Inequalities for generalized weighted mean values of convex function (Q2720280)

The authors prove an inequality of type NEWLINE\[NEWLINEM_{w,f}(r, s,a,b)< E(r+ 1,s+ 1,f(a),f(b)),NEWLINE\]NEWLINE where \(f\) is a certain differentiable function on \([a,b]\), \(w\) is a nonnegative weight function, \(M_{w,f}\) is a generalized weighted mean, and \(E\) is the Stolarsky mean. The proof is based, as usual, on the Tchebycheff integral inequality, and Cauchy's mean value theorem in integral form. The result extends an earlier theorem by \textit{M.-B. Sun} [Math. Practice Theory 27, No. 3, 193-198 (1997)].