Quadratic forms dependent on values of Neumann functions are studied. Their monotonicity under domain extension and polarization is proved. Also the behavior of these quadratic forms under univalent conformal mappings is investigated. As an application, a distortion theorem extending the results of Dubinin and Kim to finitely connected domains is obtained. Bibliography: 15 titles.
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References
G. M. Goluzin, The Geometric Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1966).
Z. Nehari, “Some inequalities in the theory of functions,” Trans. Amer. Math. Soc., 75, No. 2, 256–286 (1953).
P. Duren and M. M. Schiffer, “Robin functions and energy functionals of multiply connected domains,” Pacific J. Math., 148, No. 2, 251–273 (1991).
P. Duren and M. M. Schiffer, “Robin functions and distortion of capacity under conformal mapping,” Complex Variables, 21, 189–196 (1993).
V. N. Dubinin, “On quadratic forms involving Green’s and Robin functions,” Mat. Sb., 200, No. 10, 25–38 (2009).
V. N. Dubinin and M. Vuorinen, “Robin functions and distortion theorems for regular mappings,” Math. Nachr., 283, No. 11, 1589–1602 (2010).
V. N. Dubinin and V. Yu. Kim, “Generalized condensers and theorems on boundary distortion under conformal mapping,” Dal’nevost. Mat. Zh., 13, No. 2, 196–208 (2013).
V. N. Dubinin, Condenser Capacities and Symmetrization in the Geometric Theory of Functions of a Complex Variable [in Russian], Dal’nauka, Vladivostok (2009).
D. Karp and E. Prilepkina, “Reduced modulus with free boundary and its applications,” Ann. Acad. Sci. Fenn., 34, 353–378 (2009).
E. G. Emel’yanov, “On quadratic differentials on multiply connected domains that are perfect squares. II,” Zap. Nauchn. Semin. POMI, 350, 40–51 (2007).
P. Duren, J. Pfaltzgraff, and E. Thurman, “Physical interpretation and further properties of Robin capacity,” Algebra Analiz, 9, No. 3, 211–219 (1997).
V. N. Dubinin and E. G. Prilepkina, “On preservation of the generalized reduced module under geometric transformations of plane domains,” Dal’nevost. Mat. Zh., 6, Nos. 1–2, 39–56 (2005).
E. G. Prilepkina, “On composition principles for reduced modules,” Sib. Mat. Zh., 52, No. 6, 1357–1372 (2011).
E. G. Prilepkina, “Distortion theorems for univalent functions in multiply connected domains,” Dal’nevost. Mat. Zh., 9, Nos. 1–2, 140–149 (2009).
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag (1992).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 429, 2014, pp. 157–177.
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Prilepkina, E.G. On Quadratic Forms Generated by Neumann Functions. J Math Sci 207, 909–922 (2015). https://doi.org/10.1007/s10958-015-2414-5
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DOI: https://doi.org/10.1007/s10958-015-2414-5