Ostrowski-Gruss-Cebysev type inequalities involving several functions.Abstract In this paper, we establish some Ostrowski-Gruss-Cebysev type inequalities This page lists Wikipedia articles about named mathematical inequalities. Pure mathematics
In finance, contracts whose value is derived from another asset, which can include stocks, bonds, currencies, interest rates, commodities, and related indexes. Purchasers of derivatives are essentially wagering on the future performance of that asset. are convex functions In mathematics, a real-valued function f defined on an interval (or on any convex subset of some vector space) is called convex, or concave up, if for any two points x and y in its domain C and any t in [0,1], we have Keywords Keywords are the words that are used to reveal the internal structure of an author's reasoning. While they are used primarily for rhetoric, they are also used in a strictly grammatical sense for structural composition, reasoning, and comprehension. and Phrases: Ostrowski Ostrowski, a surname, may refer to: People
1. Introduction Throughout, let ||h'||[.sub.[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. ]] [colon colon, in anatomy colon, in anatomy: see intestine. colon, in punctuation colon, in writing: see punctuation. colon Segment that makes up most of the large intestine. , equals] [sup.[t[member of](a,b)]] |h' (t)|, S (f, g) = f (x) g (x) - [1/2(b - a)] [g (x) [[integral].sub.a.sup.b] f (t) dt + f (x) [[integral].sub.a.sup.b] g (t) dt], and T (f, g) = [1/[b - a]] [[integral].sub.a.sup.b] f (x) g (x) dx - ([1/[b - a]] [[integral].sub.a.sup.b] f (x) dx) ([1/[b - a]] [[integral].sub.a.sup.b] g (x) dx) where x [member of] [a, b], h' exists and is bounded on (a, b) and f, g are integrable on [a, b]. The Ostrowski's inequality [5] states that if f' exists and is bounded on (a, b), then, for all x [member of] [a, b], we have the inequality |[[integral].sub.a.sup.b] f(t)dt - f(x) (b - a)| [less than or equal to] [1/4 (b - a)[.sup.2] + (x - [[a + b]/2])[.sup.2]] ||f'||[.sub.[infinity]]. (1.1) The Cebysev's inequality [6] states that if f', g' [member of] [L.sub.[infinity]] [a, b], then we have the inequality |T (f, g)| [less than or equal to] [1/12] (b - a)[.sup.2] ||f'||[.sub.[infinity]] ||g'||[.sub.[infinity]]. (1.2) The Gruss's inequality [6], states that if m [less than or equal to] f (x) [less than or equal to] M and n [less than or equal to] f (x) [less than or equal to] N < [infinity] for all x [member of] [a, b], then we have the inequality |T (f, g)| [less than or equal to] [1/4] (M - m) (N - n), (1.3) where m, M, n, N are real numbers. For some recent 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 the inequalities (1.1)-(1.3), see [1 - 11]. Recall the following definitions of a convex function and a log-convex function: Let f : [a, b] [right arrow] R and g : [a, b] [right arrow] (0, [infinity]). The functions f and g are called convex Convex Curved, as in the shape of the outside of a circle. Usually referring to the price/required yield relationship for option-free bonds. on [a, b] and log-convex on [a, b], respectively, if f ([lambda]x + (1 - [lambda]) y) [less than or equal to] [lambda]f (x) + (1 - [lambda]) f (y), and f (tx + (1 - t) y) [less than or equal to] [f (x)][.sup.[lambda]] [f (y)][.sup.1-[lambda]] for all x, y [member of] [a, b] and [lambda] [member of] [0, 1]. In [10], Pachpatte established the following four theorems This is a list of theorems, by Wikipedia page. See also
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, g : [a, b] [right arrow] R be absolutely continuous functions on [a, b]. ([a.sub.1]) If |f'| and |g'| are convex on [a, b], then |S (f, g)| [less than or equal to] [1/4] [[1/4] + ([x - [[a+b]/2]]/[b - a])[.sup.2]] (b - a) x {|g (x)| [|f' (x)| + ||f'||[.sub.[infinity]]] + |f (x)| [|g' (x)| + ||g'||[.sub.