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On a property of convex functions

1992; Springer Nature; Linguagem: Inglês

10.1007/978-3-0348-7565-3_43

2296-6072

Heinz König,

Mathematical Inequalities and Applications

For an interval I ⊂ ℝ define M(I) to consist of the functions φ: I → ℝ which are either monotone or, for some c ∈ I, monotone decreasing on {x ∈ I: x≤ c} and monotone increasing on {x ∈ I : x ≥ c}. The property in question is that for a continuous convex function f: I →ℝ IR on an interval I ⊂]0, ∞[ the function x ↦ f(x)/x is in M(I). It has been presented by János Aczél as a problem attributed to Marek Kuczma, with mention of a solution due to himself and to Che Tat Ng. Here we want to present a simple proof.