Some Gruss type inequalities for vector-valued functions in Banach spaces and applications.Abstract Some Gruss type inequalities This page lists Wikipedia articles about named mathematical inequalities. Pure mathematics
The theory of vector-valued functions of vector-valued functions A vector-valued function is a mathematical function that maps real numbers onto vectors. Vector-valued functions can be defined as:
metric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the are also pointed out. 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: Gruss inequality, Bochner integral, Banach spaces, Hilbert spaces. 1. Introduction In 1934, G. Gruss [5] proved the following inequality |[1/[b - a]] [[integral].sub.a.sup.b] f (t) g (t) dt - [1/[b - a]] [[integral].sub.a.sup.b] f (t) dt x [1/[b - a]] [[integral].sub.a.sup.b] g (t) dt| [less than or equal to] [1/4](M - m) (N - n), (1.1) provided -[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. ] < m [less than or equal to] f (t) [less than or equal to] M < [infinity], -[infinity] < n [less than or equal to] g (t) [less than or equal to] N < [infinity] for a.e. t [member of] [a, b]; and showed that the constant 1/4 is the best possible. An extension of the above result to vector-valued functions in Hilbert spaces was obtained in 2001 by S.S. Dragomir Dragomir (pronounced Drah-go-meer) is a name of Slavic origin, typical for Bulgaria and Serbia, as well as Romania. It is comprised of the Slavic words drag (dear, precious) and mir (peace). It can be translated as To whom peace is precious, i.e. [3]: Let (H; <[dot],[dot]>) be a Hilbert space over K, [OMEGA 1. (programming) Omega - A prototype-based object-oriented language from Austria. ["Type-Safe Object-Oriented Programming with Prototypes - The Concept of Omega", G. Blaschek, Structured Programming 12:217-225, 1991]. 2. ] [subset A group of commands or functions that do not include all the capabilities of the original specification. Software or hardware components designed for the subset will also work with the original. ] [R.sup.n] a measurable set, f, g : [OMEGA] [right arrow] H Bochner Bochner is the surname of:
This page or section lists people with the surname Bochner. measurable functions In mathematics, measurable functions are well-behaved functions between measurable spaces. Functions studied in analysis that are not measurable are generally considered pathological. on [OMEGA] and f, g [member of] [L.sub.2,[rho]] ([OMEGA], H), where [L.sub.2,[rho]] ([OMEGA], H) [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] {f : [OMEGA] [right arrow] H; [[integral].sub.[OMEGA]] [rho] (t) ||f (t)||[.sup.2] dt < [infinity]} and [rho] : [OMEGA] [right arrow] [0, [infinity]) is a Lebesgue integrable function In mathematics, an integrable function is a function whose integral exists. Unless specifically stated, the integral in question is usually the Lebesgue integral. Otherwise, one can say that the function is "Riemann-integrable" (i.e. with [[integral].sub.[OMEGA]] [rho] (x) dx = 1. If there exist vectors Vectors Something used to transport genetic information to a cell. Mentioned in: Gene Therapy x, X, y, Y [member of] H such that either [[integral].sub.[OMEGA]] [rho] (t) Re <X - f (t), f (t) - x> dt [greater than or equal to] 0, and [[integral].sub.