[infinity]]]} for x [member of] [a, b]. ([a.sub.2]) If |f'| and |g'| are log-convex on [a, b], then |S (f, g)| [less than or equal to] [1/2 (b - a)] {|g (x)| |f' (x)| [[integral].sub.s.sup.b] |x - t| ([A - 1]/lnA) dt +|f (x)| |g' (x)| [[integral].sub.a.sup.b] |x - t| ([B - 1]/lnB) dt} for x [member of] [a, b], where A = |f' (t)|/|f' (x)| and B = |g' (t)|/|g' (x)|. (1.4) Theorem B. Let f and g : [a, b] [right arrow] R be absolutely continuous functions on [a, b]. ([b.sub.1]) If |f'| and |g'| are convex on [a, x] and [x, b], then |S (f, g)| [less than or equal to] 1/2 {|g (x)| F (x) + |f (x)| G (x)}, for x [member of] [a, b], where F (x) = [1/6] [|f' (a)|([x - a]/[b - a])[.sup.2] + |f' (b)|([b - x]/[b - a])[.sup.2] + {1 + 4 ([x - [[a+b]/2]]/[b - a])[.sup.2]} |f' (x)|] (b - a) and G (x) = [1/6] [|g' (a)| ([x - a]/[b - a])[.sup.2] + |g' (b)|([b - x]/[b - a])[.sup.2] + {1 + 4 ([x - [[a+b]/2]]/[b - a])[.sup.2]} |g' (x)|] (b - a) for x [member of] [a, b]. ([b.sub.2]) If |f'| and |g'| are log-convex on [a, x] and [x, b], then |S (f, g)| [less than or equal to] [1/2] [|g (x)| P (x) + |f (x)| Q (x)], for x [member of] [a, b], where P (x) = (b - a) [|f' (a)| ([x - a]/[b - a])[.sup.2] [[[A.sub.1] ln [A.sub.1] + 1 - [A.sub.1]]/[(ln [A.sub.1])[.sup.2]]] + |f' (b)| ([b - x]/[b - a])[.sup.2] [[[B.sub.1] ln [B.sub.1] + 1 - [B.sub.1]]/[(ln [B.sub.1])[.sup.2]]]], Q (x)=(b - a) [|g' (a)|([x - a]/[b - a])[.sup.2] [[[A.sub.2] ln [A.sub.2] + 1 - [A.sub.2]]/[(ln [A.sub.2])[.sup.2]]] + |g' (b)| ([b - x]/[b - a])[.sup.2] [[[B.sub.2] ln [B.sub.2] + 1 - [B.sub.2]]/[(ln [B.sub.2])[.sup.2]]]], and [A.sub.1] = |f' (x)|/|f' (a)|, [B.sub.1] = |f' (x)|/|f' (b)|, (1.5) [A.sub.2] = |g' (x)|/|g' (a)|, [B.sub.2] = |g' (x)|/|g' (b)|, (1.6) for x [member of] [a, b]. Theorem C. Let f and g : [a, b] [right arrow] R be absolutely continuous functions on [a, b]. ([c.sub.1]) If |f'| and |g'| are convex on [a, b], then |T (f, g)| [less than or equal to] [1/[4 (b - a)[.sup.2]]] [[integral].sub.a.sup.b] [|g (x)| [|f' (x)| + ||f'||[[.sub.[infinity]]] + |f' (x)| [|g' (x)| + ||g'||[.sub.[infinity]]]] E (x) dx, where E (x) = [(x - a)[.sup.2] + (b - x)[.sup.2]]/2 (1.7) for x [member of] [a, b]. ([c.sub.2]) If |f'| and |g'| are log-convex on [a, b], then |T (f, g)| [less than or equal to] [1/[2 (b - a)[.sup.2]]] [[integral].sub.a.sup.b] [|g (x)| [[integral].sub.a.sup.b] |x - t| |f' (x)| ([A - 1]/ln A) dt + |f (x)| [[integral].sub.a.sup.b] |x - t| |g' (x)| ([B - 1]/ln B) dt] dx where A, B are defined as in (1.4). Theorem D. Let f and g : [a, b] [right arrow] R be absolutely continuous functions on [a, b]. ([d.sub.1]) If |f'| and |g'| are convex on [a, b], then |T (f, g)| [less than or equal] [1/2] [[integral].sub.a.sup.b] [([x - a]/[b - a])[.sup.2] [|g (x)| {[1/6] |f' (a)| + [1/3] |f' (x)|} + |f (x)| {[1/6] |g' (a)| + [1/3] |g' (x)|}] + ([x - a]/[b - a])[.sup.2] |g (x)| {[1/3] |f' (x)| + [1/6] |f' (b)|} + |f (x)| {[1/3] |g' (x)| + [1/6] |g' (b)|}]] dx, ([d.sub.2]) If |f'| and |g'| are log-convex on [a, x] and [x, b], then |T (f, g)| [less than or equal to] [1/2] [[integral].sub.a.sup.b] [([x - a]/[b - a])[.sup.2] {|g (x)||f' (a)| [[[A.sub.1] ln [A.sub.1] + 1 - [A.sub.1]]/[(ln [A.sub.1])[.sup.2]]] + |f (x)||g' (a)| [[[A.sub.2] ln [A.sub.2] + 1 - [A.sub.2]]/[(ln [A.sub.2])[.sup.2]]]} + ([x - x]/[b - a])[.sup.2] {|g (x)| |f' (b)| [[[B.sub.1] ln [B.sub.1] + 1 - [B.sub.1]]/[(ln [B.sub.1])[.sup.2]]] + |f (x)||g' (b)| [[[B.sub.2] ln [B.sub.2] + 1 - [B.sub.2]]/[(ln [B.sub.2])[.sup.2]]]}] dx where [A.sub.1], [B.sub.1] and [A.sub.2], [B.sub.2] are defined as in (1.5) and (1.6), respectively. In this paper, we establish some inequalities which generalize Theorems AD. 2. Main Results Throughout in this section, let [[alpha].sub.i] [member of] R (i = 1,..., n), [alpha] = ([[alpha].sub.1],..., [[alpha].sub.n]) with [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 (i=1)] [[alpha].sub.i] = 1 and let [S.sub.[alpha]] ([f.sub.1],..., [f.sub.n]) = [n.[product].[i=1]] [f.sub.i] (x) - [1/(b - a)] [n.summation over (i=1)] ([n.[product].[j=1,j[not equal to]i]] [[alpha].sub.i][f.sub.j] (x) [[integral].sub.a.sup.b] [f.sub.i] (t) dt), and [T.sub.[alpha]] ([f.sub.1],..., [f.sub.n]) = [1/[b - a]] [[integral].sub.a.sup.b] [n.[product].[i=1]] [f.sub.i] (x) dx - [n.summation over (i=1)] [([1/[b - a]] [[integral].sub.a.sup.b] [n.[product].