Sharp Bounds For Power Mean In Terms Of Generalized Heronian Mean

Various means and their sharp estimates have many applications in physics, me-chanics and engineering. In particular, many remarkable inequalities for Heronian andgeneralized Heronian means have been found by many mathematicians. In this paper, wecontinue to study the inequalities on generalized Heronian means. We find the optimalvalues forα,β,γandτ, such that the inequalitiesH_α(a, b) < A_γ(a, b) < H_β(a, b),H_η(a, b) < P (a, b) < H_ζ(a, b)andH_γ(a, b) < L(a, b) < H_τ(a, b)hold for all a, b > 0 with a≠b. Here, H_ω(a, b), A_γ(a, b), P (a, b) and L(a, b) be thegeneralized Heronian, power, Seiffert and logarithmic means of two positive numbers aand b, respectively.