Inequalities for compound mean iterations with logarithmic asymptotes
For \(x,y>0\) let \(A(x,y)= (x+y)/2\), \(G(x,y)=\sqrt{xy}\) be the arithmetic and geometric mean of \(x\) and \(y\), respectively. Then the arithmetic- geometric mean of \(x,y>0\) is \(AG(x,y)=\lim x_ n= \lim y_ n\), where \(x_ 1=x\), \(y_ 1=y\), \(x_{n+1}= A(x_ n,y_ n)\), \(y_{n+1}=G(x_ n,y_ n)\). Furthermore, let \(L(x,y)= {{x-y} \over {\log(x/y)}}\) be the logarithmic mean for \(x,y>0\), \(x\neq y\). For \(t>0\) and \(M\in \{A,G,AG,L\}\) let \(M_ t(x,y)= M(x^ t,y^ t)^{1/t}\). Then \(M_ t\leq M_ s\) for \(0<t\leq s\) for these \(M\). It is clear that \(G(x,y)\leq AG(x,y)\leq A(x,y)\), and it was shown by \textit{B. C. Carlson} that \(G(x,y)\leq L(x,y)\leq AG(x,y)\) (1972, 1991). \textit{M. K. Vamanamurthy} and the reviewer [Inequalities for means, J. Math. Anal. Appl. (to appear)] have proved several inequalities between pairs of mean values, such as \[ AG(x,y)\leq \min\{L_ 2(x,y), {\textstyle{{\pi\over 2}}} L(x,y), A_{1/2}(x,y)\}. \] On the basis of computer experiments, we have conjectured that (1) \(AG(x,y)\leq L_{3/2}(x,y)\). The present authors prove (1) among the many new inequalities they establish between pairs of mean values. The authors point out that \(L_{3/2}(1,x)- AG(1,x)\) is very close to 0 for \(x\) close to 1 and they use sophisticated symbolic computation techniques to prove (1). They also study the asymptotic behavior of some other \(AG\)-type means and prove inequalities for such means. For the hypergeometric function \(F(a,b;c;x)\) they prove (their formula (2.9)) that (2) \(F(1/2,1/2;1;1-x^ c) \leq F(1/2- \delta,1/2+\delta; 1-x^ d)\) for \(c=2\), \(d=3\), \(\delta=1/6\), and \(0<x<1\). Let us conclude by the remark that (2) has been extended to the case \(0<x<1\), \(c,d\in(0,\infty)\), \(4c<\pi d\), \(\delta \in (0,\delta_ 0)\) for an explicit \(\delta_ 0>0\) in the paper by \textit{G. D. Anderson}, \textit{R. W. Barnard}, \textit{R. C. Richards}, \textit{M. K. Vamanamurthy} and the reviewer [Inequalities for zero-balanced hypergeometric functions, Proc. Am. Math. Soc. (to appear)].
scientific article; zbMATH DE number 480462