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Miklós Schweitzer 1957- Problem 4

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October 16, 2015
college contests

Problem Statement

4. Let Fϵ(0<ϵ<1)F_{\epsilon} (0<\epsilon<1) denote the class of non-negative piecewise continuous functions defined on [0,∞)[0,\infty) which satisfy the following condition: f(x)f(y)≤ϵ∣x−y∣(x,y≥0)f(x)f(y)\leq \epsilon^{\mid x-y\mid} (x,y \geq 0). Find the value of
sϵ=sup⁡f∈Fϵ∫0∞f(x)dxs_{\epsilon}= \sup_{f\in F_{\epsilon}} \int_{0}^{\infty} f(x) dx
(R. 5)
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