Proof of One Optimal Inequality for Generalized Logarithmic, Arithmetic, and Geometric Means

A new interpretation of Jensen's inequality and geometric properties of ϕ-means

“…So, by (4) For the case of ϕ, ψ ∈ C − sm (I), since −ϕ, −ψ ∈ C + sm (I), it follows from the above discussion that a function s → x∇ t,−ξ s y is strictly monotone increasing on [0, 1]. However, by Lemma 3-(i), x∇ t,−ξ s y = x∇ t,ξ s y, where t (0, 1) and x, y I with x ≠ y, and then we obtain the desired result.…”

“…Remark. It seems that Theorem 3 is slightly related to [3,4] which discuss a comparison between a convex linear combination of the arithmetic and geometric means and the generalized logarithmic mean.…”

Sharp Bounds for the Weighted Geometric Mean of the First Seiffert and Logarithmic Means in terms of Weighted Generalized Heronian Mean

“…If we show ( , ) < 0 for , ∈ (0, 1), then ( ) = lim → 1 − ( , ) will be the best function in (8). Simple computations lead to ( , ) < 0 which is equivalent to…”

“…Recently, means has been the subject of intensive research. In particular, many remarkable inequalities for the Seiffert, logarithmic, and Heronian mean can be found in the literature [1][2][3][4][5][6][7][8][9][10][11]. In the paper [1], authors proved the following optimal inequalities: Let > 0, > 0, ̸ = then ( , ) < ( , ) < ( , ) for ≥ − 2, ≤ 1, = − 2, = 1 are the best constants.…”

“…The purpose of this paper is to find the optimal functions. For some other details about means, see [1][2][3][4][5][6][7][8][9][10][11] and the related references cited there in.…”