[j=1,j[not equal to]i]] [[alpha].sub.i][f.sub.j] (x) dx) ([1/[b - a]] [[integral].sub.a.sup.b] [f.sub.i] (x) dx)] where x [member of] [a, b], and [f.sub.i] (i = 1,..., n) are integrable on [a, b]. In order to prove our results, we need the following identities proved in [1] and [2], respectively: f (x) = [1/[b - a]] [[integral].sub.a.sup.b]f (t) dt + [1/[b - a]] [[integral].sub.a.sup.b] (x - t) [[[integral].sub.0.sup.1]f' [(1 - [lambda]) x + [lambda]t] dt] d[lambda] (2.1) and f (x) = [1/[b - a]] [[integral].sub.a.sup.b] f (t) dt + (x - a)[.sup.2] [1/[b - a]] [[integral].sub.0.sup.1] [lambda]f' [(1 - [lambda]) a + [lambda]x] d[lambda] - (b - x)[.sup.2] [1/[b - a]] [[integral].sub.0.sup.1] [lambda]f' [[lambda]x + (1 - [lambda]) b] d[lambda], (2.2) for x [member of] [a, b] where f : [a, b] [right arrow] R is an absolutely continuous function on [a, b]. Theorem 1. Let [f.sub.i] : [a, b] [right arrow] R (i = 1,..., n) be absolutely continuous functions on [a, b]. (a) If |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], then |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [[b - a]/2] [[1/4] + ([x - [[a+b]/2]]/[b - a])[.sup.2]] x [n.summation over (i=1)] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.i] (x)| (|[f'.sub.i] (x) + ||[f'.sub.i]||[.sub.[infinity]])], (2.3) for x [member of] [a, b]. (b) If |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, b], then |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] {[1/[b - a]] [n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| |[f'.sub.i] (x)| [[integral].sub.a.sup.b] |x - t| [([[D.sub.i] - 1]/[ln [D.sub.i]])] dt} (2.4) for x [member of] [a, b], where [D.sub.i] = |[f'.sub.i] (t)|/|[f'.sub.i] (x)| (i = 1,..., n) (2.5) for x, t [member of] [a, b]. Proof. Using (2.1), we have the identities [f.sub.1] (x) - [1/[b - a]] [[integral].sub.a.sup.b] [f.sub.1] (t) dt = [1/[b - a]] [[integral].sub.a.sup.b] (x - t) [[[integral].sub.0.sup.1] [f'.sub.1] [(1 - [lambda]) x + [lambda]t] d[lambda]] dt, (2.6.1) [f.sub.2] (x) - [1/[b - a]] [[integral].sub.a.sup.b] [f.sub.2] (t) dt = [1/[b - a]] [[integral].sub.a.sup.b] (x - t) [[[integral].sub.0.sup.1] [f'.sub.2] [(1 - [lambda]) x + [lambda]t] d[lambda]] dt, (2.6.2) ... [f.sub.n] (x) - [1/[b - a]] [[integral].sub.a.sup.b] [f.sub.n] (t) dt = [1/[b - a]] [[integral].sub.a.sup.b] (x - t) [[[integral].sub.0.sup.1] [f'.sub.n] [(1 - [lambda]) x + [lambda]t] d[lambda]] dt (2.6.n) for [member of] 2 [a, b]. Multiplying mul·ti·ply 1 v. mul·ti·plied, mul·ti·ply·ing, mul·ti·plies v.tr. 1. To increase the amount, number, or degree of. 2. Mathematics To perform multiplication on. both sides of (2.6.i) by [n.[product].[j=1,j[not equal to]I]] [[alpha].sub.i][f.sub.j] (x) (i = 1,..., n) and adding the resulting identities, we get [S.sub.[alpha]] ([f.sub.1],..., [f.sub.n]) = [n.summation over (i=1)] {[1/[b - a]] [n.[product].[j=1,j[not equal to]i]] [[alpha].sub.i][f.sub.j] (x) [[integral].sub.a.sup.b] (x - t) [[[integral].sub.0.sup.1] [f'.sub.i] [(1 - [lambda])x + [lambda]t] d[lambda]] dt}. (2.7) (a) Since |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], from (2.7) we get that |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] {[1/[b - a]] [n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [[integral].sub.a.sup.b] |x - t| [[[integral].sub.0.sup.1] |[f'.sub.i] [(1 - [lambda]) x + [lambda]t]| d[lambda]] dt} [less than or equal to] [n.summation over (i=1)] {[1/[b - a]] [n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [[integral].sub.a.sup.b] |x - t| [[[integral].sub.0.sup.1] (1 - [lambda]) |[f'.sub.i] (x)| + [lambda] |[f'.sub.i](t)| d[lambda]] dt} = [n.summation over (i=1)] {[1/[b - a]] [n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [[integral].sub.a.sup.b] |x - t| [1/2] (|[f'.sub.i] (x)| + |[f'.sub.i] (t)|) dt} [less than or equal to] [1/[2 (b - a)]] [[integral].sub.a.sup.b] |x - t| dt [n.summation over (i=1)] {[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| (|[f'.sub.i] (x)| + ||[f'.sub.i]||[.sub.[infinity]])} = [[(x - a)[.sup.2] + (b - x)[.sup.2]]/[4(b - a)]] [n.summation over (i=1)] {[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| (|[f'.sub.i] (x)| + ||[f'.sub.i]||[.sub.[infinity]])} [[b - a]/2] [[1/4] + ([x - [[a+b]/2]]/[b - a])[.sup.2]] [n.summation over (i=1)] {[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| (|[f'.sub.i] (x)| + ||[f'.sub.i]||[.sub.[infinity]])} which is the inequality (2.3). (b) Since |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, b], from (2.7) we get that |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] {[1/[b - a]] [n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [[integral].sub.a.sup.b] |x - t| [[[integral].sub.0.sup.1] |[f'.sub.i] [(1 - [lambda]) x + [lambda]t]| d[lambda]] dt} [n.summation over (i=1)] {[1/[b - a]] [n.