# AN APPLICATION OF A HYPERGEOMETRIC DISTRIBUTION SERIES ON CERTAIN ANALYTIC FUNCTIONS ASSOCIATEDBWITH INTEGRAL OPERATOR.

Byline: Qaisar Mehmood, Waqas Nazeer and Absar Ul Haq

ABSTRACT

The main purpose of this current note is to introduce a Hypergeometric distribution series in associated with integral operator and obtain necessary and sufficient conditions for this integral related series belonging to the classes and T ( , ) and C( , ) .

Keywords-: Analytic function, Univalent function, hyper geometric distribution.

1. INTRODUCTION

Let A denote the class of functions f of the form

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which are analytic in the unit dics Eqs. and 1 D z z Eqs. and satisfy the normalization condition (0) (0) 1 f f Eqs. Further, we denote by S the subclass of A consisting of functions of the form (1.1) which are univelant in D and let T be the class of S consisting of the function of the form

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for all (0 1) Eqs.(0 1) Eqs. for all z D Eqs. . We also consider C Eqs. Eqs. be the subclass of T consisting of the functions satisfying the following condition

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Both Eqs. T Eqs. and C Eqs. are extensively studied by Altinates and Owa and certain conditions for hypergeometric function and generalized Bessel functionf for these classes were studied by Mostafa and Porwal and Dixit.

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It is worthy to note that (0, ) Eqs. T T Eqs. be the class starlike functions of order (0 1) Eqs. , and C(0, ) CEqs. Eqs. the class convex functions of order Eqs.. The hyper geometric distribution ( , N, ) f k m is defined

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Note: Here 0,1,2,...,min Eqs. n k m N k m n Eqs. and ( , N, ) 0 if minEqs. or . f k m n k m N k m n Eqs. We introduce a power series whose coffecients are probabiliteis of hyper geometric distribution

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Motivated by results on connection between various subclasses of analytic functions by using the hypergeometric function by many author particularly the authors (see- ) and generalized Bessel functions (see - ), S.Porrwal  obtained the necessary and sufficient conditions for a functions ( , ) F m z defined by using the poisson distribution belong to the class ( , ) T and ( , ) C . In this article, we give the analogous conditions an integral operator ( , , , ) H k N m z defined by the hypergeometric distribution belong to the ( , ) T and C .

To establish our main results, we will require the following lemmas according to Altintas and Owa .

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REFERENCES

 O. Altintas and S. Owa, On subclasses of univalent functions with negative Coefficients", Pusan Kyongnam Mathematical Journal, vol. 4, pp. 41-56, 1988.

 A. O. Mostafa, A study on starlike and convex properties for hypergeometric Functions", Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 3, article 87, pp. 1-16, 2009.

 S. Porwal and K. K. Dixit, An application of generalized Bessel functions oncertain analytic functions", Acta Universitatis Matthiae Belii. Series Mathematics, pp. 51-57, 2013.

 S. Porwal, An Application of a Poisson Distribution Series on Certain Analytic",Functions Journal of Complex Analysis, Volume 2014, Article ID 984135.

 H. Silverman, Univalent functions with negative coefficients", Proceedings of the American Mathematical Society, vol. 51, pp. 109-116, 1975.

 B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737-745, 1984.

 E. P. Merkes and W. T. Scott, Starlike hypergeometric functions, Proceedingsof the American Mathematical Society, vol. 12, pp. 885-888, 1961.

 S. Porwal and K. K. Dixit, An application of certain convolution operator involving hypergeometric functions, Journal of Rajasthan Academy of Physical Sciences, vol. 9, no. 2, pp. 173 -186, 2010.

 A. K. Sharma, S.Porwal, and K.K.Dixit, Classmappings properties of convolu-tions involving certain univalent functions associated with hypergeometric func-tions", Electronic Journal of Mathematical Analysis and Applications, vol. 1, no. 2, pp. 326- 333, 2013.

 A. Gangadharan, T. N. Shanmugam, and H. M. Srivastava, Generalized hyper- geometric functions associated with uniformly convex functions", Computers and Mathematics with Applications, vol. 44, no. 12, pp. 1515- 1526, 2002.

 A. Baricz, Generalized Bessel Functions of the First Kind, vol. 1994 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.

 S. R. Mondal and A. Swaminathan, Geometric properties of generalized Bessel Functions", Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no.1, pp. 179-194, 2012.

 S. Porwal, Mapping properties of generalized Bessel functions on some sub-Classes of univalent functions", Analele Universitatii Oradea Fasc. Matematica.