Properties Of Certain Combinations In Terms Of R(x) And Some Elementary Functions

It is well known that gamma function ?(x),its logarithmic derivative ?(x),the beta func-tion B(x),Ramanujan constant R(x)and Gaussian hypergeometric function have extensive and important applications in number theory,quasiconformal theory,geometry and many other fields of mathematics,as well as in some other subjects and engineering.In the study of zero balance Gauss hypergeometric function and modular equations,the monotonicity concavity of Ranmanujan constant R(x)is essential.In this thesis,we study the analyt-ic properties of ?(x)and ?(x),show the monotonicity and convexity properties of certain combinations defined in terms of the Ramanujan constant R(x)and the beta function B(x),and obtain asymptotically sharp lower and upper bounds for R(x)-B(x),thus presenting comparisons of the values between R(x)and B(x)and improving some known related results for R(x).This thesis is divided into three chapters.In the first chapter,we introduce some definitions,notation and several known results of the gamma function ?(x),psi function ?(x),the beta function B(x)and Ramanujan constant R(x).In the second chapter,we study some monotonicity and concavity properties of gamma function ?(x),psi function ?(x),and obtain several new results of psi function ?(x).In the third chapter,we present some analytic properties of Ramanujan constant R(x),show some monotonicity and concavity properties of certain combinations defined in term of R(x)and some elementary functions,thus improving some known inequalities for R(x),and obtain asymptotically sharp lower and upper bounds for R(x)-B(x).