Maximum value of Ratio

[Tolaso J Kos]

Find the maximum value of the ratio: \( \displaystyle {\rm R}=\frac{\displaystyle \left ( \int_{0}^{1}f(x)\, dx\right )^3}{\displaystyle \int_{0}^{1}f^3 (x)\, dx} \) as \( f \) ranges over all positive continuous functions on \( [0, 1] \).

[Demetres]

By Hölder's inequality, and since \(f\) is positive, we have \[ \int f(x) \, dx \leqslant \left( \int f(x)^3 \, dx\right)^{1/3} \left( \int 1^{3/2} \, dx\right)^{2/3}.\] It immediately follows that \(R \leqslant 1\) and we can have equality when \(f\) is (for example) constant.

[In fact, since we are looking only at continuous functions, we have equality if and only if \(f\) is constant.]