[OMEGA]] [rho] (t) Re <Y - g (t), g (t) - y> dt [greater than or equal to] 0, (1.2) or, equivalently, [1], either, [[integral].sub.[OMEGA]] [rho] (t) ||f (t) - [[x + X]/2]||[.sup.2] dt [less than or equal to] [1/4] ||X - x||[.sup.2], and [[integral].sub.[OMEGA]] [rho] (t) ||g (t) - [[y + Y]/2]||[.sup.2] dt [less than or equal to] [1/4] ||Y - y||[.sup.2] (1.3) then ||[[integral].sub.[OMEGA]] [rho] (t) <f (t), g (t)> dt - <[[integral].sub.[OMEGA]] [rho] (t) f (t) dt, [[integral].sub.[OMEGA]] [rho] (t) g (t) dt>| [less than or equal to] [1/4] ||X - x|| ||Y - y||. (1.4) The constant 1/4 in (1.4) is again the best possible. This result was improved in [1], where the authors, on using a finer argument, proved that [[integral].sub.[OMEGA]] [rho] (t) <f (t), g (t)> dt - <[[integral].sub.[OMEGA]] [rho] (t) f (t) dt, [[integral].sub.[OMEGA]] [rho] (t) g (t) dt>| [less than or equal to] [1/4] ||X - x|| ||Y - y|| - [[[integral].sub.[OMEGA]] [rho] (t) Re <X - f (t), f (t) - x> dt x [[integral].sub.[OMEGA]] [rho] (t) Re <Y - g (t), g (t) - y> dt][.sup.1/2] [less than or equal to] [1/4] ||X - x|| ||Y - y||, (1.5) provided f and g satisfy either (1.2) or, equivalently, (1.3). Under the same type of hypothesis An assumption or theory. During a criminal trial, a hypothesis is a theory set forth by either the prosecution or the defense for the purpose of explaining the facts in evidence. , the authors of [1] also established the following result: ||[[integral].sub.[OMEGA]] [rho] (t) [alpha] (t) f (t) dt - [[integral].sub.[OMEGA]] [rho] (t) [alpha] (t) dt [[integral].sub.[OMEGA]] [rho] (t) f (t) dt|| [less than or equal to] [1/4] |A - a| ||X - x|| - ([[integral].sub.[OMEGA]] [rho] (t) Re [(A - [alpha] (t)) ([bar.[alpha] (t)] - [bar.a])] dt x [[integral].sub.[OMEGA]] [rho] (t) Re <X - f (t), f (t) - x> dt)[.sup.1/2] [less than or equal to] [1/4] |A - a| ||X - x||, (1.6) provided f satisfies either (1.2) or (1.3) and the scalar scalar, quantity or number possessing only sign and magnitude, e.g., the real numbers (see number), in contrast to vectors and tensors; scalars obey the rules of elementary algebra. Many physical quantities have scalar values, e.g. function [alpha] : [OMEGA] [right arrow] K satisfies the equivalent conditions: Re [(A - [alpha] (t)) ([bar.[alpha] (t)] - [bar.a])] [greater than or equal to] 0 and |[alpha] (t) - [[A + a]/2]| [less than or equal to] [1/2] |A - a|, for a.e. t [member of] [OMEGA], where A, a [member of] K are given constants. Note that in both inequalities (1.5) and (1.6) the quantity 1/4 is again the best possible. The main aim of this paper is to establish some Gruss type inequalities for Bochner integrable functions taking values in a Banach space. Applications for the case of Hilbert spaces and in connection with the Heisenberg inequality are also given. 2. Inequalities in Banach Spaces 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. 1. Let (X, ||dot||) be a Banach space over the real or complex number field K, [OMEGA] [member of] [R.sup.n] a measurable set and [rho] : [OMEGA] [right arrow] [0, [infinity]) a Lebesgue integrable function with [[integral].sub.[OMEGA]] [rho] (x) dx = 1. If [alpha] : [OMEGA] [right arrow] K is a Lebesgue integrable function such that there exists [gamma], [GAMMA] [member of] K with |[alpha] (x) - [[[gamma] + [GAMMA]]/2]| [less than or equal to] [1/2] |[GAMMA] - [gamma]| (2.1) or, equivalently, Re [([GAMMA] - [alpha] (x)) ([bar.