By Dr. Dragoslav S. Mitrinović (auth.)

The idea of Inequalities started 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 beginning for approximative meth ods. round the finish of the nineteenth and the start of the 20 th century, various inequalities have been proyed, a few of which grew to become vintage, whereas such a lot remained as remoted and unconnected effects. it really is virtually in most cases stated that the vintage paintings "Inequali ties" by means of G. H. HARDY, J. E. LITTLEWOOD and G. POLYA, which seemed in 1934, remodeled the sector of inequalities from a suite of remoted formulation right into a systematic self-discipline. the fashionable idea of Inequalities, in addition to the continued and turning out to be curiosity during this box, surely stem from this paintings. the second one English version of this e-book, released in 1952, was once unchanged apart from 3 appendices, totalling 10 pages, additional on the finish of the ebook. at the present time inequalities playa major position in all fields of arithmetic, and so they current a really lively and tasty box of analysis. J. DIEUDONNE, in his booklet "Calcullnfinitesimal" (Paris 1968), attri buted specified importance to inequalities, adopting the strategy of exposi tion characterised by way of "majorer, minorer, approcher". considering 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, quite a few inequalities ,vere associated through discovering their universal resource, whereas another papers gave lots of miscellaneous applications.

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Papers [19J and [20J of G. VALIROX. MONTEL see particularly the Theorem 3 (Three lines theorem). Let f be a complex-valued and an analytic function of a complex variable z = x + iy. Suppose that f is defined and bounded in the strip a < x < b, - (Xl < y < + (Xl. vI is logarithmically convex on [a, bJ. This theorem is due to M. RIEsz. Theorem 4 (Three circles theorem). Let f be a complex-valued and an analytic function in the annulus domain a < Izl < b. Then log M (r) is a convex function of log r (a < r < b), where M (r) = max If(z)!.

Anb n can be written as follows: n ~akbk = Sl b1 k~l ABEL'S + (S2 - Sl) b2 + ... 3 jordan's Inequality Since m (b n- 1 - bn) < (b n- 1 Sn_l mb n < snbn < bn) < M (b n_ 1 - - bn), Mb n , by addition we get inequality (1). 3 jordan's Inequality Integrating both sides of the inequality sec2 e> 1 (0 < e< n/2) over the interval (0, e), we get e> e tan (0 < e< n/2). Therefore we obtain :() Ci~ ()) = CO;2 () (e - tan e) < 0 (0 < e< n/2) . We infer that . n Sln- 2 sin(»_~ (1) () "-- (0 n < e< n/2). 2 e= Equality holds in (1) if and only if On the other side we have sin e< e n/2.

The theory of J -convex functions as opposed to the theory of convex functions, was only partially transposed to the functions of several variables. Some properties of these functions which correspond to those cited by F. BERNSTEIN and G. DOETSCH [10J, were given by H. BLUMBERG [33J and E. MOHR [34]. S. MARCUS [35J gave an analogue of the theorem of A. OSTROWSKI [12J and of M. HUKUHARA [36J for the functions of several variables. As a natural generalization of the concept of convexity for functions of two real variables, the concepts of subharmonic and of double convex functions have been studied.