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On generalized subharmonic functions

Published online by Cambridge University Press:  24 October 2008

F. F. Bonsall
F. F. Bonsall
Affiliation:
King's CollegeNewcastle-on-Tyne
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Extract

In a recent paper (1) I studied a class of generalized convex functions of a single real variable which I called sub-(L) functions. Given an ordinary linear differential equation of the second order L(y) = 0, a function f(x) is sub-(L) in (a, b) if it is majorized there by the solutions of the equation. More precisely, for every x1, x2 in (a, b),f(x) ≤ F12(x) in (x1, x2), where F12 is that solution of L(y) = 0 (supposed unique) which takes the values f(xi) at xi. It was found that sub-(L) functions are characterized in a manner closely analogous to ordinary convex functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 3 , July 1950 , pp. 387 - 395
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

REFERENCES

(1)Bonsall, F. F. The characterization of generalized convex functions. Quart. J. Math. (Oxford, 2nd series) 1 (1950), 100–111.Google Scholar
(2)Littlewood, J. E.Lectures on the theory of functions (Oxford, 1944), pp. 152–62, 208, 215.Google Scholar
(3)Goursat, E.Cours d'analyse Mathematique, 3 (Paris, 1915), 225.Google Scholar
(4)Lichtenstein, L.Encyk. Math. Wiss. (Leipzig, 1923), IIC, 12, 1277–1334, particularly 1280–90.Google Scholar