Chu, Y., Shi, M., Jiang, Y. (2012). Optimal inequalities for the power, harmonic and logarithmic means. Bulletin of the Iranian Mathematical Society, 38(3), 597-606.

Yuming Chu; Mingyu Shi; Yueping Jiang. "Optimal inequalities for the power, harmonic and logarithmic means". Bulletin of the Iranian Mathematical Society, 38, 3, 2012, 597-606.

Chu, Y., Shi, M., Jiang, Y. (2012). 'Optimal inequalities for the power, harmonic and logarithmic means', Bulletin of the Iranian Mathematical Society, 38(3), pp. 597-606.

Chu, Y., Shi, M., Jiang, Y. Optimal inequalities for the power, harmonic and logarithmic means. Bulletin of the Iranian Mathematical Society, 2012; 38(3): 597-606.

Optimal inequalities for the power, harmonic and logarithmic means

Receive Date: 16 October 2009,
Revise Date: 12 March 2011,
Accept Date: 12 March 2011

Abstract

For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers $a$ and $b$, respectively.