Logarithmic convexity of Gini means

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...

Schur-convexity and Schur-geometrically concavity of Gini means

The monotonicity and the Schur-convexity with parameters (s, t) in R2 for fixed (x, y) and the Schur-convexity and the Schur-geometrically convexity with variables (x, y) in R++ for fixed (s, t) of Gini mean G(r, s;x, y) are discussed. Some new inequalities are obtained.

Logarithmic Convexity of Area Integral Means for Analytic Functions Ii

For 0 < p < ∞ and −2 ≤ α ≤ 0 we show that the L integral mean on rD of an analytic function in the unit disk D with respect to the weighted area measure (1−|z|) dA(z) is a logarithmically convex function of r on (0, 1).

Logarithmic Convexity of Area Integral Means for Analytic Functions

We show that the L integral mean on rD of an analytic function in the unit disk D with respect to the weighted area measure (1 − |z|) dA(z), where −3 ≤ α ≤ 0, is a logarithmically convex function of r on (0, 1). We also show that the range [−3, 0] for α is best possible.

Research Article On Logarithmic Convexity for Differences of Power Means