By Percival Frost Joseph Wolstenholme

ISBN-10: 140217098X

ISBN-13: 9781402170980

This Elibron Classics ebook is a facsimile reprint of a 1863 variation via Macmillan and Co., Cambridge - London.

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Tucuman (A) 5, 147-151 (1946). 3. : On aresult of S. SHERMAN eoncerning doubly stochastie matriees. Proc. Am. Math. Soe. 9, 350-353 (1958). BibIiographical Notes 53 4. : Proofs of two theorems on doubly-stoehastie matrices. 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. Math. Soe. 8, 875-879 (1957). § 32. 1. , and G. G. LORENTZ: An integraI inequality. Am. Math. Monthly 61, 626-631 (1954). § 33. 1. , and L. LOPES: InequaIities for symmetrie functions and Hermitian matriees. Canad. J. Math. 8, 524-531 (1956). 2. , and M. MARCUS: Symmetrie means and matrix inequalities. 1961, to appear. § 35.

4) More general results can be obtained by using the inverse operators associated with the linear differential equation L(u) = J(t), where L(u) = d"ujdt"+ Pl(t) d,,-l ujdt,,-l+ ... -+- P,,(t) u. § 47. 4) is a particular case of more general inequalities that can systematically be generated in the following way. Let F (u, v) ~ 0 (1) 47 § 49. Inequalities with Alternating Signs be an inequality with the property than equality holds only for u = v. We now proceed to study the perturbed inequality obtained by setting u=w+rJ, (2) v = w + sg, where w,J and g are element s of the same nature as u and v, and r and s are scalar quantities.

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