[product].[i=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [[integral].sub.a.sup.b] |x - t| [[[integral].sub.0.sup.1] [|[f'.sub.i] (x)|][.sup.1-[lambda]] [|[f'.sub.i] (t)|][.sup.[lambda]] d[lambda]] dt} [n.summation over (i=1)] {[1/[b - a]] [n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [[integral].sub.a.sup.b] |x - t| [|[f'.sub.i] (x)| [[integral].sub.0.sup.1] [|[f'.sub.i] (t)]/|[f'.sub.i] (x)|][.sup.[lambda]] d[lambda] dt} = [n.summation over (i=1)] {[1/[b - a]] [n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| |[f'.sub.i] (x)| [[integral].sub.a.sup.b] |x - t| [([[|[f'.sub.i](t)|/|[f'.sub.i](x)|] - 1]/[ln [|[f'.sub.i] (t)|/|[f'.sub.i] (x)|]])] dt} [n.summation over (i=1)] {[1/b - a] [n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| |[f'.sub.i] (x)| [[integral].sub.a.sup.b] |x - t| [([[D.sub.i] - 1]/[ln [D.sub.i]])] dt} which is the inequality (2.4), where [D.sub.i] (i = 1,..., n) are defined as in (2.5).. This completes the proof. Let [alpha] = (1/n,..., 1/n) in Theorem 1, then we have the following corollary corollary: see theorem. : Corollary 1. Let [f.sub.i] : [a, b] [right arrow] R (i = 1,..., n) be absolutely continuous functions on [a, b]. (a) If |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], then |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [[b - a]/2n] [[1/4] + ([x - [[a+b]/2]]/[b - a])[.sup.2]] x [n.summation over (i=1)] [[n.[product].[j=1,j[not equal to]i]] |[f.sub.j] (x)| (|[f'.sub.i] (x)| + ||[f'.sub.i]||[.sub.[infinity]])], for x [member of] [a, b]. (b) If |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, b], then |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] [1/n] {[1/[b - a]] [n.[product].[j=1,j[not equal i]]] |[f.sub.i] (x)| |[f'.sub.i] (x)| [[integral].sub.a.sup.b] |x - t| [([[D.sub.i] - 1]/[ln [D.sub.i]])] dt} for x [member of] [a, b], where [D.sub.i] (i = 1,..., n) are defined as in (2.5). Remark 1. If we choose n = 2, then Corollary 1 reduces to Theorem A. Theorem 2. Let [f.sub.i] : [a, b] [right arrow] R (i = 1,..., n) be absolutely continuous functions on [a, b]. (a) If |[f'.sub.i]| (i = 1,..., n) are convex on [a, x] and [x, b], then |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] ([n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [F.sub.i] (x)) (2.8) where [F.sub.i] (x) = [[b - a]/6] [|[f'.sub.i] (a)| ([x - a]/[b - a])[.sup.2] + |[f'.sub.i] (b)| ([b - x]/[b - a])[.sup.2] + (1 + 4 ([x - [[a+b]/2]]/[b - a])[.sup.2]) |[f'.sub.i] (x)|] (2.9) for x [member of] [a, b] and i = 1,..., n. (b) If |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, x] and [x, b], then |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] ([n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [H.sub.i] (x)) (2.10) where [I.sub.i] = |[f'.sub.i] (x)|/|[f'.sub.i] (a)|, [J.sub.i] = [|[f'.sub.i] (x)|/|[f'.sub.i] (b)|] (i = 1,..., n) (2.11) and [H.sub.i] (x) = (b - a) [|[f'.sub.i] (a)| ([x - a]/[b - a])[.sup.2] [[[I.sub.i] ln [I.sub.i] + 1 - [I.sub.i]]/[(ln [I.sub.i])[.sup.2]]] + |[f'.sub.i] (b)| ([b - x]/[b - a])[.sup.2] [[[J.sub.i] ln [J.sub.i] + 1 - [J.sub.i]]/[(ln [J.sub.i])[.sup.2]]]] (2.12) for x [member of] [a, b] and i = 1,..., n. Proof. Using (2.2), we have the identities [f.sub.1] (x) - [1/[b - a]] [[integral].sub.a.sup.b] [f.sub.1] (t) dt = [[(x - a)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.1] [(1 - [lambda])a + [lambda]x] d[lambda] - [[(b - x)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.1] [[lambda]x + (1 - [lambda]) b] d[lambda], (2.13.1) [f.sub.2] (x) - [1/[b - a]] [[integral].sub.a.sup.b] [f.sub.2] (t) dt = [[(x - a)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.2] [(1 - [lambda]) a + [lambda]x] d[lambda] - [[(b - x)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.2] [[lambda]x + (1 - [lambda]) b] d[lambda], (2.13.2) ... [f.sub.n] (x) - [1/[b - a]] [[integral].sub.a.sup.b] [f.sub.n] (t) dt = [[(x - a)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.n] [(1 - [lambda]) a + [lambda]x] d[lambda] - [[(b - x)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.n] [[lambda]x + (1 - [lambda]) b] d[lambda], (2.13.n) for x [member of] [a, b]. Multiplying both sides of (2.13.i) by [n.[product].[j=1,j[not equal to]i]] [[alpha].sub.i][f.sub.j] (x) (i = 1,..., n) and adding the resulting identities, we get [S.sub.