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 \[ J_ c(f)=((c+1)/z^ c)\int^{z}_{0}t^{c- 1}f(t)dt\quad (c\geq 0). \] The operator \(J_ c\), when \(c\in N=\{1,2,3,\ldots \}\), 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(\(\alpha\),\(\beta)\) 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.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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