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Turkish MO 1994 P4

Source: Turkish Mathematical Olympiad 2nd Round 1994

September 27, 2006
algebra

Problem Statement

Let f:R+→R+f: \mathbb{R}^{+}\rightarrow \mathbb{R}+ be an increasing function. For each u∈R+u\in\mathbb{R}^{+}, we denote g(u)=inf⁡{f(t)+u/t∣t>0}g(u)=\inf\{ f(t)+u/t \mid t>0\}. Prove that:
(a)(a) If x≤g(xy)x\leq g(xy), then x≤2f(2y)x\leq 2f(2y);
(b)(b) If x≤f(y)x\leq f(y), then x≤2g(xy)x\leq 2g(xy).
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