SCHUR-CONVEXITY OF THE GENERALIZED HERONIAN MEANS INVOLVING TWO POSITIVE NUMBERS

Schur Convexity of Generalized Heronian Means Involving Two Parameters

Schur Power Convexity of the Daróczy Means

In this paper, the Schur convexity is generalized to Schur f -convexity, which contains the Schur geometrical convexity, harmonic convexity and so on. When f : R+ →R is defined by f (x) = (xm−1)/m if m = 0 and f (x) = lnx if m = 0 , the necessary and sufficient conditions for f -convexity (is called Schur m -power convexity) of Daróczy means are given, which improve, generalize and unify Shi et...

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.

On Generalized Schur Numbers

Let L(t) represent the equation x1 + x2 + · · · + xt−1 = xt. For k > 1, 0 6 i 6 k − 1, and ti > 3, the generalized Schur number S(k; t0, t1, . . . , tk−1) is the least positive integer m such that for every k-colouring of {1, 2, . . . ,m}, there exists an i ∈ {0, 1, . . . , k − 1} such that there exists a solution to L(ti) that is monochromatic in colour i. In this paper, we report twenty-six p...

Off-diagonal Generalized Schur Numbers

We determine all values of the 2-colored off-diagonal generalized Schur numbers (also called Issai numbers), an extension of the generalized Schur numbers. These numbers, denoted S(k, l), are the minimal integers such that any red and blue coloring of the integers from 1 to S(k, l) must admit either a solution to ∑k−1 i=1 xi = xk consisting of only red integers, or a solution to ∑l−1 i=1 xi = x...