Abstract
Let \({n\in\mathbb{N}}\). For \({k\in\{1,\dots,n\}}\) let \({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let \({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary \({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let E r(Ωn) be the holomorphic Smirnov class on Ωn with index r. We show that the generalized isoperimetric inequality
holds for arbitrary \({f_1\in E^p(\Omega^n)}\) and \({f_2\in E^q(\Omega^n)}\), where 0 < p, q < ∞. We also determine necessary and sufficient conditions for equality.
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References
Aronszajn N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)
Beckenbach E.F., Radó T.: Subharmonic functions and surfaces of negative curvature. Trans. Am. Math. Soc. 35(3), 662–674 (1933)
Bénéteau C., Khavinson D.: The isoperimetric inequality via approximation theory and free boundary problems. Comput. Methods Funct. Theory 6(2), 253–274 (2006)
Burbea J.: Sharp inequalities for holomorphic functions. Ill. J. Math. 31, 248–264 (1987)
Bläsjö V.: The isoperimetric problem. Am. Math. Mon. 112(6), 526–566 (2005)
Carleman T.: Zur Theorie der Minimalflächen. Math. Z. 9(1–2), 154–160 (1921)
Duren P.: Theory of H p Spaces. Pure and applied mathematics, vol. 38, pp. xii+258. Academic Press, New York-London (1970)
Fuks, B.A.: Special Chapters in the Theory of Analytic Functions of Several Complex Variables. (Russian) Gosudarstv. Izdat. Fiz.-Mat. Lit., 427 pp. Moscow (1963)
Gamelin T.W., Khavinson D.: The isoperimetric inequality and rational approximation. Am. Math. Mon. 96, 18–30 (1989)
Hayman W.K., Kennedy P.B.: Subharmonic Functions, pp. xvii+284. Academic Press, London, New York (1976)
Keldysh M., Lavrentiev M.: Sur la représentation conforme des domaines limités par des courbes rectifiables. Ann. Sci. École Norm. Sup. 54, 1–38 (1937)
Kolaski C.J.: Isometries of Bergman spaces over bounded Runge domains. Can. J. Math. 33(5), 1157–1164 (1981)
Mateljević M., Pavlović M.: New proofs of the isoperimetric inequality and some generalizations. J. Math. Anal. Appl. 98(1), 25–30 (1984)
Mateljević M., Pavlović M.: Some inequalities of isoperimetric type concerning analytic and subharmonic functions. Publ. Inst. Math. (Beograd) (N.S.) 50(64), 123–130 (1991)
Pavlović M., Dostanić M.: On the inclusion \({H^2(\mathbb{U}^n)\subset H^{2n}(B_n)}\) and the isoperimetric inequality. J. Math. Anal. Appl. 226(1), 143–149 (1998)
Osserman R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84(6), 1182–1238 (1978)
Rudin W.: Function theory in polydiscs. Benjamin, New York (1969)
Saitoh S.: The Bergman norm and the Szegö norm. Trans. Am. Math. Soc. 249(2), 261–279 (1979)
Strebel, K.: Quadratic Differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5, xii+184 pp. Springer, Berlin (1984)
Vukotić D.: The isoperimetric inequality and a theorem of Hardy and Littlewood. Amer. Math. Monthly 110(6), 532–536 (2003)
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Kalaj, D. Isoperimetric inequality for the polydisk. Annali di Matematica 190, 355–369 (2011). https://doi.org/10.1007/s10231-010-0153-2
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DOI: https://doi.org/10.1007/s10231-010-0153-2