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Improved bounds for Hermite-Hadamard inequalities in higher dimensions. (English) Zbl 1462.26022

In this paper, it iss proved that: If \(\Omega \subset \mathbb{R}^n\) is a convex domain and \(f: \Omega \to \mathbb{R}\) is positive and subharmonic function, then \[\frac1{\mid \Omega \mid}\int_{\Omega} f(x)dx \le \frac{c_n}{\mid \partial \Omega\mid}\int_{\partial \Omega} f\sigma d\sigma,\] where \(c_n \le 2n^{3/2}\). This inequality has been known for convex functions with much larger constant, however in this paper it was shown that the optimal constant satisfies \(c_n \ge n-1.\) As a consequence of the above result, the authors established a sharp geometric inequality for two convex domains where one contains the other \(\Omega_2 \subset \Omega_1 \subset \mathbb{R}^n\): \[\frac{\mid\partial\Omega_1\mid}{\mid\Omega_1\mid} \frac{\mid \Omega_2\mid}{\mid \partial\Omega_2 \mid} \le n.\]

MSC:

26D15 Inequalities for sums, series and integrals
26B25 Convexity of real functions of several variables, generalizations
28A75 Length, area, volume, other geometric measure theory
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35B50 Maximum principles in context of PDEs
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References:

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