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
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, 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.
CHICAGO
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, 39 2 (2013): 259–269,
VANCOUVER
Chu Y., Hou S., Xia W. Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means. BIMS. 2013;39(2):259–269.
