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Find the smallest constant c such that the ineq. is true

Source: Romanian District Olympiad 2006, Grade 12, Problem 4

March 11, 2006
inequalitiesintegrationfunctionreal analysisreal analysis unsolved

Problem Statement

Let F={f:[0,1]→[0,∞)∣f\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f continuous }\} and nn an integer, n≥2n\geq 2. Find the smallest real constant cc such that for any f∈Ff\in \mathcal F the following inequality takes place ∫01f(xn)dx≤c∫01f(x)dx. \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx.
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