New Opial type finite difference inequalities *.Abstract In this paper, we establish some new Opial type finite difference A finite difference is a mathematical expression of the form f(x + b) − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. inequalities This page lists Wikipedia articles about named mathematical inequalities. Pure mathematics
Keywords and Phrases: Opial type inequality, Finite difference inequalities, Forward differences, Holder's inequality, Nondecreasing functions of several variables. 1. Introduction In this paper, we denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. by [R.sup.n] the n-dimensional Euclidean space Euclidean space In geometry, a two- or three-dimensional space in which the axioms and postulates of Euclidean geometry apply; also, a space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension) and the distance between and by N the set of natural numbers. For a sequence [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .], we denote the operators [??] and [nabla] by [??][u.sub.m] = [u.sub.m+1] - [u.sub.m] and [nabla][u.sub.]m = [u.sub.m] - [u.sub.m-1]. We shall a dopt the usual convention that the empty sum is zero. In 1960, Opial [7] esatblished the following important integral inequality: Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. A. Let f (x) [member of] [C.sup.1] [0, h] be such that f (0) = f (h) = 0, and f (x) > 0 in (0, h). Then [[integral].sup.h.sub.0]|f (x) [f.sup.1](x)|dx [less than or equal to] h/4 [[integral].sup.h.sub.0](f'[(x)).sup.2] dx, (1) where h/4 is the best possible. The inequality (1) is known in the literature as Opial inequality. For some results which generalize generalize /gen·er·al·ize/ (-iz) 1. to spread throughout the body, as when local disease becomes systemic. 2. to form a general principle; to reason inductively. , improve and extend this famous integral inequality see [1] and [3]-[10]. In [10], Wong established the following discrete Opial type inequality about the backward difference operator: Theorem B. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be a nondecreasing sequence of nonnegative non·neg·a·tive adj. Of, relating to, or being a quantity that is either positive or zero. Adj. 1. nonnegative - either positive or zero real numbers with [u.sub.0] = 0. Then for l > 1 and n [member of] N, [n.summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (m=1)] [u.sup.l.sub.m] [??] [u.sub.m] [less than or equal to] [(n+1).sup.l]/;+1 [n.summation over (m=1] [([??][u.sub.m]).sup.l+1]. (2) In [9], Pachpatte established the following three discrete Opial type inequalities: Theorem C. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be a sequence of real numbers with u0 = 0. Let [f.sub.k] (x), k = 1, 2,..., p be nonnegative real-valued nondecreasing functions defined for x [member of] R such that [f.sub.k] (0) = 0, k = 1, 2,..., p. Then for n [member of] N, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Theorem D. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be defined as in Theorem C. Let [f.sub.k] (x), k = 1, 2,..., p be continuous real-valued nondecreasing functions defined for x [member of] R such that |[f.sub.k] (x)| [less than or equal to]| [f.sub.k] (|x|) (x [member of] R) and [f.sub.k] (y) [less than or equal to] [f.sub.k] (z) (0 [less than or equal to] y [less than or equal to] z, y, z [member of] R). Let F [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] dt, s > 0. Then for n [member of] N, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4) Theorem E. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be defined as in Theorem B. Then for p > 1 in N and n [member of] N, we have [n-1.summation over (m=0)] [u.sup.p-1.sub.m] [??] [u.sub.m] [less than or equal to [n.sup.p-1]/p [n-1.summation over (m=0)] [([??][u.sub.m]).sup.p] (5) and [n-1.summation over (m=0)] [u.sup.p.sub.m] [??] [u.sub.m] [less than or equal to [n.sup.p]/p+1 [n-1.summation over (m=0)] [([??][u.sub.m]).sup.p+1] (5) Remark 1. (a) A misprint mis·print tr.v. mis·print·ed, mis·print·ing, mis·prints To print incorrectly. n. An error in printing. of (6) in the orginal paper has been corrected here. (b) It's obvious that the inequality (6) can be obtained from the inequality (5). In this paper, we establish some results which generalize Theorems This is a list of theorems, by Wikipedia page. See also
C-E Communications-Electronics C-E Combustion Engineering, Inc . 2. Main Results The following concept has been introduced in [2]. Definition. Let p be a positive integer integer: see number; number theory and let x = ([x.sub.1],..., [x.sub.p]) and y = ([y.sub.1],..., [y.sub.p]) belong to [R.sup.p]. We write x [less than or equal to] y if and only if [x.sub.i], [less than or equal to] [y.sub.i], I = 1,..., p. If G: [R.sup.p] [right arrow] R, we say that G(x) is a nondecreasing function in [R.sup.p] if x, y [member of] [R.sup.p] and x [less than or equal to] y imply G(x) [less than or equal to] G(y). Theorem 1. Let p be a positive integer and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [f.sub.k], k = 1, 2,..., p be defined as in Theorem C. Let G : [R.sup.p] [right arrow] [0,[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ]) be a continuous and nondecreasing function with