By Tsuyoshi Ando (auth.), Themistocles M. Rassias, Hari M. Srivastava (eds.)

Analytic and Geometric Inequalities and purposes is dedicated to fresh advances in numerous inequalities of Mathematical research and Geo metry. topics handled during this quantity contain: Fractional order inequalities of Hardy style, differential and quintessential inequalities with preliminary time fluctuate ence, multi-dimensional vital inequalities, Opial sort inequalities, Gruss' inequality, Furuta inequality, Laguerre-Samuelson inequality with extensions and functions in facts and matrix thought, distortion inequalities for ana lytic and univalent services linked to convinced fractional calculus and different linear operators, challenge of infimum within the optimistic cone, alpha-quasi convex services outlined by means of convolution with incomplete beta features, Chebyshev polynomials with integer coefficients, extremal difficulties for poly nomials, Bernstein's inequality and Gauss-Lucas theorem, numerical radii of a few significant other matrices and boundaries for the zeros of polynomials, measure of convergence for a category of linear operators, open difficulties on eigenvalues of the Laplacian, fourth order challenge boundary worth difficulties, bounds on entropy measures for combined populations in addition to controlling the rate of Brownian movement by way of its terminal price. A wealth of purposes of the above can also be integrated. we want to show our appreciation to the prestigious mathematicians who contributed to this quantity. ultimately, it truly is our excitement to recognize the high-quality cooperation and tips supplied via the workers of Kluwer educational Publishers. June 1999 Themistocles M. Rassias Hari M.

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The following result is a compact generalization of the inequalities (1) and (2). Theorem 2. 01=1 for The result is best possible and equality in (5) holds for P(z) Izl ~ 1. (5) = azn , a:# O. Remark 2. Dividing the two sides of (5) by R-1 and taking the limits as R -+ 1 with 13 = 1, we get the inequality (1). For 13 = 0, the inequality (5) reduces to (2). Next we use Theorem 2 to prove the following interesting result. 31 Theorem 3. If P(z) is a polynomial of degree nand Q(z) = zn P(I/z), then, for every real or complex number (3 with 1(31 ~ 1 and R ~ 1, IP(Rz) - (3P(z)1 IQ(Rz) - (3Q(z) I + (3llzln + 11 - (31} max ~ {IR n - IP(z)1 Izl=1 for The result is sharp and equality in (6) holds for P(z) = zn Izl (6) ~ 1.

3 [3, 6, 7]. with 2:, l/POt = 1, Ot Proof IIp,,, For any r E cJ(n) = 1 for all a. 5 [3, 6, 7]. D. 1 by letting POt = q for all a. D. 4 with qOt = 1 for all a. D. 1 we need the following basic lemma. 1. If f E cJ(n), then for any tEn, i L . II;(t 1, ... ,ti-1,ui,ti+1, ... ,tn)ldui . If(t)l:s; -1 2n ,. Proof. Since f b ; a. = 0 on an, for each i = 1, ... , n, we have f(t) = i t; fi(t1, ... D. 1. 1 and Lemma B, we have for all tEn. Since POt ~ 1, c(pOt, n) = nP,,-l and so III f Ot (t) Iq" a :s; -C",qOt(COt)P"", LJ- LJ n a POt 2 By Holder's Inequality, we have IIlr(tW" .

2) Ko(a,b):= inf{J(y): y E BCo} 1991 Mathematics Subject Classification. Primary: 26DlO; Secondary: 34C10, 34L05. Key words and phrases. Opial inequality, extremals, eigenvalue problems, disconjugacy. M. M. ), Analytic and Geometric Inequalities and Applications, 37-52. © 1999 Kluwer Academic Publishers. 2). 2) have been considered by FitzGerald [2]. However, there is a point in FitzGerald's proof with which we have difficulty. It deals with an approximation procedure. In FitzGerald's proof, as well as ours below, a critical step is to reduce the problem to the analysis of functions for which y' can change sign only once in the interval [a, b].