Note on Wirtinger’s inequality

Abstract

In this note we refine the following theorem due to W. Wirtinger: If f has period 27r and satisfies J0 2 " f(x)dx = 0, then 1 2 " l(x)dx:::; 127r j'2(x)dx with strict inequality unless f(x) = a cos(x) + bsin(x), (a, bE ]E.). One of the most fascinating subjects in the theory of inequalities are integral inequalities involving a function and its derivative. Among the theorems in this field the so-called Wirtinger inequality has attracted special interest, because of its close connection to linear differential equations and differential geometry;see [1-5]. Wirtinger's inequality, which compares the integral of a square of a function with that of a square of its first derivative, appeared in 1916 in W. Blaschke's classical book "Kreis und Kugel" [3, p. 105]. Proposition. Let f be a real function with period 27r and J0 27r f (x)dx = O. If f' E L 2 , then (1) with strict inequality unless f(x) = acos(x) + bsin(x), where a and b are real constants. Three elegant proofs of (1) are given in [1]. An interesting survey on Wirtin-ger's and related inequalities can be found in a recently published monograph [5], which represents numerous extensions, refinements, variants, discrete analogues and applications of (1), and provides more than 200 references on this subject. It is the aim of this paper to establish a refinement of Wirtinger's inequality which we couldn't locate in the literature. More precisely we investigate the following problem: Let r, s, t and q be non-negative integers with r > max{ s, t} and q 2 2. What is the best possible value K(r, s, t; q) such that the inequality