Some Properties Of The Zero-Balanced Hypergeometric Functions

The Guassian hypergeometric function F(a,b;c;x),which is said to be zero-balanced if c=a+b,and its special case the complete elliptic integrals K(r)and K'(r)of the first kind play very important roles in the theory of special functions.Many other special functions are particular or limiting cases of F(a,b;a+b;x).The zero-balanced hypergeometric function F(a,b;a+b;x)has wide and important applications in many fields of mathematics such as quasiconformal theory,Ramanujan modular equation and number theory,and in physics and engineering as well.In the study of the zero-balanced hypergeometric functions,the monotonicity and concavity properties of the Ramanujan constant R(x,c-x)and its related special functions are essential.In this thesis,we reveal some new analytic properties of the zero-balanced hypergeomet-ric function,the complete elliptic integrals of the first kind,and the Ramanujan constant,from which some sharp inequalities follow,thus developing this field.This thesis is divided into three chapters.In Chapter 1,we introduce some related concepts,notations and several known results,state the developing history of the zero-balanced hypergeometric function and Ramanujan constant,and give the background of this thesis.In Chapter 2,we extend some known properties of R(x)to R(x,c-x)by studying the analytic properties of certain combinations defined in terms of R(x,c-x)and some elementary functions,and obtain some sharp upper and lower bounds of the R(x,c-x).In Chapter 3,we obtain several monotonicity properties and sharp inequalities for the zero-balanced hypergeometric function,by studying the analytic properties of certain combi-nations defined in terms of F(a,b;a+b;x)and some elementary functions such as trigonomet-ric functions,thus extending several known related results for the complete elliptic integrals of the first kind to zero-balanced hypergeometric functions.And then,we obtain some new sharp upper and lower bounds of the quotient of the zero-balanced hypergeometric functions F(a,b;+b;x2)/F(a,b;a+b;x)by studying the analytic properties of certain combinations defined in terms of F(a,b;a+b;x2)/F(a,b;a+b;x)and polynomials.In particular,H.Alzer and K.C.Richards' inequalities for the ratios of complete elliptic integrals are substantially improved,and a complete answer to M.E.H.Ismail's question is given.