Optimal convex combinations bounds of centrodial and harmonic means for logarithmic and identric means

we find the greatest values alpha1alpha1 and alpha2alpha2, and the least values beta1beta1 and beta2beta2 such that the inequalities alpha1C(a,b)+(1−alpha1)H(a,b)0a,b>0 with aneqbaneqb. Here, C(a,b)C(a,b), H(a,b)H(a,b), L(a,b)L(a,b), and I(a,b)I(a,b) are the centroidal, harmonic, logarithmic, and identric means of two positive numbers aa and bb, respectively.