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