[alpha]] ([f.sub.1],..., [f.sub.n]) = [n.summation over (i=1)] {[n.[product].[j=1,j[not equal to]i]] [[alpha].sub.i][f.sub.j] (x) [[[(x - a)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.i] [(1 - [lambda]) a + [lambda]x] d[lambda] - [[(b - x)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.i] [[lambda]x + (1 - [lambda]) b] d[lambda]]} (2.14) Using (2.14), we get that |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)|[M.sub.i] (x)] (2.15) where [M.sub.i] (x) = [[(x - a)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda] |[f'.sub.i] [(1 - [lambda]) a + [lambda]x]| d[lambda] + [[(b - x)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda] |[f'.sub.i] [[lambda]x + (1 - [lambda]) b]| d[lambda] (2.16) for x [member of] [a, b] and i = 1,..., n. (a) Since |[f'.sub.i]| (i = 1,..., n) are convex on [a, x] and [x, b], we have that [[integral].sub.0.sup.1] [lambda] |[f'.sub.i] [(1 - [lambda]) a + [lambda]x]| d[lambda] [less than or equal to] |[f'.sub.i] (a)| [[integral].sub.0.sup.1] [lambda] (1 - [lambda]) d[lambda] + |[f'.sub.i] (x)| [[integral].sub.0.sup.1] [[lambda].sup.2]d[lambda] = [1/6] |[f'.sub.i] (a)| + [1/3] |[f'.sub.i] (x)| (2.17) and [[integral].sub.0.sup.1] [lambda] |[f'.sub.i] [[lambda]x + (1 - [lambda]) b]| d[lambda] [less than or equal to] |[f'.sub.i] (x)| [[integral].sub.0.sup.1] [[lambda].sup.2]d[lambda] + |[f'.sub.i] (b)| [[integral].sub.0.sup.1] [lambda] (1 - [lambda]) d[lambda] = [1/3] |[f'.sub.i] (x)| + [1/6] |[f'.sub.i] (b)| (2.18) where x [member of] [a, b] and i = 1,..., n. From (2.16)-(2.18), we get that [M.sub.i] (x) [less than or equal to] [[(x - a)[.sup.2]]/[b - a]] [[1/6] |[f'.sub.i] (a)| + [1/3] |[f'.sub.i] (x)|] + [[(b - x)[.sup.2]]/[b - a]] [[1/3] |[f'.sub.i] (x)| + [1/6] |[f'.sub.i] (b)|] = [[b - a]/6] [|[f'.sub.i] (a)|([x - a]/[b - a])[.sup.2] + |[f'.sub.i] (b)|([b - x]/[b - a])[.sup.2] + 2 |[f'.sub.i] (x)| (([x - a]/[b - a])[.sup.2] + ([b - x]/[b - a])[.sup.2])] = [F.sub.i] (x) (2.19) where x [member of] [a, b] and i = 1,..., n. Using (2.15) and (2.19), we get the inequality (2.8). (b) Since |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, x] and [x, b], we have that [[integral].sub.0.sup.1] [lambda] |[f'.sub.i] [(1 - [lambda]) a + [lambda]x]| d[lambda] [less than or equal to] [[integral].sub.0.sup.1] [lambda] |[f'.sub.i] (a)|[.sup.1-[lambda]] |[f'.sub.i] (x)|[.sup.[lambda]] d[lambda] = |[f'.sub.i] (a)| [[integral].sub.0.sup.1] [lambda][I.sub.i.sup.[lambda]] d[lambda] = |[f'.sub.i] (a)| [[[I.sub.i] ln [I.sub.i] + 1 - [I.sub.i]]/[(ln [I.sub.i])[.sup.2]]] (2.20) and [[integral].sub.0.sup.1] [lambda] |[f'.sub.i] [[lambda]x + (1 - [lambda]) b]| d[lambda] [less than or equal to] [[integral].sub.0.sup.1] [lambda] |[f'.sub.i] (x)|[.sup.[lambda]] |[f'.sub.i] (b)|[.sup.1-[lambda]] d[lambda] = |[f'.sub.i] (b)| [[integral].sub.0.sup.1] [lambda][J.sub.i.sup.[lambda]]d[lambda] = |[f'.sub.i] (b)| [[[I.sub.i] ln [J.sub.i] + 1 - [J.sub.i]]/[(ln [J.sub.i])[.sup.2]]] (2.21) where x [member of] [a, b] and i = 1,..., n. From (2.16), (2.20) and (2.21), we get [M.sub.i] (x) [less than or equal to] [H.sub.i] (x) (2.22) where x [member of] [a, b] and i = 1,..., n. Using (2.15) and (2.22), we get the inequality (2.10). This completes the proof. Let [alpha] = (1/n,..., 1/n) in Theorem 2, then we have the following corollary: Corollary 2.Let [f.sub.i] : [a, b] [right arrow] R (i = 1,..., n) be absolutely continuous functions on [a, b]. (a) If |[f'.sub.i]| (i = 1,..., n) are convex on [a, x] and [x, b], then |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/n] [n.summation over (i=1)]([n.[product].[j=1,j[not equal to]i]] |[f.sub.j] (x)| [F.sub.i] (x)) where [F.sub.i] (x) is defined as in (2.9) for x [member of] [a, b] and i = 1,..., n. (b) If |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, x] and [x, b], then |[S.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/n] [n.summation over (i=1)] ([n.[product].[j=1,j[not equal to]i]] |[f.sub.j] (x)| [H.sub.i] (x)) where [H.sub.i] (x) is defined as in (2.12) for x [member of] [a, b] and i = 1,..., n. Remark 2. If we choose n = 2, then Corollary 2 reduces to Theorem B. Theorem 3. Let [f.sub.i] : [a, b] [right arrow] R (i = 1,..., n) be absolutely continuous functions on [a, b]. (a) If |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], then |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/[2 (b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| (|[f'.sub.i] (x)| + ||[f'.sub.i]||[.sub.