[alpha] (x)] - [bar.[gamma]])] [greater than or equal to] 0 (2.2) for a.e. x [member of] [OMEGA], and f : [OMEGA] [right arrow] X is a Bochner measurable function such that [rho][alpha]f and [rho]f are Bochner integrable on [OMEGA], then, ||[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) f (x) dx - [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx x [[integral].sub.[OMEGA]] [rho] (x) f (x) dx|| [less than or equal to] [1/2] |[GAMMA] - [gamma]| [[integral].sub.[OMEGA]] [rho] (x)||f (x) - [[integral].sub.[OMEGA]] [rho] (y) f (y) dy|| dx. (2.3) The constant 1/2 in (2.3) is the best possible. Proof. The following Sonin Suo Ni, (1601-1667) also known as Soni, and rarely Sony (Manchu:  ; Chinese: 索尼; Pinyin: Suǒní type identity for the Bochner integral
holds:
[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) f (x) dx - [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx x [[integral].sub.[OMEGA]] [rho] (x) f (x) dx = [[integral].sub.[OMEGA]] [rho] (x) ([alpha] (x) - [[[gamma] + [GAMMA]]/2]) (f (x) - [[integral].sub.[OMEGA]] [rho] (y) f (y) dy) dx. (2.4) (for the scalar case, see [6, p. 246]). Taking the norm in (2.4), we deduce de·duce tr.v. de·duced, de·duc·ing, de·duc·es 1. To reach (a conclusion) by reasoning. 2. To infer from a general principle; reason deductively: ||[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) f (x) dx - [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx x [[integral].sub.[OMEGA]] [rho] (x) f (x) dx|| [less than or equal to] [[integral].sub.[OMEGA]] [rho] (x) - |[alpha] (x) - [[[gamma] + [GAMMA]]/2]| ||f (x) - [[integral].sub.[OMEGA]] [rho] (y) f (y) dy|| dx [less than or equal to] [1/2] |[GAMMA] - [gamma]| [[integral].sub.[OMEGA]] [rho] (x) ||f (x) - [[integral].sub.[OMEGA]] [rho] (y) f (y) dy|| dx. and the inequality (2.3) is obtained. Now, to prove the sharpness of the constant 1/2, assume that (2.3) holds for [OMEGA] = [a, b], X = R, [rho] [equivalent to] 1/[b-a], with a constant c > 0. That is: |[1/[b - a]] [[integral].sub.a.sup.b] [alpha] (t) f (t) dt - [1/[b - a]] [[integral].sub.a.sup.b] [alpha] (t) dt x [1/[b - a]] [[integral].sub.a.sup.b] f (t) dt| [less than or equal to] c ([GAMMA] - [gamma]) [1/[b - a]] [[integral].sub.a.sup.b] |f (t) - [1/[b - a]] [[integral].sub.a.sup.b] f (s) ds| dt, (2.5) where -[infinity] < [gamma] [less than or equal to] [alpha] (t) [less than or equal to] [GAMMA] < [infinity] for a.e. t [member of] [a, b], and [[integral].sub.a.sup.b] is the usual Lebesgue integral on [a, b]. If we choose, in (2.5), [alpha] = f and f : [a, b] [right arrow] R defined by [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. 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. ], then, obviously [gamma] = -1, [GAMMA] = 1, [1/[b - a]] [[integral].sub.a.sup.b] [f.sup.2] (t) dt - ([1/[b - a]] [[integral].sub.a.sup.b] f (t) dt)[.sup.2] = 1, [1/[b - a]] [[integral].sub.a.sup.b] |f (t) - [1/[b - a]] [[integral].sub.a.sup.b] f (s) ds| dt = 1, and by (2.5) we get c [greater than or equal to] 1/2. Remark 1. If [alpha] takes real values and there exist constants m, M such that -[infinity] < m [less than or equal to] [alpha] [less than or equal to] M < [infinity] for a.e. x [member of] [OMEGA], then (2.3) becomes: ||[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) f (x) dx - [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx x [[integral].sub.