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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 (f)=((c+1)/z c ) 0 z t c-1 f(t)dt(c0)·

The operator J c , when cN={1,2,3,...}, 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(α,β) 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:
30C45Special classes of univalent and multivalent functions
33C05Classical hypergeometric functions, 2 F 1