[infinity]]) E (x)] dx} (2.23) where E (x) is defined as in (1.7) for x [member of] [a, b]. (b) If |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, b], then |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/[(b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| x [[integral].sub.a.sup.b] (|x - t| |[f'.sub.i] (x)| ([[D.sub.i] - 1]/[ln [D.sub.i]])) dt] dx} (2.24) where [D.sub.i] (i = 1,..., n) are defined as in (2.5). Proof. From the hypotheses of [f.sub.i] (i = 1,..., n), the identity (2.7) holds. Integrating both sides of (2.7) with respect to x from a to b and rewriting re·write v. re·wrote , re·writ·ten , re·writ·ing, re·writes v.tr. 1. To write again, especially in a different or improved form; revise. 2. it, we have [T.sub.[alpha]] ([f.sub.1],..., [f.sub.n]) = [1/[(b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] [[alpha].sub.i][f.sub.j] (x) x [[integral].sub.a.sup.b] (x - t) ([[integral].sub.0.sup.1] [f'.sub.i] [(1 - [lambda]) x + [lambda]t] d[lambda]) dt] dx}. (2.25) (a) Since |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], from (2.25) we get that |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/[(b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| x [[integral].sub.a.sup.b] (|x - t| [[integral].sub.0.sup.1] [(1 - [lambda]) |[f'.sub.i] (x)| + [lambda] |[f'.sub.i] (t)|] d[lambda]) dt] dx} = [1/[(b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| x [[integral].sub.a.sup.b] (|x - t| [[|[f'.sub.i] (x)| + |[f'.sub.i] (t)|]/2]) dt] dx} [less than or equal to] [1/[2 (b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| x (|[f'.sub.i] (x)| + ||[f'.sub.i]||[.sub.[infinity]]) [[integral].sub.a.sup.b] |x - t| dt] dx} = [1/[2 (b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| (|[f'.sub.i] (x)| + ||[f'.sub.i]||[.sub.[infinity]]) E (x)] dx} (2.26) which is the inequality (2.23), where E (x) is defined as in (1.7). (b) Since |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, b], from (2.26) we get that |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/[(b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| x [[integral].sub.a.sup.b] (|x - t| [[integral].sub.0.sup.1] [[|[f'.sub.i] (x)|][.sup.1-[lambda]] [|[f'.sub.i] (t)|][.sup.[lambda]]] d[lambda]) dt] dx} = [1/[(b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| x [[integral].sub.a.sup.b] (|x - t| |[f'.sub.i] (x)| [[integral].sub.0.sup.1] [|[f'.sub.i] (t)|/|[f'.sub.i] (x)|][.sup.[lambda]] d[lambda]) dt] dx} = [1/[(b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| x [[integral].sub.a.sup.b] (|x - t| |[f'.sub.i] (x)| ([[D.sub.i] - 1]/[ln [D.sub.i]])) dt] dx} which is the inequality (2.4), where [D.sub.i] (i = 1,..., n) are defined as in (2.5). This completes the proof. Let [alpha] = (1/n,..., 1/n) in Theorem 3, then we have the following corollary: Corollary 3. Let [f.sub.i] : [a, b] [right arrow] R (i = 1,..., n) be absolutely continuous functions on [a, b]. (a) If |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], then |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/[2n (b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[f.sub.j] (x)|(|[f'.sub.i] (x)| + ||[f'.sub.i]||[.sub.[infinity]]) E (x)] dx} where E (x) is defined as in (1.7) for x [member of] [a, b]. (b) If |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, b] , then |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/[n (b - a)[.sup.2]]] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[f.sub.j] (x)| x [[integral].sub.a.sup.b] (|x - t| |[f'.sub.i] (x)| ([[D.sub.i] - 1]/[ln [D.sub.i]])) dt] dx} where [D.sub.i] (i = 1,..., n) are defined as in (2.5). Remark 3. If we choose n = 2, then Corollary 3 reduces to Theorem C. Theorem 4. Let [f.sub.i] : [a, b] [right arrow] R (i = 1,..., n) be absolutely continuous functions on [a, b]. (a) If |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], then |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| {([x - a]/[b - a])[.sup.2] ([1/6] |[f'.sub.i] (a)| + [1/3] |[f'.sub.i] (x)|) + ([b - x]/[b - a])[.sup.2] ([1/3] |[f'.sub.i] (x)| + [1/6] |[f'.sub.i] (b)|)}] dx}. (2.27) (b) If |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, x] and [x, b], then |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| {([x - a]/[b - a])[.sup.2] |[f'.sub.i]| (a)| [[[I.sub.i] ln [I.sub.i] + 1 - [I.sub.i]]/[(ln [I.sub.i])[.sup.2]]] + ([b - x]/[b - a])[.sup.2] |[f'.sub.i] (b)| [[[J.sub.i] ln [J.sub.i] + 1 - [J.sub.i]]/[(ln [J.sub.i])[.sup.2]]]}] dx} (2.28) where [I.sub.i], [J.sub.i] (i = 1,..., n) are defined as in (2.11). Proof. From the hypotheses of [f.sub.i] (i = 1,..., n), the identity (2.14) holds. Integrating both sides of (2.14) with respect to x from a to b and rewriting it, we have [T.sub.