[OMEGA]] [rho] (x) f (x) dx|| [less than or equal to] [1/2] (M - m) [[integral].sub.[OMEGA]] [rho] (x) ||f (x) - [[integral].sub.[OMEGA]] [rho] (y) f (y) dy|| dx. Note that a scalar version of this inequality has been obtained previously by Cerone and Dragomir in [2], using a different technique. Remark 2. A slightly more general result for [alpha] (t) [member of] [bar.B] (c, r) [colon, equals] {z [member of] C| |z - c| [less than or equal to] r} for a.e. x [member of] [OMEGA], is: ||[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) f (x) dx - [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx x [[integral].sub.[OMEGA]] [rho] (x) f (x) dx|| [less than or equal to] r [[integral].sub.[OMEGA]] [rho] (x) ||f (x) - [[integral].sub.[OMEGA]] [rho] (y) f (y) dy|| dx. (2.6) Here the inequality (2.6) is also sharp. The following dual result may be stated as well. Theorem 2. Let (X, ||dot||) and [OMEGA], [rho] be as above. If f : [OMEGA] [right arrow] X is Bochner measurable on [OMEGA] and there exist vector v [member of] X and r > 0 such that f (x) [member of] [bar.B] (v, r) [colon, equals] {y [member of] X| ||y - v|| [less than or equal to] r} for a.e. x [member of] [OMEGA] and [alpha] : [OMEGA] [right arrow] K a Lebesgue integrable function with [rho][alpha]f, [rho]f Bochner integrable functions on [OMEGA], then we have the sharp inequalities ||[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) f (x) dx - [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx x [[integral].sub.[OMEGA]] [rho] (x) f (x) dx|| [less than or equal to] r [[integral].sub.[OMEGA]] [rho] (x) |[alpha] (x) - [[integral].sub.[OMEGA]] [rho] (y) [alpha] (y) dy| dx [less than or equal to] r [[[integral].sub.[OMEGA]] [rho] (x) |[alpha] (x)|[.sup.2] dx - |[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx|[.sup.2]][.sup.1/2]. (2.7) Proof. The first inequality in (2.7) is obvious from the Sonin type identity: [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) f (x) dx - [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx x [[integral].sub.[OMEGA]] [rho] (x) f (x) dx = [[integral].sub.[OMEGA]] [rho] (x)([alpha] (x) - [[integral].sub.[OMEGA]] [rho] (y) [alpha] (y) dy)(f (x) - v) dx. The second inequality follows by Schwarz's integral inequality: [[integral].sub.[OMEGA]] [rho] (x)|[alpha] (x) - [[integral].sub.[OMEGA]] [rho] (y) [alpha] (y) dy|dx [less than or equal to] [[[integral].sub.[OMEGA]] [rho] (x)|[alpha] (x) - [[integral].sub.[OMEGA]] [rho] (y) [alpha] (y) dy|[.sup.2] dx][.sup.1/2] = [[[integral].sub.[OMEGA]] [rho] (x)|[alpha] (x)|[.sup.2] dx - |[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx|[.sup.2]][.sup.1/2]. The details are omitted. The following particular case holding for Hilbert spaces may be useful for applications. Corollary corollary: see theorem. 1. Let (H; <[dot],[dot]>) be a Hilbert space over the real or complex number field and [OMEGA], [rho] and [alpha] as in Theorem 2. If there exist vectors v, V [member of] H such that for the Bochner measurable function [rho] : [OMEGA] [right arrow] H either Re <V - f (x), f (x) - v> [greater than or equal to] 0, (2.8) or, equivalently, ||f (x) - [[v + V]/2]|| [less than or equal to] [1/2] ||V - v|| (2.9) for a.e. x [member of] [OMEGA] and [rho][alpha]f, [rho]f Bochner integrable on [OMEGA], then, ||[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) f (x) dx - [[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx x [[integral].sub.