## 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.

### MSC:

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

### Keywords:

Libera integral operator; close-to-convex functions

### Citations:

Zbl 0172.097; Zbl 0158.077
Full Text:

### References:

 [1] S. D. Bernardi: Convex and starlike univalent functions. Trans. Amer. Math. Soc, 135, 429-446 (1969). · Zbl 0172.09703 [2] I. S. Jack: Functions starlike and convex of order a. J. London Math. Soc, (2)3, 469-474 (1971). · Zbl 0224.30026 [3] R. J. Libera: Some classes of regular univalent functions. Proc. Amer. Math. Soc, 16, 755-758 (1965). JSTOR: · Zbl 0158.07702 [4] A. E. Livingston: On the radius of univalence of certain analytic functions, ibid., 17, 352-357 (1966). JSTOR: · Zbl 0158.07701 [5] T. H. MacGregor: The radius of convexity for starlike functions of order J. ibid., 12, 885-888 (1961). · Zbl 0113.05505 [6] S. S. Miller and P. T. Mocanu: Second order differential inequalities in the complex plane. J. Math. Anal. Appl., 65, 289-305 (1978). · Zbl 0367.34005 [7] B. Pinchuk: On starlike and convex functions of order a. Duke Math. J., 35, 721-734 (1968). · Zbl 0167.36101 [8] M. S. Robertson: On the theory of univalent functions. Ann. of Math., 37, 374- 408 (1936). JSTOR: · Zbl 0014.16505 [9] A. Schild: On starlike functions of order a. Amer. J. Math., 87, 65-70 (1965). JSTOR: · Zbl 0163.09901
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