Geodesic
Inequalities for Positive Definite Functions
Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 823-836

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Positive definite kernels and functions are considered. The key tool in the paper is the well-known main inequality for such kernels, namely, the Cauchy–Bunyakovskii inequality for the special inner product generated by a given positive definite kernel. It is shown that Ingham's inequality (and, in particular, Hilbert's inequality) is, essentially, the main inequality for the positive definite function $\sin(\pi x)/x$ on $\mathbb{R}$ and for a system of integer points. Using the main inequality, we prove new inequalities of Krein–Gorin type and Ingham's inequality.
Keywords: positive definite kernels and functions, Ingham's inequality, Hilbert's inequality, Krein's inequality, Weil's inequality, Gorin's inequality. 
V. P. Zastavnyi. Inequalities for Positive Definite Functions. Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 823-836. http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a1/
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