G((0,..., 0)) = 0. Then for n 2 N, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7) Proof. Using the definition of the operator [DELTA], we have the following identities [u.sub.i] = [i-1.summation over (r=0)] [??][u.sub.r] (8) and [DELTA]G([f.sub.1](|[u.sub.m]|),..., [f.sub.p](|u.sub.m|)) = G([f.sub.1]( (|[u.sub.m+1]|),..., [f.sub.p](|u.sub.m+1|)) - G([f.sub.1]|([u.sub.m]|),..., [f.sub.p](|u.sub.m|)) (9) where i = 1, 2,..., and m = 1,..., n - 1. Now from the definition of G, the properties of the functions [f.sub.k] (k = 1, 2,..., p) and the identities (8) and (9), we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] This completes the proof. Remark 2. In Theorem 1, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Then the inequality (7) reduces to the inequality (3). Theorem 2. Let p and G be defined as in Theorem 1 and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [f.sub.k], k = 1, 2,..., p be defined as in Theorem D. Define F(s) = [[integral.sup.s.sub.0] G([f.sub.1](t),..., [f.sub.p](t))dt, s > 0. Then for n [member of] N, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10) Proof. As in the proof of Theorem1, we have the identity (8). Using the definition of G, the hypotheses on the functions [f.sub.k] (k = 1, 2,..., p) and the identity (8), we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11) From the definition of F, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12) where m = 0, 1,..., n - 1. Using (12) in (11), we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13) Now by changing the variable r to m on (13), we obtain the inequality (10). This completes the proof. Remark 3. In Theorem 2, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Then the inequality (10) reduces to the inequality (4). 3. Some Special Cases In this section, we give a theorem which provides some estimates on these type of the inequality (2) about the forward difference operator. Theorem 3. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be defined as in Theorem B. Then for r > 1 in R and n [member of] N, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14) Proof. Let G(([x.sub.1],..., xp)) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Using the hypotheses on the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and the inequality (7), we have the following inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15) On the other hand, using the Mean Value Theorem In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the "average" derivative of the section. , we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16) Now applying Holder's inequality with indices r and r/r-1 for [n-1.summation over (m=0)] [??][u.sub.m], we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17) The combination of (15), (16) and (17) immediately gives the inequality (14). This completes the proof. Remark 4. In Theorem, let r = p > 1 in N. Then the inequality (14) reduces to the inequality (5). Received October 29, 2005, Accepted January 16, 2006. References [1] Drumi Bainov and Pavel Simeonov, Integral Inegualities and Applications, Kluwer Acadlmic Publishers (1992). [2] P. R. Beesack, On certain discrete inequalities involving partial sums, Canadian Jour. Math. 21 (1969), 222-234. [3] L. K. Hua, On an inequality of Opial, Scientia Sinica 14 (1965), 789-790. 124 Shiow-Ru Hwang and Kuei-Lin Tseng [4] C. M. Lee, On a discrete analogue (electronics) analogue - (US: "analog") A description of a continuously variable signal or a circuit or device designed to handle such signals. The opposite is "discrete" or "digital". of inequalities of Opial and Yang yang (yang) [Chinese] in Chinese philosophy, the active, positive, masculine principle that is complementary to yin; see yin, under principle. , Canadian Math. Bull. 11 (1968), 73-77. [5] J. Myjak, Boundary value problems In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. for nonlinear A system in which the output is not a uniform relationship to the input. nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input. differential and difference equations of the second order, Zeszyty Nauk. Univ. Jagiellonsi, Prace Math. 15 (1971), 113-123. [6] C. Olech, A simple proof of a certain result of Z. Opial, Ann. Polon. Math. 8 (1960), 61-63. [7] Z. Opial, Sur une inegalite, Ann. Polon. Math. 8 (1960), 29-32. [8] B. G. Pachpatte, On Opial like discrete inequalities, An. Sti. Univ. A1. I. Cuza Iasi, Mat. 36 (1990), 237-240. [9] B. G. Pachpatte, A Note on Opial Type Finite Difference Inequalities, Tamsui Oxford J. Math. Sci. 21(1) (2005) 33-39. [10] J. S. W. Wong, A discrete analogue of Opial's inequality, Canadian Math. Bull. 10 (1967), 115-118. * 2000 Mathematics Subject Classification. 26D15. Shiow-Ru Hwang ([dagger]) China Institute of Technology, Nankang, Taipei, Taiwan 11522. Kuei-Lin Tseng ([double dagger double dagger n. A reference mark (  ) used in printing and writing. Also called diesis.Noun 1. ]) Department of Mathematics, Aletheia Universty, Tamsui, Taiwan 25103. ([dagger]) E-mail: hsru@cc.chit chit 1 n. 1. A statement of an amount owed for food and drink; a check. 2. A short letter; a note. 3. .edu.tw ([double dagger]) E-mail: kltseng@email.au.edu.tw |
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