An analytical investigatıon on starlikeness and convexity properties for hypergeometric functions

In this study, we analytically investigate hypergeometric functions having some properties such as convexity and starlike. We fundamentally focus on obtaining desired conditions on the parameters \ a,b\ , and $c$ in order that a hypergeometric function to be in various subclasses of starlike and convex functions of order \ \alpha=2^{-r}\ and order \ \alpha=2^{-r}\ type $\beta=2^{-1}$, with $r$ is a positive integer.

Keywords:

Analytic function, univalent function, starlike function, convex function hypergeometric function, gamma function,

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[1] Carlson BC, Shaffer DB. Starlike and presatarlike hypergeometric functions. SIAM Journal on Mathematical Analysis 1984; 15: 737-745.
[2] Duren PL. Univalent functions. Berlin, Germany: Springer-Verlag, 1983.
[3] Eenigenburg PJ. A class of starlike mappings of the unit disk. Compositio Mathematica 1972; 24: 235-238.
[4] Jain PK, Gupta VP. Certain classes of univalent functions with negative coefficients. Bulletin of the Australian Mathematical Society 1976; 14: 409-416.
[5] Merkes E, Scott BT. Starlike hypergeometric functions. Proceedings of the American Mathematical Society 1961; 12: 885-888.
[6] Miller SS, Mocanu PT. Differential Subordinations Theory and Applications. New York, NY, USA: Marcel Dekker Inc., 2000.
[7] Miller SS, Mocanu PT. Univalence of Gaussiaan and Confluent Hypergeometric Functions. Proceedings of The American Mathematical Society 1990
[8] Mostafa AO. Starlikenees and convexity results for hypergeometric functions. Computers and Mathematics with Applications 2010; 59: 2821-2826.
[9] Robertson MS. On the theory of univalent functions. Annals of Mathematics 1936; 37: 374-408.
[10] Ruscheweyh ST, Singh V. On the order of starlikeness of hypergeometric functions. Journal of Mathematical Analysis and Applications 1986; 113: 1-11.
[11] Sakar FM,Aydoğan M. Subclass of m-quasiconformal harmonic functions in association with Janowski starlike functions. Applied Mathematics and Computation 2018; 319: 461-468.
[12] Sansone G, Gerretsen J. Lectures on the theory of functions of a complex variable. Groningen, the Netherlands: Noordhoff Publishing, 1969.
[13] Silverman H. Univalent functions with negative coefficients. Proceedings of the American Mathematical Society 1975; 15: 109-116.
[14] Silverman H. Subclasses of Starlike functions. Revue Roumaine de Mathématique Pures et Appliquées 1978; 23: 1093-1099.
[15] Silverman H. Starlike and convexity properties for hypergeometric functions. Journal of Mathematical Analysis and Applications 1993; 172: 574-581.
[16] Teubner BC. E. Study, Konforme Abbildung Einfachzusammenhangender Bereiche. Leipzig und Berlin, 1913 (in German).
[17] Wright DJ. On a class of starlike functions. Compositio Mathematica 1969; 21: 122-124.