Some theorems on the cos 𝜋𝜆 inequality

Lewis, John L.

doi:10.1090/S0002-9947-1972-0294671-8

Some theorems on the cos $\pi \ \lambda$ inequality
HTML articles powered by AMS MathViewer

by John L. Lewis

Trans. Amer. Math. Soc. 167 (1972), 171-189

DOI: https://doi.org/10.1090/S0002-9947-1972-0294671-8

PDF | Request permission

Abstract:

In this paper we consider subharmonic functions $u \leqq 1$ in the unit disk whose minimum modulus and maximum modulus satisfy a certain inequality. We show the existence of an extremal member of this class with largest maximum modulus. We then obtain an upper bound for the maximum modulus of this function in terms of the logarithmic measure of a certain set. We use this upper bound to prove theorems about subharmonic functions in the plane.

References

J. M. Anderson, Asymptotic properties of integral functions of genus zero, Quart. J. Math. Oxford Ser. (2) 16 (1965), 151–165. MR 179357, DOI 10.1093/qmath/16.2.151
P. D. Barry, On a theorem of Besicovitch, Quart. J. Math. Oxford Ser. (2) 14 (1963), 293–302. MR 156993, DOI 10.1093/qmath/14.1.293
A. S. Besicovitch, On integral functions of order $<1$, Math. Ann. 97 (1927), no. 1, 677–695. MR 1512383, DOI 10.1007/BF01447889

Études sur un problème de majorisation

Maurice Heins, Selected topics in the classical theory of functions of a complex variable, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1962. MR 0162913
Maurice Heins, Entire functions with bounded minimum modulus; subharmonic function analogues, Ann. of Math. (2) 49 (1948), 200–213. MR 23342, DOI 10.2307/1969122
Ulf Hellsten, Bo Kjellberg, and Folke Norstad, Subharmonic functions in a circle, Ark. Mat. 8 (1969), 185–193. MR 273040, DOI 10.1007/BF02589557
Alfred Huber, Über Wachstumseigenschaften gewisser Klassen von subharmonischen Funktionen, Comment. Math. Helv. 26 (1952), 81–116 (German). MR 49395, DOI 10.1007/BF02564294
Bo Kjellberg, On certain integral and harmonic functions. A study in minimum modulus, University of Uppsala, Uppsala, 1948. Thesis. MR 0027065
Bo Kjellberg, On the minimum modulus of entire functions of lower order less than one, Math. Scand. 8 (1960), 189–197. MR 125967, DOI 10.7146/math.scand.a-10608
Bo Kjellberg, A theorem on the minimum modulus of entire functions, Math. Scand. 12 (1963), 5–11. MR 159942, DOI 10.7146/math.scand.a-10666
Frédéric Riesz, Sur les fonctions subharmoniques et leur rapport à la théorie du potentiel, Acta Math. 54 (1930), no. 1, 321–360 (French). MR 1555311, DOI 10.1007/BF02547526
M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894