A note on some inequalities for means. (English) Zbl 0693.26005

The logarithmic and identric means of two positive numbers a and b are defined by \(L=L(a,b):=(b-a)/(\ln b-\ln a)\) for \(a\neq b;\quad L(a,a)=a,\) and \(I=I(a,b):=\frac{1}{e}(b^ b/a^ a)^{1/(b-a)}\) for \(a\neq b,\quad I(a,a)=a,\) respectively. Let \(A=A(a,b):=(a+b)/2\) and \(G=G(a,b):=\sqrt{ab}\) denote the arithmetic and geometric means of a and b, respectively. Recently, in two interesting papers, H. Alzer has obtained the following inequalities: \((1)\quad A.G<L.I\) and \(L+I<A+G;\quad (2)\quad \sqrt{G.I}<L<\frac{1}{2}(G+I)\) which hold true for all positive \(a\neq b.\) In our paper we prove, by using new methods, that the left side of (1) is weaker than the left side of (2), while the right side of (1) is stronger than the right side of (2).
Reviewer: J.Sándor


26D15 Inequalities for sums, series and integrals
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[1] H. Alzer, Ungleichungen f?r Mittelwerte. Arch. Math.47, 422-426 (1986). · Zbl 0585.26014
[2] H. Alzer, Two inequalities for means. C.R. Math. Rep. Acad. Sci. Canada9, 11-16 (1987). · Zbl 0615.26015
[3] B. C. Carlson, The logarithmic mean. Amer. Math. Monthly79, 615-618 (1972). · Zbl 0241.33001
[4] K. S.Kunz, Numerical analysis. New York 1957. · Zbl 0079.33601
[5] E. B. Leach andM. C. Sholander, Extended mean values II. J. Math. Anal. Appl.92, 207-223 (1983). · Zbl 0517.26007
[6] T. P. Lin, The power mean and the logarithmic mean. Amer. Math. Monthly81, 879-883 (1974). · Zbl 0292.26015
[7] J. S?ndor, Some integral inequalities. Elem. Math.43, 177-180 (1988).
[8] J.S?ndor, Inequalities for means. Submitted.
[9] H. J. Seiffert, Eine Integralungleichung f?r streng monotone Funktionen mit logarithmisch konvexer Umkehrfunktion. Elem. Math.44, 16-18 (1989). · Zbl 0721.26010
[10] K. B. Stolarsky, The power and generalized means. Amer. Math. Monthly.87, 545-548 (1980). · Zbl 0455.26008
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