Some Properties Of The Gamma And Psi Functions

The gamma function r(x)psi function ?(x)and beta function B(x,y)play a very important role in many branches of mathematics,and are widely applied in other disciplines such as physics and engineering.For example,these special functions are essential in the studies of the analytic properties of the Gaussian hypergeometric functions and the Ramanu-jan R-function.Therefore,it is quite significant for us to study the properties of the special functions ?(x)and ?(x).In this thesis,we obtain some analytic:properties of ?(x)and ?(x),and improve some related known results for these functions,by studying the monotonicity,concavity and log-arithmic concavity properties of certain combinations defined in terms of ?(x),?(x)?'(x),?"(x)and some elementary functions.This thesis is divided into three chapters.In the first chapter,we introduce some concepts and notations required in this thesis,and state the background of our studies.In the second chapter,we reveal some properties of the gamma function.In the third chapter,we present some new properties of ?(x)and its derivatives by studying the monotonicity and concavity propersties of the certain combinations defined in terms of ?(x)and some elementary functions.