Some Roperties Of The Psi Function And Omplete Elliptic Integrals

In this thesis,the author studies the analytic properties of the gamma and psi functions and complete elliptic integrals,by which some sharp inequalities and improvements on some related known results are attained for these functions.It is well known that the Gaussian hypergeometric functions,complete elliptic integrals and generalized elliptic integrals play a very important role in geometric function theory and quasiconformal theory,as well as in some other fields of mathematics,physics and engineering.Many properties of Gaussian hypergeometric functions are related to the gamma and psi functions.Therefore,it is quite significant for us to study the properties of the special functions above-mentioned.The thesis consists of the following three chapters.In chapter 1,we give the background of this thesis,and introduce some needed concepts,notations and related known results.In chapter 2,we give a simple proof of the following Anderson and Qiu’s conjecture:The function x→[log Γ(x +1)]/(x log x)is strictly concave on(1,∞).Moreover,we present some monotonicity properties of certain combinations defined in terms of ψ(n)and elementary functions,from which some lower and upper bounds of the series Σk=1∞(nk-x)(x+k)-n-2 for x ∈(0,oo)follow.In chapter 3,we obtain some monotonicity properties of the ratio κ(r)/κ(√r)of com-plete elliptic integrals of the first kind,and establish a double inequality for κ(r)/κ(√r),thus substantially improving the result recently proved by H.Alzer and K.Richards.In ad-dition,we extend the known monotonicity property of the function r→4√1+xκ(r)/κ(√r)for r ∈(0,1)to the generalized elliptic integral of the first kind.