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Putnam 2013 B4

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December 9, 2013
Putnamfunctionintegrationinequalitiescalculuscollege contests

Problem Statement

For any continuous real-valued function ff defined on the interval [0,1],[0,1], let μ(f)=∫01f(x) dx,Var(f)=∫01(f(x)−μ(f))2 dx,M(f)=max⁡0≤x≤1∣f(x)∣.\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|. Show that if ff and gg are continuous real-valued functions defined on the interval [0,1],[0,1], then Var(fg)≤2Var(f)M(g)2+2Var(g)M(f)2.\text{Var}(fg)\le 2\text{Var}(f)M(g)^2+2\text{Var}(g)M(f)^2.
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