We find the greatest values $alpha_{1} $ and $alpha_{2} $, and the least values $beta_{1} $ and $beta_{2} $ such that the inequalities $alpha_{1} C(a,b)+(1-alpha_{1} )H(a,b)<L(a,b)<beta_{1} C(a,b)+(1-beta_{1} )H(a,b)$ and $alpha_{2} C(a,b)+(1-alpha_{2}) H(a,b)<I(a,b)<beta_{2} C(a,b)+(1-beta_{2} )H(a,b)$ hold for all $a,b>0$ with $aneq b$. Here, $C(a,b)$, $H(a,b)$, $L(a,b)$, and $I(a,b)$ are the centroidal, harmonic, logarithmic, and identric means of two positive numbers $a$ and $b$, respectively.
Chu, Y., Hou, S., & Xia, W. (2013). Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means. Bulletin of the Iranian Mathematical Society, 39(2), 259-269.
MLA
Y. Chu; S. Hou; W. Xia. "Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means". Bulletin of the Iranian Mathematical Society, 39, 2, 2013, 259-269.
HARVARD
Chu, Y., Hou, S., Xia, W. (2013). 'Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means', Bulletin of the Iranian Mathematical Society, 39(2), pp. 259-269.
VANCOUVER
Chu, Y., Hou, S., Xia, W. Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means. Bulletin of the Iranian Mathematical Society, 2013; 39(2): 259-269.