[OMEGA]] (x) f (x) dx|| [less than or equal to] [1/2] ||V - v|| [[integral].sub.[OMEGA]] [rho] (x)|[alpha] (x) - [[integral].sub.[OMEGA]] [rho] (y) [alpha] (y) dy|dx [less than or equal to] [1/2] ||V - v|| [[[integral].sub.[OMEGA]] [rho] (x)|[alpha] (x) - [[integral].sub.[OMEGA]] [rho] (y) [alpha] (y) dy|[.sup.2] dx][.sup.1/2]. (2.10) The quantity 1/2 is the best possible in both inequalities in (2.10). Proof. The proof is obvious by Theorem 2 on taking into account that in the Hilbert space (H; <[dot],[dot]>) the following two statements are equivalent (i) ||y - [[V +v]/2]|| [less than or equal to] [1/2] ||V - v|| (ii) Re <V - y, y - v> [greater than or equal to] 0, where y, v, V [member of] H. The following result is similar to (1.5). Theorem 3. Let (H; <[dot],[dot]>) be a Hilbert space over the real or complex number field and f, g : [OMEGA] [right arrow] H Bochner measurable on [OMEGA] while [rho] : [OMEGA] [right arrow] [0, [infinity]) is Lebesgue integrable and [[integral].sub.[OMEGA]] [rho] (x) dx = 1. If there exist vectors v, V [member of] H such that either (2.8) or, equivalently, (2.9) hold for a.e. x [member of] [OMEGA] and [alpha] f, [rho]g are Bochner integrable on [OMEGA], then, |[[integral].sub.[OMEGA]] [rho] (x) <f (x), g (x)> dx - <[[integral].sub.[OMEGA]] (x) f (x) dx, [[integral].sub.[OMEGA]] [rho] (x) g (x) dx>| [less than or equal to] [1/2] ||V - v|| [[integral].sub.[OMEGA]] [rho] (x)|| g (x) - [[integral].sub.[OMEGA]] [rho] (y) g (y) dy|| dx [less than or equal to] [1/2] ||V - v|| [[[integral].sub.[OMEGA]] [rho] (x) ||g (x)||[.sup.2] - ||[[integral].sub.[OMEGA]] [rho] (y) g (y) dy||[.sup.2] dx][.sup.1/2] (provided g [member of] [L.sub.2,[rho]] ([OMEGA], H)). (2.11) Again, the constant 1/2 is the best possible. Proof. The following Sonin type identity may be stated as well. [[integral].sub.[OMEGA]] [rho] (x) <f (x), g (x)> dx - <[[integral].sub.[OMEGA]] [rho] (x) f (x) dx, [[integral].sub.[OMEGA]] [rho] (x) g (x) dx> = [[integral].sub.[OMEGA]] [rho] (x)<f (x) - [[V + v]/2], g (x) - [[integral].sub.[OMEGA]] [rho] (y) g (y) dy>dx. (2.12) Taking the modulus See modulo. , using the hypothesis and the Schwarz Schwarz is a common surname, derived from the German schwarz, meaning black. It may refer to: People
|[[integral].sub.[OMEGA]] [rho] (x) <f (x), g (x)> dx - <[[integral].sub.[OMEGA]] [rho] (x) f (x) dx, [[integral].sub.[OMEGA]] [rho] (x) g (x) dx>| [less than or equal to] [[integral].sub.[OMEGA]] [rho] (x)| <f (x) - [[V + v]/2], g (x) - [[integral].sub.[OMEGA]] [rho] (y) g (y) dy>| dx [less than or equal to] [[integral].sub.[OMEGA]] [rho] (x) ||f (x) - [[V + v]/2]|| ||g (x) - [[integral].sub.[OMEGA]] [rho] (y) g (y) dy|| dx [less than or equal to] [1/2] ||V - v|| [[integral].sub.[OMEGA]] [rho] (x) ||g (x) - [[integral].sub.[OMEGA]] [rho] (y) g (y) dy|| dx [less than or equal to] [1/2] ||V - v|| [[[integral].sub.[OMEGA]] [rho] (x)||g (x) - [[integral].sub.[OMEGA]] [rho] (y) g (y) dy||[.sup.2] dx][.sup.1/2] = [1/2] ||V - v|| [[[integral].sub.[OMEGA]] [rho] (x) ||g (x)||[.sup.2] - ||[[integral].sub.