By George Boole
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Additional info for A Treatise on Differential Equations
NOLL: Eine Bemerkung zur Schwarzschen Ungleichheit. Math. Nachr. 7, 55-59 (1952). § 48. 1. : über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 1, 377 -402 (1918). 2. SIEGEL, C. : The trace of totally positive and real algebraic integers. Ann. of Math. 46, 302-312 (1945). 3. : On the arithmetic and geometric means and on HÖLDER'S inequality. Proc. Am. Math. Soe. 9, 452-459 (1958). 4. : Zur Abschätzung arithmetischer Mittel reeller Zahlen durch Differenzprodukte derselben.
It is defined geometrically by means of the transformation of reeiprocal polars with respeet to the sphere (x, x) = 1. 1]. It follows that we have (x, y) ~ F(x) G(y) , (4) an inequality apparently first explicitly stated and used by MAHLER  in the geometrical theory of numbers. So me interesting extensions of the pole-polar relationship are discussed in L. C. YOUNG , which we have followed in the preceding treatment; see also the paper by FENcHEL , which contains other referenees. 2], where many interesting results may be found, showing the intimate relation between inequalities, convexity, and the "mixed volume" concept of MINKOWSKI.
An) be aset of real quantities. If the equation in s, P (sa + x) = P (sa l + Xl' sa2 + XI"'" san + X n) = 0, has m real zeros for all real Xi' we say that P (x) is hyperbolie with respeet to a. An equivalent definition is that P (sa + x) = '" [s + Ä,,(a, x)] , P(a) II (1) II-I where P(a) 9= 0 and the Ä,,(a, x) are real whenever X is real. 45. Starting with one hyperbolic polynomial, we can form new hyperbolic polynomials by means of the following resuIt: § 38. Examples 37 Lemma 2. If P(x) is hyperbolie with respeet to a, and m > 1, then Õ m Q(x) =k~lak ÕXk P(x) (2) is also hyperbolie with respeet to a.
A Treatise on Differential Equations by George Boole