A new refinement of the arithmetic mean -- geometric mean inequality (Q1384862)

The following inequality is offered as main result. Let \(x_{j}\) be positive reals, \(p_{j}\) \((j=1,\dots,n)\) nonnegative reals with sum 1, \(A\) the arithmetic, \(G\) the geometric mean of the \(x_{j}\) with weight \(p_{j}\) and \(M\) the maximum of the \(x_{j}\) \((j=1,\dots,n)\). Then \[ \sum_{j=1}^{n} p_{j}(x_{j}-G)^{2}\leq 2M(A-G) \] with equality iff all \(x_j\) with positive coefficients are equal.