6 10 appendix proof of schwarz s inequality

... In this note we give a new proof of Blundon s inequality by making use of the following preliminary result: Lemma Any positive real numbers x, y, z such that x + y + z = xyz satisfy the inequality ... − 10 Proof Since the numbers are positive, from the given condition it follows immediately that x < xyz ⇔ yz > 1, and similarly xz > and yz > 1, which shows that it is not possible for two of ... occurs if and only if ABC is equilateral Proof According to the well-known formulae cot A = s( s − a) , (s − b) (s − c) cot B = we deduce that cot s( s − b) , (s − c) (s − a) A = cot cot C = s( s −...

Ngày tải lên: 15/08/2014, 17:07

... conjugacy classes and P orbits In this section we describe the P = Pn (K) conjugacy classes of nilpotent elements in g = gln (K) We shall also describe certain Jacobson-Morosov triples that are associate ... is an ri ×rj block of A Fix i0 ∈ S( α) a and j0 ∈ S( α) Then Ai0 ,j0 = (˜l ,s ) satisfies al ,s = if s − l > pj0 − ti0 ˜ Proof The proof is a straightforward application of Lemma 6. 10 and Lemma 6. 11 ... proof The main difficulty is to study P conjugacy classes of nilpotent elements, their tangent spaces and the transversals to these tangent spaces We recall some of the results: Let X be a nilpotent...

Ngày tải lên: 28/03/2014, 22:20

... function is strictly concave (or convex) on a suitable convex subset of Csm,f(I) (see Theorem 3) We also observe a certain boundedness of -means, more precisely, sup M(1 s) ϕ +s (f ) = Mψ−ϕ (f ) s 0 ... geometric-arithmetic mean inequality as an application of our results Lemmas This section is devoted to collecting some lemmas which we will need in the proof of our main results The first lemma is to describe ... result asserts that a -mean function: ∇ ® M (f) is well defined and order preserving, and this assertion simultaneously gives a new interpretation of Jensen s inequality However, this assertion...

Ngày tải lên: 21/06/2014, 00:20

... and p α is the best constant, if α ∈ , , then αA a, b and p α is the best constant Proof of Theorem 2.1 Because Lp a, b is increasing with respect to p ∈ R for fixed a and b, it suffices to prove ... easy to see that this interval must be of the form t0 , , for some t0 ∈ 0, This follows from the fact that s2 is strictly positive on 0, and s1 is strictly increasing on this interval ±∞, we will ... pp 60 2 60 6, 1995 J S ndor, “On certain inequalities for means II,” Journal of Mathematical Analysis and Applications, vol a 199, no 2, pp 62 9 63 5, 19 96 J S ndor, “On certain inequalities for...

Ngày tải lên: 21/06/2014, 07:20

... s1 , r s2 , and s s1 s2 /2 3.28 for t s1 , s s2 , and r if s2 < s1 Similarly by setting m in 3. 36 , we have special case of 3.29 for r s1 , s s1 s2 /2 if s1 < s2 and for r s2 , s s1 , t s1 s2 ... t s1 , r s2 , and s s1 s2 /2 if 3.39 for t s1 , s s2 , r s2 < s1 Similarly by setting m in 3.44 , we have special case of 3.40 for r s1 , s s2 , s1 s2 /2 if s2 < s1 and t s1 s2 /2 if s1 < s2 ... matrices H sk sl /2, s1 m , H sk sl /2, s1 s2 /2 m are positive semi-definite matrices Particularly k,l k,l det H det H sl sk sk m , s1 sl s1 , k,l s2 ≥ 0, m k,l ≥ 0, 3.35 3. 36 where H s, t is defined...

Ngày tải lên: 21/06/2014, 17:20

... 1.14 k Define the set S by S p1 , , pn | pk ≥ 0, ∀k 1.15 We wish to show h p ≥ on the set S subject to the constraint p1 · · · pn Suppose the minimum of h p is in the interior of S By the Lagrange ... minimum of h p in S subject to the constraint p1 · · · pn occurs on the boundary of S, where some of the pk must be zero At the minimum, say po , we are back to the case of n − or less, and by ... within the domain of the function f for various unknown weight vectors p This is why the statement of Theorem 1.2 makes the assumption that this is true for all possible weight vectors The following...

Ngày tải lên: 22/06/2014, 03:20

... when ε is sufficiently small, which contradicts the hypothesis Hence the constant factor D(A,B) in (2 .6) is the best possible and T = D(A,B) This completes the proof ∞ Theorem 2.3 Suppose that ... Assume that the constant factor D(A,B) in (2 .6) is not the best possible, then there exist a positive real number K Journal of Inequalities and Applications with K < D(A,B), such that (2 .6) is ... Li: Institute of Logic and Cognition, Department of Mathematics, Sun Yat-Sen University, Guangzhou 5102 75, China Email address: stslyj@mail.sysu.edu.cn Zhiping Wang: Department of Mathematics, Guangdong...

