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)

In this paper, optimal convex combination bounds of centroidal and harmonic means for weighted geometric mean of logarithmic and identric means are proved. We find the greatest value λ(α) and the least value Δ(α) for each α ∈ (0,1) such that the double inequality: λC(a,b)+(1−λ)H(a,b) < Lα (a,b)I1−α (a,b) < ΔC(a,b)+(1−Δ)H(a,b) holds for all a,b > 0 with a = b. Here, C(a,b), H(a,b) , L(a,b) and I...

In the paper, the authors survey integral representations (including the Lévy– Khintchine representations) and applications of some bivariate means (including the logarithmic mean, the identric mean, Stolarsky’s mean, the harmonic mean, the (weighted) geometric means and their reciprocals, and the Toader–Qi mean) and the multivariate (weighted) geometric means and their reciprocals, derive inte...

Let us consider the logarithmic mean L; the identric mean I; the trigonometric means P and T de…ned by H. J. Sei¤ert, the hyperbolic mean M de…ned by E. Neuman and J. Sándor, and the Gini mean J . The optimal estimations of these means by power means Ap and also some of the optimal estimations by Lehmer means Lq are known. We prove two new optimal estimation by Lehmer means L0 < M < L1=6 and L1...