Otto Hölder's 1889 paper begins with a weighted-mean inequality for a function with increasing derivative. In current terminology, its opening statement is a weighted form of Jensen's inequality: applying a convex function after taking a weighted mean gives a lower bound for the weighted mean of its values. This connects product estimates to the geometry of convex functions, rather than treating them as isolated formulas.[ref]
Ueber einen Mittelwerthssatz by Otto Hölder. Otto Hölder, Ueber einen Mittelwerthssatz (1889), §§1–2, printed pp. 38–39: weighted means and increasing derivative
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