For other inequalities named after Wirtinger, see Wirtinger's inequality. In mathematics, historically Wirtinger's inequality for real functions was an inequality used in Fourier analysis. It was named after Wilhelm Wirtinger. It was used in 1904 to prove the isoperimetric inequality. A variety of closely related results are today known as Wirtinger's inequality. Theorem First version Let be a periodic function of period 2π, which is continuous and has a continuous derivative throughout R, and such that Then with equality if and only if f(x) = a sin(x) + b cos(x) for some a and b (or equivalently f(x) = c sin (x + d) for some c and d). This version of the Wirtinger inequality is the one-dimensional Poincaré inequality, with optimal constant. Second version The following related inequality is also called Wirtinger's inequality : whenever f is a C1 function such that f(0) = f(a) = 0. In this form, Wirtinger's inequality is seen as the one-dimensional version of Friedrichs' inequality. Proof The proof of the two versions are similar. Here is a proof of the first version of the inequality. Since Dirichlet's conditions are met, we can write and moreover a0 = 0 since the integral of f vanishes. By Parseval's identity, and and since the summands are all ≥ 0, we get the desired inequality, with equality if and only if an = bn = 0 for all n ≥ 2. References Paul J. Nahin (2006) Dr. Euler's Fabulous Formula, page 183, Princeton University Press Komkov, Vadim (1983) Euler's buckling formula and Wirtinger's inequality. Internat. J. Math. Ed. Sci. Tech. 14, no. 6, 661–668. Fourier analysis Inequalities Theorems in analysis