[alpha]] ([f.sub.1],..., [f.sub.n]) = [n.summation over (i=1)] {[[integral].sub.a.sup.b] ([n.[product].[j=1,j[not equal to]i]] [[alpha].sub.i][f.sub.j] (x) [[[(x - a)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.i] [(1 - [lambda]) a + [lambda]x] d[lambda] - [[(b - x)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda][f'.sub.i] [[lambda]x + (1 - [lambda]) b] d[lambda]] dx)}. (2.29) (a) Since |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], from (2.29) we get that |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than equal to] [n.summation over (i=1)] {[[integral].sub.a.sup.b] ([n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [[[(x - a)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda]|[f'.sub.i] [(1 - [lambda]) a + [lambda]x]| d[lambda] + [[(b - x)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] [lambda]|[f'.sub.i] [[lambda]x + (1 - [lambda]) b]| d[lambda]] dx)}. (2.30) [less than equal to] [n.summation over (i=1)] {[[integral].sub.a.sup.b] ([n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| x [[[(x - a)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] ([lambda] (1 - [lambda]) |[f'.sub.i] (a)| + [[lambda].sup.2] |[f'.sub.i] (x)|) d[lambda] + [[(b - x)[.sup.2]]/[b - a]] [[integral].sub.0.sup.1] ([[lambda].sup.2] |[f'.sub.i] (x)| + [lambda] (1 - [lambda]) |[f'.sub.i] (b)|) d[lambda]] dx)} = [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| {([x - a]/[b - a])[.sup.2] ([1/6] |[f'.sub.i] (a)| + [1/3] |[f'.sub.i] (x)|) + ([b - x]/[b - a])[.sup.2] ([1/3] |[f'.sub.i] (x)| + [1/6] |[f'.sub.i] (b)|)}] dx}. which is the inequality (2.27). (b) Since |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, x] and [x, b], from (2.30) we get that |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [n.summation over (i=1)] {[[integral].sub.a.sup.b] ([n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [([x - a]/[b - a])[.sup.2] [[integral].sub.0.sup.1] ([lambda][|[f'.sub.i] (a)|][.sup.1 - [lambda]] [|[f'.sub.i] (x)|][.sup.[lambda]]) d[lambda] + ([b - x]/[b - a])[.sup.2] [[integral].sub.0.sup.1] ([lambda][|[f'.sub.i] (x)|][.sup.[lambda]] [|[f'.sub.i] (b)|][.sup.1 - [lambda]]) d[lambda]] dx)} = [n.summation over (i=1)] {[[integral].sub.a.sup.b] ([n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [([x - a]/[b - a])[.sup.2] (|[f'.sub.i] (a)| [[integral].sub.0.sup.1] [lambda][I.sub.i.sup.[lambda]]d[lambda]) + ([b - x]/[b - a])[.sup.2] (|[f'.sub.i] (a)| [[integral].sub.0.sup.1] [lambda][J.sub.i.sup.[lambda]]d[lambda])] dx)} = [n.summation over (i=1)] {[[integral].sub.a.sup.b] ([n.[product].[j=1,j[not equal to]i]] |[[alpha].sub.i][f.sub.j] (x)| [([x - a]/[b - a])[.sup.2] |[f'.sub.i] (a)| [[[I.sub.i] ln [I.sub.i] + 1 - [I.sub.i]]/[(ln [I.sub.i])[.sup.2]]] + ([b - x]/[b - a])[.sup.2] |[f'.sub.i]| (b)| [[[J.sub.i] ln [J.sub.i] + 1 - [J.sub.i]]/[(ln [J.sub.i])[.sup.2]]]] dx)} which is the inequality (2.28), where [I.sub.i], [J.sub.i] (i = 1,..., n) are defined as in (2.11). This completes the proof. Let [alpha] = (1/n,..., 1/n) in Theorem 4, then we have the following corollary: Corollary 4. Let [f.sub.i] : [a, b] [right arrow] R (i = 1,..., n) be absolutely continuous functions on [a, b]. (a) If |[f'.sub.i]| (i = 1,..., n) are convex on [a, b], then |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/n] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[f.sub.j] (x)| {([x - a]/[b - a])[.sup.2] ([1/6] |[f'.sub.i] (a)| + [1/3] |[f'.sub.i] (x)|) + ([b - x]/[b - a])[.sup.2] ([1/3] |[f'.sub.i] (x)| + [1/6] |[f'.sub.i] (b)|)}] dx}. (b) If |[f'.sub.i]| (i = 1,..., n) are log-convex on [a, x] and [x, b], then |[T.sub.[alpha]] ([f.sub.1],..., [f.sub.n])| [less than or equal to] [1/n] [n.summation over (i=1)] {[[integral].sub.a.sup.b] [[n.[product].[j=1,j[not equal to]i]] |[f.sub.j] (x)| {([x - a]/[b - a])[.sup.2] |[f'.sub.i]| (a)| [[[I.sub.i] ln [I.sub.i] + 1 - [I.sub.i]]/[(ln [I.sub.i])[.sup.2]]] + ([b - x]/[b - a])[.sup.2] |[f'.sub.i]| (b)| [[[J.sub.i] ln [J.sub.i] + 1 - [J.sub.i]]/[(ln [J.sub.i])[.sup.2]]]}] dx} where [I.sub.i], [J.sub.i] (i = 1,..., n) are defined as in (2.11). Remark 4. If we choose n = 2, then Corollary 4 reduces to Theorem D. References [1] N. S. Barnett Barnett as a personal name can refer to:
