Some applications of the generalized Libera integral operator
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 \textit{S. D. Bernardi} [Trans. Am. Math. Soc. 135, 429-446 (1969; Zbl 0172.097)]. In particular, the operator \(J_ 1\) was studied earlier by \textit{R. J. Libera} [Proc. Am. Math. Soc. 16, 755-758 (1965; Zbl 0158.077)] and \textit{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.
scientific article; zbMATH DE number 3933391