Note on Wirtinger’s inequality

Abstract

In this note we refine the following theorem due to W. Wirtinger: If f has period 2π and satisfies $\int_0^{{2\pi }} {f(x)dx = 0}$, then

$$ \int_0^{{2\pi }} {{f^2}(x)dx \leqslant {{\int_0^{{2\pi }} {f'} }^2}(x)dx} $$

with strict inequality unless f(x) = a cos(x) + b sin(x), (a, b ∈ ℝ).