By Dr. Dragoslav S. Mitrinović (auth.)

The concept of Inequalities started its improvement from the time whilst C. F. GACSS, A. L. CATCHY and P. L. CEBYSEY, to say in basic terms an important, laid the theoretical starting place for approximative meth ods. round the finish of the nineteenth and the start of the 20 th century, a number of inequalities have been proyed, a few of which grew to become vintage, whereas so much remained as remoted and unconnected effects. it really is virtually mostly said that the vintage paintings "Inequali ties" by means of G. H. HARDY, J. E. LITTLEWOOD and G. POLYA, which seemed in 1934, reworked the sphere of inequalities from a suite of remoted formulation right into a systematic self-discipline. the trendy thought of Inequalities, in addition to the continued and starting to be curiosity during this box, absolutely stem from this paintings. the second one English variation of this booklet, released in 1952, was once unchanged aside from 3 appendices, totalling 10 pages, additional on the finish of the e-book. this present day inequalities playa major function in all fields of arithmetic, they usually current a really lively and tasty box of analysis. J. DIEUDONNE, in his booklet "Calcullnfinitesimal" (Paris 1968), attri buted distinctive importance to inequalities, adopting the tactic of exposi tion characterised through "majorer, minorer, approcher". considering the fact that 1934 a large number of papers dedicated to inequalities were released: in a few of them new inequalities have been found, in others classical inequalities ,vere sharpened or prolonged, quite a few inequalities ,vere associated through discovering their universal resource, whereas another papers gave loads of miscellaneous applications.

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The idea of Inequalities begun its improvement from the time while C. F. GACSS, A. L. CATCHY and P. L. CEBYSEY, to say in basic terms an important, laid the theoretical origin for approximative meth ods. round the finish of the nineteenth and the start of the 20 th century, a number of inequalities have been proyed, a few of which grew to become vintage, whereas so much remained as remoted and unconnected effects.

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**Example text**

184-185): GRAM'S determinant (see [2J, + Yl' x 2' ... , xn)1/2 < F(xl' x 2' ... , xn)1/2 + F(Yl' x 2, ... , xn)1/2. F(Xl If the vectors Xl' ... , Xn are elements of an arbitrary then we have (see [3J and [4J): Theorem 2. Given the vectors Xl' ... , Xn and a projection P with Yi (i = 1, ... ,12), then with equality il and only il Xi = Yi (i = r(Yl' ... , Y n -l) r(Yl' .. ·,Yn ) = PXi 1, ... ,12). Theorem 3. Given the vectors Xl' ... , Xn and a projection P with Yi (i = 1, ... ,12), then ~;-;------:-- space, HILBERT = PXi r(xl , ...

21. : Sur les fonctions sousharmoniques et leurs rapports avec les fonctions convexes. C. R Acad. Sci. Paris 185, 633-635 (1927). 22. THORIN, G. : Convexity theorems generalizing those of M. Riesz and Hadamard with some applications. Medd. Lunds Univ. Mat. Sem. 9, 1-57 (1948). 23. SALEM, R: Convexity theorems. Bull. Amer. Math. Soc. 55, 851-859 (1949), or Oeuvres mathematiques de R SALEM. Paris 1967, pp. 432-440. 24. OVCARENKO, 1. : On three types of convexity (Russian). Zap. -Mat. Fak. Har'kov.

The following inequality (1) holds with equality if and only if the vectors Xl' This inequality is called GRAM'S ... , Xn are linearly dependent. inequality. Proof. If the vectors Xl, ... , Xn are linearly dependent, then there exist scalars IXl' ... , IXn not all zero, such that (2) Multiplying the above equation by Xk (3) (k = 1, ... , n), we get (k = 1, ... , n). System (3) will have nontrivial solutions for IXi if r(Xl' ... , xn) = O. Therefore, ifthe vectors Xv"'' Xn are linearly dependent, then r(Xl"'" xn) = O.