A refinement of the discrete Wirtinger inequality
In the present paper, the following refinement of Tang's discrete Wirtinger inequality [\textit{D. Tang}, Bull. Aust. Math. Soc. 43, No. 3, 467-474 (1991; Zbl 0751.26006)] is established: Theorem A. Let \(f(\theta)\) be a positive \(C^2\)-function on \((0,\ell)\) such that \(f'(\theta) f''(\theta) \neq 0\) and \(f'(\theta)^2 - f(\theta)f''(\theta) = \mu\), for \(\theta \in (0,\ell)\), where \(\mu\) is a constant. Then, if \(f''(\theta) < 0\) (or \(> 0)\)) on \((0,\ell)\), we have \[ \left( \sum^n_{i = 1} f(\theta_i)\right)^2 - c_n \sum^n_{i = 1} f(\theta_i) f'(\theta_i) \geq [\text{or } \leq] \left( nf(\sigma) - \sum^n_{i = 1} f(\theta_i)\right)^2 \] where \(\theta_i \in (0,\ell)\) with \(\theta_1 + \dots + \theta_n = m\ell\), \(\sigma = m\ell/n\), \(c_n = nf(\sigma)/f'(\sigma)\), and \(m(\leq n)\) is a positive constant. Equality in the last inequality holds if and only if \(\theta_1 = \dots = \theta_n = \sigma\). Two notable consequences of Theorem A for the cases when \(m =2\), \(\mu = 1\), \(\ell = \pi/2\), and \(f(\theta) = \sin \theta\) and \(f(\theta) = \cos \theta\) are derived in the Theorems B and \(\text{B}'\), respectively. As a geometric application of Theorem A, a Benneson-style isoperimetric inequality for plane polygons is obtained in Theorem C. The classification of all the functions verifying Theorem \(A\) is given. Finally, the main result (Theorem A) is restated as a weighted exponential inequality.
scientific article; zbMATH DE number 908599
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