By Dr. Dragoslav S. Mitrinović (auth.)

The conception of Inequalities begun its improvement from the time while C. F. GACSS, A. L. CATCHY and P. L. CEBYSEY, to say purely crucial, laid the theoretical beginning for approximative meth ods. round the finish of the nineteenth and the start of the twentieth century, quite a few inequalities have been proyed, a few of which turned vintage, whereas so much remained as remoted and unconnected effects. it's nearly regularly stated that the vintage paintings "Inequali ties" through G. H. HARDY, J. E. LITTLEWOOD and G. POLYA, which seemed in 1934, reworked the sector of inequalities from a suite of remoted formulation right into a systematic self-discipline. the trendy idea of Inequalities, in addition to the ongoing and starting to be curiosity during this box, certainly stem from this paintings. the second one English version of this e-book, released in 1952, used to be unchanged apart from 3 appendices, totalling 10 pages, additional on the finish of the e-book. this present day inequalities playa major position in all fields of arithmetic, they usually current a really energetic and engaging box of study. J. DIEUDONNE, in his publication "Calcullnfinitesimal" (Paris 1968), attri buted targeted importance to inequalities, adopting the tactic of exposi tion characterised by means of "majorer, minorer, approcher". on the grounds that 1934 a mess 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, numerous inequalities ,vere associated by means of discovering their universal resource, whereas another papers gave quite a few miscellaneous applications.

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MITRINOVIC, D. , and P. M. VASIC: Sredine. Matematicka Biblioteka, vol. 40. Beograd 1969, 124 pp. 2. BECKEN BACH, E. , and R. BELLMAN: An Introduction to Inequalities. New York 1961, 133 pp. 3. HARDY, G. , J. E. LITTLEWOOD and G. P6LYA: Inequalities. , Cambridge 1952, 324 pp. 4. : Cours d' Analyse de l'Ecole Royale Polytechnique 1. Analyse algebrique. Paris 1821, or Oeuvres completes II, vol. 3. Paris 1897, pp. 375-377. 5. : Sur la moyenne arithmetique et la moyenne geometrique de plusieurs quantites positives.

P. 54. 2. General Inequalities Proof. Assume that p < 0, and put P = -P/q, Q = l/q. Then l/P + l/Q = 1 with P > 0 and Q > O. Therefore, according to (1), we have 1 1 c~Afr C~B~)Q > k~AkBk' where Ak > 0 and Bk > 0 for k = 1, ... , n. The last inequality for Ak = ak"q and Bk = aZbZ becomes (4). J. L. W. V. JENSEN [2J proved the following generalization of HOLDER'S inequality: Theorem 3. Let aij (i = 1, ... , n; j = 1, ... , m) be positive numbers and let iXv ... , iX", be positive numbers such that ~ IXI + ...

II I and g are monotone in the opposite sense, inequality in (10) reverses. From (10) for P(x) 1 follows (9). 1). M. BIERNACKI [8J proved inequality (10) under different conditions from those given in Theorem 10. In fact, he proved: Theorem 11. Let I, g and p be integrable lunctions on (a, b), and let p be positive on that interval. II the lunctions . _ __ gl (x) = f Pix) dx f P (x) g (x) dx _a- x - - - f Pix) dx a a reach extreme values only in a linite number 01 common points Irom (a, b), and il they are monotone in the same sense on (a, b), then (10) holds.