On The Monotonicity And Convexity For Generalized Elliptic Integral Of The First Kind

In the theory of quasiconformal mapping,the Gauss hypergeometric function F(a,6;c;x)and the complete elliptic integral K(r)of the first kind play a very important role.In addition,they are also widely used in physics,engineering and other disciplines.Since zero-balanced hypergeometric function F(a,b;a+b;x)has a logarithmic singularity at x=1,Ramanujan gave the asymptotic formula of zero-balanced hypergeometric function near x=1.In the case a=b=1/2,Ramanujan’s asymptotic formula degenerates to K(r)～log(4/r’)(r → 1-).Inspired by the above asymptotic formula,many comparison inequalities between K(r)and log(4/r’)have been established,which also were extended to general hypergeometric functions.Obviously,log(1+4/r’)is closer to K(r)than log(4/r’)at r=0,and thereby it makes more sense to establish the inequalities between K(r)and log(1+4/r’).In this thesis,we mainly study the monotonicity,convexity and absolute monotonicity of the ratio(difference)of the generalized elliptic integral of the first kind Ka((?))and log(1+c/(?)),and establish several inequalities between Ka(r)and log(1+c/r’).The main innovation of this thesis is to deduce the recurrence formula for the power series coefficients of F2(a,b;c;x),and study the monotonicity and convexity of several complicated combination of Ka((?))and log(1+c/(?))by using the monotonicity rule of the ratio of two power series.This thesis consists of the following six chapters:In Chapter 1,we describe the research background and significance of this topic,and introduce the development history and research status of Gauss hypergeometric function and complete elliptic integrals.At the same time,we show the innovation of this thesis.In Chapter 2,we give the definition and some related properties of Gauss hypergeometric function and generalized elliptic integrals,and introduce an auxiliary function Hf,g and four technical lemmas,which are used to prove our main results.In Chapter 3,we study the monotonicity and convexity of the ratio of the generalized elliptic integral of the first kind Ka((?))and log(1+c/(?)),which generalizes some known results of complete elliptic integral of the first kind.In Chapter 4,we discuss the convexity and absolute monotonicity of the difference between the generalized elliptic integral of the first kind Ka((?))and log(1+c/(?)).However,the study of absolute monotonicity is only for the case of a=1/2,and the results have not been extended to the generalized elliptic integral of the first kind,but this is also an improvement of the known results.In Chapter 5,we obtain some sharp lower and upper bounds of the(generalized)complete elliptic integral of the first kind in terms of elementary functions,which improve previous known inequalities.In Chapter 6,we point out the problems that have not been solved,and give the corresponding ideas and conjectures based on the current conclusions.