Logarithmic convexity of extended mean values

THE FUNCTION (bx − ax)/x: LOGARITHMIC CONVEXITY AND APPLICATIONS TO EXTENDED MEAN VALUES

In the present paper, we first prove the logarithmic convexity of the elementary function b x −a x x , where x 6= 0 and b > a > 0. Basing on this, we then provide a simple proof for Schur-convex properties of the extended mean values, and, finally, discover some convexity related to the extended mean values.

Necessary and Sufficient Conditions for the Schur Harmonic Convexity or Concavity of the Extended Mean Values

In this paper, we prove that the extended values E(r, s;x, y) are Schur harmonic convex (or concave, respectively) with respect to (x, y) ∈ (0,∞) × (0,∞) if and only if (r, s) ∈ {(r, s) : s ≥ −1, s ≥ r, s+ r + 3 ≥ 0} ∪ {(r, s) : r ≥ −1, r ≥ s, s+r+3 ≥ 0} (or {(r, s) : s ≤ −1, r ≤ −1, s+r+3 ≤ 0}, respectively).

Logarithmic Convexity of Gini Means

where x and y are positive variables and r and s are real variables. They are also called sum mean values. There has been a lot of literature such as [3, 4, 5, 6, 9, 10, 11, 12, 13, 19, 20, 21] and the related references therein about inequalities and properties of Gini means. The aim of this paper is to prove the monotonicity and logarithmic convexity of Gini means G(r, s;x, y) and related fun...

Optimising heat exchanger network synthesis using convexity properties of the logarithmic mean temperature difference

A New Proof of Monotonicity for Extended Mean Values

In this article, a new proof of monotonicity for extended mean values is given.