[OMEGA]] [rho] (y) g (y) dy||[.sup.2] dx][.sup.1/2], provided g [member of] [L.sub.2,[rho]] ([OMEGA], H). Remark 3. Assume that for the Lebesgue integrable function [alpha] : [OMEGA] [right arrow] K there exist [gamma], [GAMMA] [member of] K such that either (2.1) or, equivalently, (2.2) hold, then, 0 [less than or equal to] [[integral].sub.[OMEGA]] [rho] (x) |[alpha] (x)|[.sup.2] dx - |[[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx|[.sup.2] [less than or equal to] [1/2] |[GAMMA] - [gamma]| [[integral].sub.[OMEGA]] [rho] (x)|[alpha] (x) - [[integral].sub.[OMEGA]] [rho] (y) [alpha] (y) dy|dx, (2.13) and [1] 0 [less than or equal to] |[[integral].sub.[OMEGA]] [rho] (x) [[alpha].sup.2] (x) dx - ([[integral].sub.[OMEGA]] [rho] (x) [alpha] (x) dx)[.sup.2]| [less than or equal to] |[GAMMA] - [gamma]| [[integral].sub.[OMEGA]] [rho] (x) |[alpha] (x) - [[integral].sub.[OMEGA]] [rho] (y) [alpha] (y) dy| dx. (2.14) The quantity 1/2 is sharp in both instances. 3. Applications for Some Integral Inequalities of the Heisenberg Type In the following we use the Gruss type inequality |[[integral].sub.[OMEGA]] [rho] (t)Re <f (t), g (t)> dt - Re<[[integral].sub.[OMEGA]] [rho] (t) f (t) dt, [[integral].sub.[OMEGA]] [rho] (t) g (t) dt>| [less than or equal to] [1/2] ||V - v|| [[integral].sub.a.sup.b] [rho] (t)||g (t) - [[integral].sub.a.sup.b] [rho] (s) g (s) ds|| dt, (3.1) provided [rho] [member of] L ([a, b]), [[integral].sub.a.sup.b] [rho] (t) dt = 1, [rho]f, [rho]g [member of] L ([a, b],H), (H, <[dot],[dot]>) is a real or complex Hilbert space and f : [a, b] [right arrow] H is Bochner measurable and such that either Re <V - f (t), f (t) - v> [greater than or equal to] 0 for a.e. t [member of] [a, b], (3.2) or, equivalently, ||f (t) - [[v + V]/2]|| [less than or equal to] [1/2] ||V - v|| for a.e. t [member of] [a, b]. Notice that the inequality (3.1) follows by (2.10) on taking into account that, for complex numbers z [member of] C, |Rez| [less than or equal to] |z|. It is well known that if (H; <[dot],[dot]>) is a real or complex Hilbert space and f : [a, b] [subset] R [right arrow] H is an absolutely continuous vector-valued function, then f is differentiable dif·fer·en·tia·ble adj. 1. That can be differentiated: differentiable species. 2. Mathematics Possessing a derivative. almost everywhere on [a, b], the derivative derivative: see calculus. derivative In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function. f' : [a, b] [right arrow] H is Bochner integrable on [a, b] and f (t) = [[integral].sub.a.sup.t] f' (s) ds for any t [member of] [a, b]. (3.3) The following theorem provides a version of the Heisenberg inequality in the general setting of Hilbert spaces and has been obtained by S.S. Dragomir in [4]. Theorem 4. Let [phi] : [a, b] [right arrow] H be an absolutely continuous function with the property that b ||[phi] (b)||[.sup.2] = a ||[phi] (a)||[.sup.2], then, [[integral].sub.a.sup.b] ||[phi] (t)||[.sup.2] dt [less than or equal to] 2 [[[integral].sub.a.sup.b] ||[phi]' (t)||[.sup.2] dt x [[integral].sub.a.sup.b] [t.sup.2] ||[phi] (t)||[.sup.2] dt][.sup.1/2]. (3.4) The constant 2 is the best possible. Remark 4. It is obvious that a sufficient condition for (3.4) to hold is that [phi] (a) = [phi] (b) = 0. In the following we point out different upper bounds from (3.4), for the integral [[integral].sub.a.sup.b] ||[phi] (t)||[.sup.2] dt. Proposition 1. Let [phi] : [a, b] [right arrow] H be an absolutely continuous function