AN APPLICATION OF A HYPERGEOMETRIC DISTRIBUTION SERIES ON CERTAIN ANALYTIC FUNCTIONS ASSOCIATEDBWITH INTEGRAL OPERATOR.
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.
Let A denote the class of functions f of the form
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
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
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.
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
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
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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|Date:||Aug 31, 2015|
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