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Some characterizations of univalent, starlike and convex hypergeometric functions. (English) Zbl 0567.30010
E. P. Merkes and W. T. Scott [Proc. Am. Math. Soc. 12, 885- 888 (1961; Zbl 0102.065)] proved an interesting result characterizing starlike hypergeometric functions. More recently, B. C. Carlson and D. B. Shaffer [SIAM J. Math. Anal. 15, 737-745 (1984; reviewed above)] introduced a linear operator and demonstrated how this operator can be applied systematically to the study of various interesting classes of starlike, convex, and prestarlike hypergeometric functions. The object of the present paper is to establish several characterization theorems involving univalent, starlike, and convex hypergeometric functions in the unit disk. Various inequalities involving hypergeometric series are also proved.

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