Monotonicity of weighted averages of convex functions. (English) Zbl 1444.26032
Summary: We consider weighted averages of the form \(B_n(W,f)=\sum^ n_{r=0}w_{n,r}f(r/n)\), where \(W\) is a summability matrix and \(f\) is convex. Conditions are given for \(B_n(W,f)\) to increase or decrease withn. It decreases wheneverWis a Hausdorff mean. The sequence of Bernstein polynomials for a convex function is a special case.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 26A51 | Convexity of real functions in one variable, generalizations |
| 26E60 | Means |
| 40G05 | Cesàro, Euler, Nörlund and Hausdorff methods |
Keywords:
monotonicity; weighted average; convex; summability matrix; Hausdorff mean; Bernstein polynomialReferences:
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