Analytic Function Classes Involving The Generalized Hypergeometric Functions

As we all know,generalized hypergeometric functions play an important role in the theory of geometric functions,as an important tool especially in solving the famous Bieberbach conjecture by L.de Branges.Based on this,the properties of various hypergeometric functions are studied by many scholars.As early as the eighties and nineties of last century,many scholars began to study all kinds of analytic function classes related to generalized hypergeometric functions.In 1985,L.de Branges proved Bieberbach’s conjecture by using hypergeometric functions,and set off a research upsurge of function classes related to generalized hypergeometric functions;In 1990,S.S.Miller and P.T.Mocanu studied the local univalent,starlikeness and convex properties by using differential subordination method;In 1999,J.Dziok and H.M.Srivastava studied some properties of negative coefficients analytic functions,such as coefficient estimation,distortion theorem,etc.In recent years,many scholars also use differential subordination to define function classes,and study the related properties of these new function subclasses.In 2003,J.Dziok and H.M.Srivastava used Montel normalization to study the coefficient estimation,distortion theorem,convex radius and starlikeness radius of certain subclasses of analytic functions;In 2007,J.Patel,A.K.Mishra and H.M.Srivastava studied the class of multivalent analytic functions described by Dziok-Srivastava operator,and gave the inclusion relation,real part estimation,module estimation,radial angle estimation,sufficient conditions for functions in and generalized neighborhoods of function classes,etc.;In 2009,M.K.Aouf and H.E.Darwiah studied the distortion theorem,extreme point,coefficient estimation,convex radius and close-to-convex radius,modified Hadamard convolution and other properties of multivalent analytic function classes;In the same year,Z.G.Wang,Y.P.Jiang and H.M.Srivastava studied the inclusion relation and integral representation of some subclases of meromorphic multivalent functions and convolution property.Inspired by the above,we obtain some sufficient conditions and properties of generalized hypergeometric functions in some classes of analytic functions by using differential subordination in this paper.The full text is divided into four parts:In the first part,we introduce hypergeometric functions,differential subordinations and multiple analytic function classes.In the second part,we study the sufficient conditions and the third order differential subordination of a transformation(called generalized Bessel function),which is a special generalized hypergeometric function.In the third part,we obtain the sufficient conditions and other properties of analytic functions.In the forth part,we investigate the extremum problem,coefficient estimation,close-to-convex and convex radius of the class of analytic functions defined by Dziok-Srivastava operator.