The Properties And Applications Of The Guassian Hypergeometric Function And Ramanujan R-function

The Gaussian hypergeometric function F(a,b;c;x)and its special case-zero-balanced hypergeometric function F(a,b;a+b;x)play extremely important roles in the theory of special functions.The Gaussian hypergeometric function has wide and important applications in geometric function theory,quasiconformal theory and number theory in mathematics and many other fields and subjects such as physics,engineering technology as well.For example,the solution of Legendre function,Jacobi polynomials,special spherical polynomials,Chebyshev polynomials and other equations can be expressed by the Gaussian hypergeometric functions.It has close relations with many other types of special functions,and many special functions of mathematics and physics are particular or limiting cases of F(a,b;c;x).In addition,the study of the analytical properties of Ramanujan R-function R(a,b)and its related special functions can not only improve the known results of R(a,b),but also promote the study of the Gaussian hypergeometric functions and special functions in some other fields.In this thesis,we study and reveal some new analytic properties of the zero-balanced Gaussian hypergeometric function F(a,b;a+b;x),Ramanujan R-function R(a,b),complete p-elliptic integrals and generalized Gr(?)tzsch ring function ?a,b(r)and generalized Hübner upper bound function ma,b(r)+log r,from which some sharp inequalities follow.This thesis is divided into five chapters.In Chapter 1,we introduce some related concepts,notations and definitions,and state the development history,research status and some known results of the zero-balanced Gauss hypergeometric function,Ramanujan R-function,complete p-elliptic integrals,generalized Gr(?)tzsch ring function and generalized Hübner upper bound function.Then,we give the research significance and main research content of this thesis.In Chapter 2,we get some new properties of the zero-balance Gauss hypergeometric function F(a,b;a+b;x)by studying the analytic properties of certain combinations defined by F(a,b;a+b;x)and some elementary functions,and obtain some sharp upper and lower bounds of F(a,b;a+b;x).And according to its special form,we can get some known results of complete elliptic integral k(r)and the generalized elliptic integral ka(r).In chapter 3,we reveal the analytic properties of Ramanujan R-function R(a)which defined with Beta function B(a)and some elementary functions.We improve the known results and obtain some sharp upper and lower bounds.In Chapter 4,we apply some results of the zero-balanced Gaussian hypergeometric function F(a,b;a+b;x)to generalized Gr(?)tzsch ring function ?a,b(r)and generalized Hübner upper bound function ma,b(r)+log r,and obtain some related properties.In Chapter 5,we apply some results of the zero-balanced Gaussian hypergeometric function F(a,b;a+b;x)to complete p-elliptic integral kp(r),and obtain some sharp upper and lower bounds of kp(r)expressed by elementary functions.