Geodesic
Green energy of discrete signed measure on concentric circles
Izvestiya. Mathematics , Tome 87 (2023) no. 2, pp. 265-283.

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We show that the difference between the Green energy of a discrete signed measure relative to a circular annulus concentrated at some points on concentric circles and the energy of the signed measure at symmetric points is non-decreasing during the expansion of the annulus. As a corollary, generalizations of the classical Pólya–Schur inequality for complex numbers are obtained. Some open problems are formulated.
Keywords: Green function, Green energy, capacity of condensers, dissymmetrization, inequality. 
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V. N. Dubinin. Green energy of discrete signed measure on concentric circles. Izvestiya. Mathematics , Tome 87 (2023) no. 2, pp. 265-283. http://geodesic.mathdoc.fr/item/IM2_2023_87_2_a2/

[1] J. S. Brauchart, “Optimal logarithmic energy points on the unit sphere”, Math. Comp., 77:263 (2008), 1599–1613 | DOI | MR | Zbl

[2] J. S. Brauchart, D. P. Hardin, and E. B. Saff, “The Riesz energy of the $N$th roots of unity: an asymptotic expansion for large $N$”, Bull. Lond. Math. Soc., 41:4 (2009), 621–633 | DOI | MR | Zbl

[3] J. S. Brauchart, D. P. Hardin, and E. B. Saff, “The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere”, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., 578, Amer. Math. Soc., Providence, RI, 2012, 31–61 | DOI | MR | Zbl

[4] D. P. Hardin, A. P. Kendall, and E. B. Saff, “Polarization optimality of equally spaced points on the circle for discrete potentials”, Discrete Comput. Geom., 50:1 (2013), 236–243 | DOI | MR | Zbl

[5] S. V. Borodachov, D. P. Hardin, A. Reznikov, and E. B. Saff, “Optimal discrete measures for Riesz potentials”, Trans. Amer. Math. Soc., 370:10 (2018), 6973–6993 | DOI | MR | Zbl

[6] S. V. Borodachov, D. P. Hardin, and E. B. Saff, Discrete energy on rectifiable sets, Springer Monogr. Math., Springer, New York, 2019 | DOI | MR | Zbl

[7] N. S. Landkof, Foundations of modern potential theory, Nauka, Moscow, 1966 ; English transl. Grundlehren Math. Wiss., 180, Springer-Verlag, New York–Heidelberg, 1972 | MR | Zbl | MR | Zbl

[8] V. N. Dubinin, Condenser capacities and symmetrization in geometric function theory, Springer, Basel, 2014 | DOI | MR | Zbl

[9] V. N. Dubinin, “Green energy and extremal decompositions”, Probl. Anal. Issues Anal., 8(26):3 (2019), 38–44 | DOI | MR | Zbl

[10] V. N. Dubinin, “Pólya–Schur inequality and the Green energy of a discrete charge”, Dokl. Akad. Nauk. Matem., inform., proc. upr., 492 (2020), 24–26 ; English transl. Dokl. Math., 101:3 (2020), 192–194 | DOI | Zbl | DOI | MR

[11] N. I. Akhiezer, Elements of the theory of elliptic functions, 2nd ed., Nauka, Moscow, 1970 ; English transl. Transl. Math. Monogr., 79, Amer. Math. Soc., Providence, RI, 1990 | MR | Zbl | DOI | MR | Zbl

[12] V. N. Dubinin, “Asymptotics for the capacity of a condenser with variable potential levels”, Sibirsk. Mat. Zh., 61:4 (2020), 796–802 ; English transl. Siberian Math. J., 61:4 (2020), 626–631 | DOI | MR | Zbl | DOI

[13] M. Schiffer, “Some recent developments in the theory of conformal mapping”, Appendix to: R. Courant, Dirichlet's principle, conformal mapping, and minimal surfaces, Interscience Publishers, Inc., New York, N.Y., 1950, 249–323; Russian transl. Inostr. Lit., Moscow, 1953, 234–301 | DOI | MR | Zbl

[14] S. P. Suetin, “An analogue of the Hadamard and Schiffer variational formulas”, Teor. Mat. Fiz., 170:3 (2012), 335–341 ; English transl. Theoret. and Math. Phys., 170:3 (2012), 274–279 | DOI | MR | Zbl | DOI

[15] I. Schur, “Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten”, Math. Z., 1:4 (1918), 377–402 | DOI | MR | Zbl

[16] L. Fejes Tóth, “On the sum of distances determined by a pointset”, Acta Math. Acad. Sci. Hungar., 7 (1956), 397–401 | DOI | MR | Zbl

[17] V. N. Dubinin, “Some properties of the reduced inner modulus”, Sibirsk. Mat. Zh., 35:4 (1994), 774–792 ; English transl. Siberian Math. J., 35:4 (1994), 689–705 | MR | Zbl | DOI

[18] V. N. Dubinin, “On the change in harmonic measure under symmetrization”, Mat. Sb., 124(166):2(6) (1984), 272–279 ; English transl. Sb. Math., 52:1 (1985), 267–273 | MR | Zbl | DOI