(networking) edu - ("education") The top-level domain for educational establishments in the USA (and some other countries). E.g. "mit.edu". The UK equivalent is "ac.uk". .au/v5n2.html] [2] P. Cerone and S. S. Dragomir, Ostrowski type inequalities for functions whose derivatives satisfy certain convexity Convexity A measure of the curvature in the relationship between bond prices and bond yields. Notes: Positive convexity corresponds to curvature that opens upward. Negative convexity corresponds to curvature that opens downward. assumptions, Demonstratio Math., 37(2) (2004), 299-308. [3] S. S. Dragomir and A. Sofo, Ostrowski type inequalities for functions whose derivatives are convex, Proceedings of the 4th International Conference on Modelling and Simulation The mathematical representation of the interaction of real-world objects. See scientific application and simulator. Simulation A broad collection of methods used to study and analyze the behavior and performance of actual or theoretical systems. , November November: see month. 11-13, 2002. Victoria University, Melbourne Melbourne, city, Australia Melbourne, city (1991 pop. 2,761,995), capital of Victoria, SE Australia, on Port Phillip Bay at the mouth of the Yarra River. Melbourne, Australia's second largest city, is a rail and air hub and financial and commercial center. , Australia Australia (ôstrāl`yə), smallest continent, between the Indian and Pacific oceans. With the island state of Tasmania to the south, the continent makes up the Commonwealth of Australia, a federal parliamentary state (2005 est. pop. . RGMIA Res. Rep. Coll., 5(Supp) (2002), Art 30. [ONLINE: http://rgmia.vu.edu.au/v5(E).html] [4] S. S. Dragomir and Th. M. Rassias (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. , Kluwer Academic Publishers, Dordrecht Dordrecht (dôr`drĕkht) or Dort (dôrt), city (1994 pop. 113,394), South Holland prov., SW Netherlands, at the point where the Lower Merwede divides to form the Noord and Oude Maas (Old Meuse) rivers. , 2002. [5] D. S. Mitrinovic, J. E. Pecaric and A. M. Fink fink Slang n. 1. A contemptible person. 2. An informer. 3. A hired strikebreaker. intr.v. finked, fink·ing, finks 1. To inform against another person. , Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991. [6] D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993. [7] B. G. Pachpatte, A note on integral inequalities involving two log-convex functions, Math. Inequal In`e´qual a. 1. Unequal; uneven; various. . Appl., 7(4) (2004),511-515. [8] B. G. Pachpatte, A note on Hadamard type integral inequalities involving several log-convex functions, Tamkang J. Math., 36(1) (2005), 43-47. [9] B. G. Pachpatte, Mathematical Inequalities, North-Holland Mathematical Library, Vol.67 Elsevier Elsevier, the world's largest publisher of medical and scientific literature, forms part of the Reed Elsevier group. Based in Amsterdam, the company has substantial operations in the UK, USA and elsewhere. , 2005. [10] B. G. Pachpatte, On Ostrowski-Gruss-Cebysev type inequalities for functions whose modulus of derivatives are convex, J. Inequal. Pure Appl. Math., 6(4) (2005), Article 128. [11] J. E. Pecaric, F. Proschan and Y. L. Tang tang, in zoology tang: see butterfly fish. , Convex Functions, Partial Orderings partial ordering - A relation R is a partial ordering if it is a pre-order (i.e. it is reflexive (x R x) and transitive (x R y R z => x R z)) and it is also antisymmetric (x R y R x => x = y). and Statistcal Applications, Academic Press, New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of , 1991. Shiow-Ru Hwang Hwang can refer to:
China Institute of Technology, Nankang Nankang may refer to:
Received July July: see month. 28, 2005, Accepted August 10, 2005. * 2000Mathematics Subject Classification. 26D15, 26D20. ([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 |
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