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Some applications of the generalized Libera integral operator. (English) Zbl 0583.30016

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 ${J}_{c}$ by

${J}_{c}\left(f\right)=\left(\left(c+1\right)/{z}^{c}\right){\int }_{0}^{z}{t}^{c-1}f\left(t\right)dt\phantom{\rule{1.em}{0ex}}\left(c\ge 0\right)·$

The operator ${J}_{c}$, when $c\in N=\left\{1,2,3,...\right\}$, was introduced by S. D. Bernardi [Trans. Am. Math. Soc. 135, 429-446 (1969; Zbl 0172.097)]. In particular, the operator ${J}_{1}$ was studied earlier by R. J. Libera [Proc. Am. Math. Soc. 16, 755-758 (1965; Zbl 0158.077)] and A. E. Livingston [Proc. Am. Math. Soc. 17, 352-357 (1966; Zbl 0158.077)].

The object of the present paper is to prove several interesting characterization theorems involving the generalized Libera integral operator ${J}_{c}$ and a general class C($\alpha$,$\beta \right)$ of close-to- convex functions in the unit disc. An application of the integral operator ${J}_{c}$ to a class of generalized hypergeometric functions is also considered.

##### MSC:
 30C45 Special classes of univalent and multivalent functions 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$
##### Keywords:
Libera integral operator; close-to-convex functions