) belonging to a class A of normalized analytic functions in the unit disc, we define the generalized Libera integral operator $J\sb c$ by $$J\sb c(f)=((c+1)/z\sp c)\int\sp{z}\sb{0}t\sp{c- 1}f(t)dt\quad (c\ge 0).$$ The operator $J\sb c$, when $c\in N=\{1,2,3,\ldots \}$, was introduced by {\it S. D. Bernardi} [Trans. Am. Math. Soc. 135, 429-446 (1969; Zbl 0172.097)]. In particular, the operator $J\sb 1$ was studied earlier by {\it R. J. Libera} [Proc. Am. Math. Soc. 16, 755-758 (1965; Zbl 0158.077)] and {\it A. E. Livingston} [Proc. Am. Math. Soc. 17, 352-357 (1966; Zbl 0158.077)]. \par The object of the present paper is to prove several interesting characterization theorems involving the generalized Libera integral operator $J\sb c$ and a general class C($\alpha$,$\beta)$ of close-to- convex functions in the unit disc. An application of the integral operator $J\sb c$ to a class of generalized hypergeometric functions is also considered.
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
33C05 Classical hypergeometric functions

Keywords: Libera integral operator; close-to-convex functions

Citations: Zbl 0172.097; Zbl 0158.077

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