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By Edwin F. Beckenbach, Richard Bellman (auth.)

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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. The proof follows from ROLLE'S theorem as in § 12. Repeated application of this lemma shows that the polynomials {Pk} defined by m P(sa+x)= :EskPk(X) (3) k =1 are hyperbolic. § 37. Garding's Inequality Let M x m ), where each of the Xi is an m-dimensional vector Xi = (x~, x~, ...

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. Rend. Circ. Mat. Palermo (2) 2, 177-202 (1953). § 49. 1. : über eine Verallgemeinerung des Dirichletschen Integrals. Math. Z. 52,676-685 (1950). 2. : On an inequality of WEINBERGER. Am. Math. Monthly 60, 402 (1953). 3. WEINBERGER, H. : An inequality with alternating signs. Proc.

Proe. Am. Math. Soe. 9, 371-374 (1958). 5. RYSER, H. : Matriees of zeros and ones. Bull. Am. Math. Soe. 66, 442-464 (1960). 1]. §31. 1. : A new proof of an inequaIity of HARDy-LITTLEWOOD-PÕLYA. Mat. Tisdkr. (B) 1947, 53-54. 2. HARDY, G. , J. E. LITTLEWOOD and G. PÕLVA: Some simple inequalities satisfied by eonvex funetions. Messenger of Math. 58, 145-152 (1929). 3. RUDERMAN, H. : Two new inequalities. Am. Math. Monthly 59,29-32 (1952). 4. BEESACK, P. : A note on an integraI inequality. Proe. Am.

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