with the property that [phi] (a) = [phi] (b) = 0. If there exist vectors v, V [member of] H such that either ||[phi]' (t) - [[v + V]/2]|| [less than or equal to] [1/2] ||V - v|| for a.e. t [member of] [a, b] (3.5) or, equivalently, Re <V - [phi]' (t), [phi]' (t) - v> [greater than or equal to] 0 for a.e. t [member of] [a, b], (3.6) then, [[integral].sub.a.sup.b] ||[phi] (t)||[.sup.2] dt [less than or equal to] ||V - v|| [[integral].sub.a.sup.b] ||t[phi] (t) - [1/[b - a]] [[integral].sub.a.sup.b] s[phi] (s) ds|| dt. (3.7) Proof. Applying the inequality (3.1) for [rho] (t) = 1/[b - a], f (t) = [phi]' (t) and g (t) = t[phi] (t), t [member of] [a, b], we can write: |[1/[b - a]] [[integral].sub.a.sup.b] tRe <[phi]' (t), [phi] (t)> dt - Re<[1/[b - a]] [[integral].sub.a.sup.b] [phi]' (t) dt, [1/[b - a]] [[integral].sub.a.sup.b] t[phi] (t) dt>| [less than or equal to] [1/2] ||V - v|| [1/[b - a]] [[integral].sub.a.sup.b] ||t[phi] (t) - [1/[b - a]] [[integral].sub.a.sup.b] s[phi] (s) ds|| dt. (3.8) Since [phi] (a) = [phi] (b) = 0, hence [[integral].sub.a.sup.b] [phi]' (t) dt = 0, (3.9) [[integral].sub.a.sup.b] tRe <[phi]' (t), [phi] (t)> dt = -[1/2] x [[integral].sub.a.sup.b] ||[phi] (t)||[.sup.2] dt, (3.10) where, for the last equality equality Generally, an ideal of uniformity in treatment or status by those in a position to affect either. Acknowledgment of the right to equality often must be coerced from the advantaged by the disadvantaged. Equality of opportunity was the founding creed of U.S. we have used an identity obtained in [4] (see the Eq. (5.3) from [4]) under the more general assumption, i.e., b ||[phi] (b)||[.sup.2] = a ||[phi] (a)||[.sup.2]. Making use of (3.9), (3.10) and (3.8), we conclude that (3.7) holds true and the proposition is proven. Proposition 2. Let [phi] : [a, b] [right arrow] H be an absolutely continuous function with the property that [phi] (a) = [phi] (b) = 0. If there exist vectors w, W [member of] H so that either ||t[phi]' (t) - [[w +W]/2]|| [less than or equal to] [1/2] ||W - w|| for a.e. t [member of] [a, b], (3.11) or, equivalently, Re <W - t[phi]' (t), t[phi]' (t) - w> [greater than or equal to] 0 for a.e. t [member of] [a, b], (3.12) then |||[[integral].sub.a.sup.b] [phi] (t) dt||[.sup.2] - [1/2] (b - a) [[integral].sub.a.sup.b] ||[phi] (t)||[.sup.2] dt| [less than or equal to] [1/2] ||W - w|| [[integral].sub.a.sup.b]|| [phi] (t) - [1/[b - a]] [[integral].sub.a.sup.b] [phi] (s) ds|| dt. (3.13) Proof. Applying the inequality (3.1) for [rho] (t) = 1/[b - a], f (t) = t[phi]' (t) and g (t) = [phi] (t), t [member of] [a, b], we can write: |[1/[b - a]] [[integral].sub.a.sup.b] tRe <[phi]' (t), [phi] (t)> dt - Re<[1/[b - a]] [[integral].sub.a.sup.b] t[phi]' (t) dt, [1/[b - a]] [[integral].sub.a.sup.b] [phi] (t) dt>| [less than or equal to] [1/2] ||W - w|| [[integral].sub.a.sup.b] ||[phi] (t) - [1/[b - a]] [[integral].sub.a.sup.b] [phi] (s) ds|| dt. (3.14) Since [phi] (a) = [phi] (b) = 0, hence [[integral].sub.a.sup.b] t[phi]' (t) dt = -[[integral].sub.a.sup.b] [phi] (t) dt. (3.15) Therefore, by (3.10), (3.15) and (3.14), we deduce |-[1/[2 (b - a)]] [[integral].sub.a.sup.b] ||[phi] (t)||[.sup.2] dt + Re<[1/[b - a]] [[integral].sub.a.sup.b] [phi] (t) dt, [1/[b - a]] [[integral].sub.a.sup.b] [phi] (t) dt>| [less than or equal to] [1/2] ||W - w|| x [[integral].sub.a.sup.b]||[phi] (t) - [1/[b - a]] [[integral].sub.a.sup.b] [phi] (s) ds|| dt, which is clearly equivalent to (3.13). References [1] C. Buse, P. Cerone, S. S. Dragomir and J. Roumeliotis, A refinement of Gruss type inequality for the Bochner integral of vector-valued functions in Hilbert spaces and applications, RGMIA RGMIA Research Group in Mathematical Inequalities and Applications Res. Rep (programming) REP - A directive used in IBM object code card decks (and later PTF Tapes) to REPlace fragments of already assembled or compiled object code prior to link edit. . Coll v. t. 1. To embrace. ., 7(3)(2004), Art. 9. [ONLINE http://rgmia.vu.edu See .edu. (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/v7n3.html]. [2] P. Cerone and S. S. Dragomir, A refinement of the Gruss inequality and applications, RGMIA Res. Rep. Coll., 5 (2002), No. 2, Article 14. [ONLINE http://rgmia.vu.edu.au/v5n2.html]. [3] S. S. Dragomir, Integral Gruss inequality for mappings with values in Hilbert spaces and applications, J. Korean Korean, language of uncertain ancestry. It is thought by some scholars to be akin to Japanese, by others to be a member of the Altaic subfamily of the Ural-Altaic family of languages (see Uralic and Altaic languages), and by still others to be unrelated to any known Math. Soc., 38(6)(2001), 1261-1273. [4] S. S. Dragomir, Refinements of Schwarz and Heisenberg inequalities in Hilbert spaces, J. Inequal In`e´qual a. 1. Unequal; uneven; various. . Pure & Appl. Math., 5(3)(2004), Art. 60. [ONLINE http://jipam.vu.edu.au/article.php?sid=446]. [5] G. Gruss, Uber das maximum des absoluten Betrages von [1/[b-a]] [[integral].sub.a.sup.b] f (x) g (x) dx - [1/[(b-a)[.sup.2]]] [[integral].sub.a.sup.b] f (x) dx x [[integral].sub.a.sup.b] g (x) dx, Math. Z., 39(1935), 215-226. [6] 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. , Classical and New Inequalities in Analysis, Kluwer Acad. Publ., 1993. N. S. Barnett Barnett as a personal name can refer to:
n. A reference mark (  ) used in printing and writing. Also called diesis.Noun 1. ]), and S. S. Dragomir ([section]) School of Computer Science and Mathematics Victoria University of Technology PO Box 14428, MCMC MCMC Markov Chain Monte Carlo MCMC Malaysian Communications and Multimedia Commission MCMC Mid-Continent Mapping Center McMC McMaster-Carr MCMC Marine Corps Maintenance Contractor 8001, Victoria, 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. . and C. Buse ([paragraph]) Department of Mathematics West University of Timisoara Timişoara (tēmēshwä`rä), Hung. Temesvár, city (1990 pop. 351,293), W Romania, in the Banat, on the Beja Canal. Timisoara, 1900, Bd. V. Parvan Parvan may refer to:
 –), republic (v), 91,699 sq mi (237,500 sq km), SE Europe.
Received June June: see month. 27, 2005, Accepted October October: see month. 5, 2005. *2000 Mathematics Subject Classification. 46B05, 46C05, 26D15, 26D10. ([dagger]) E-mail: neil@csm.vu.edu.au ([double dagger]) E-mail: pc@csm.vu.edu.au ([section]) E-mail: sever TO SEVER, practice. When defendants who are sued jointly have separate defences, they may in general sever, that is, each one rely on his own separate defence; each may plead severally and insist on his own separate plea. See Severance. .dragomir@vu.edu.au ([paragraph]) E-mail: buse@math.uvt.ro |
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