Birkhoff's Inequality

In homogeneous coordinates, the first positive quadrant joins (0,1) with (1,0) by "points" (f_1,f_2), and is mapped onto the hyperbolic line -infty<u<+infty by the correspondence Ln(f_2/f_1)=u. Now define


Let P be any bounded linear transformation of a Banach space B that maps a closed convex cone C of B onto itself. Then the C-norm N(P;C) of P is defined by


for pairs f,g in C with finite theta(f,g;C). Birkhoff's inequality then states that if the transform CP of C under P has finite diameter Delta under theta(f,g;C), then


(Birkhoff 1957).

Explore with Wolfram|Alpha


Birkhoff, G. "Extensions of Jentzsch's Theorem." Trans. Amer. Math. Soc. 85, 219-227, 1957.Jentzsch, R. "Über Integralgleichungen mit positivem Kern." J. reine angew. Math. 141, 235-244, 1912.Schmeidler, W. Integralgleichungen mit Anwendungen in Physik und Technik, Vol. 1. Lineare Integralgleichungen. Leipzig, Germany: Geest & Portig, p. 298, 1955.

Referenced on Wolfram|Alpha

Birkhoff's Inequality

Cite this as:

Weisstein, Eric W. "Birkhoff's Inequality." From MathWorld--A Wolfram Web Resource.

Subject classifications