Ngày tải lên: 22/06/2014, 06:20

... when ε is sufficiently small, which contradicts the hypothesis Hence the constant factor D(A,B) in (2 .6) is the best possible and T = D(A,B) This completes the proof ∞ Theorem 2.3 Suppose that ... Assume that the constant factor D(A,B) in (2 .6) is not the best possible, then there exist a positive real number K Journal of Inequalities and Applications with K < D(A,B), such that (2 .6) is ... Li: Institute of Logic and Cognition, Department of Mathematics, Sun Yat-Sen University, Guangzhou 5102 75, China Email address: stslyj@mail.sysu.edu.cn Zhiping Wang: Department of Mathematics, Guangdong...

Ngày tải lên: 22/06/2014, 06:20

... also j= integrable on [a,b] Passing the limit as n → ∞ in both sides of inequality (4 .6) , we obtain the inequality (4.1) The proof of Theorem 4.1 is complete Remark 4.2 Motivated by the proof of ... applications of functional equations,” Uspekhi Matematicheskikh Nauk (N .S. ), vol 11, no 3 (69 ), pp 3 68 , 19 56 (Russian) [2] Y J Cho, M Mati´ , and J Peˇ ari´ , “Improvements of some inequalities of Acz´ ... (1.4) In this paper, we show a new sharp and generalized version of Popoviciu s inequality, which is a unified improvement of Acz´ l s inequality and Popoviciu s inequality In e Section 4, the...

Ngày tải lên: 22/06/2014, 18:20

... p } Proof (i) Since the case p = is trivial, we assume that < p < For any (s, t) ∈ Ω, we have ks, t ≥ and ks + t ≥ By Minkowski s inequality, we obtain k p s p + p t p ≥ Also, an easy consideration ... (2 .10) Then pk(1 − ks) p−1 ∂F b (a,b ,s) = ps p−1 a − p s (2.11) Consider the simultaneous equations F(a,b ,s) = and (∂F/ s) (a,b ,s) = If < s < 1/k, then the equation (∂F/ s) (a,b ,s) = yields b= ... { (s, (1 − ks)/ ) : ≤ s ≤ 1/k}, the family of lines ᏸ is represented by p − ks spa + b=1 0 s k (2.9) We here remark that each line of ᏸ has no singular point Now, put F(a,b ,s) = s p a + (1 − ks)...

Ngày tải lên: 22/06/2014, 21:20

... variants of Peˇ ari´ s c c generalizations of the discrete Ostrowski s inequalities In Section we use results from Section to obtain generalizations of Milne s inequality and its converse Mercer s ... variants of Milne s inequality and its converse are also given ˇ Variants of Cebyˇev s inequality s ˇ A classic result due to Cebyˇev (1882, 1883) is stated as follows Let w be a nonnegative s n-tuple ... opposite directions, the inequality (2.7) is reversed Proof Without any loss of generality we may suppose that n-tuples x and y are both monotonically decreasing (in other cases the proof is similar)...

Ngày tải lên: 22/06/2014, 21:20

... means of q-Lagrange inversion Several authors have given quite distinct bijective proofs of q-series identities, many of which may be interpreted as statements about partitions – see Andrews [1][3], ... bijective proof of the Pfaff-Saalsch¨tz identity [6] u In view of the fact that so many q-series identities may be derived by means of q-Lagrange inversion as well as by bijective methods, it is desirable ... [1][3], Bressoud [4], Garsia and Milne [8], Joichi and Stanton [14], Sylvester [19] An exceptional example is Garsia and Milne s proof of the Rogers-Ramanujan identities [ 16] , making use of the involution...

Ngày tải lên: 07/08/2014, 06:22

... produce σ is to transpose two of the rows or two of the columns in the corresponding diagram One must be careful, however, to preserve row and column sums If two of the row sums are the same, or ... sets of horizontal, the electronic journal of combinatorics 11 (2004), #R48 vertical, southeast, southwest, northeast, and northwest vertices of the six-vertex model of A Then for indeterminants ... setting a = in the Izergin-Korepin formula [5] [6] quoted in Theorem 1.5 below This determinant evaluation, expressed as a sum of weights of n × n alternating sign matrices, formed the basis of...

Ngày tải lên: 07/08/2014, 08:20

... Richard Stanley s 60 th Birthday, Massachusetts Institute of Technology, Cambridge, Massachusetts, June 22– 26, 2004, slide available at: http://www-math.mit.edu/~apost/talks/perm-slides.pdf [4] R P Stanley, ... edge of F , then j is called a child of i Note that i is also a descendant of i itself Let S( F, i) be the induced subtree of F on descendants of i, rooted at i We call this tree the descendant subtree ... the smallest among the labels of the descendants of v If we pick a labeling ω uniformly at random, the probability that ω(v) is the smallest among the labels of the descendants of v is 1/h(v) So...

Ngày tải lên: 07/08/2014, 08:22

... out that this result provides an alternate short proof of Aggarwal s theorem, as we shall discuss in Subsection 2.1 The proof of Theorem 1.5 is similar in spirit to that in [6] We consider a graph ... number of holes For the case of more holes, let us first consider the dual graph of the quadrilateralization of the polygon from Fig It consists of four cycles connected by single edges, as illustrated ... edges of those quadrilaterals are either edges or internal diagonals of P joining pairs of vertices — see Fig 1(c); the existence of a quadrilateralization was established by Kahn et al [6] In...

Ngày tải lên: 07/08/2014, 13:21

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Ngày tải lên: 19/06/2014, 15:20

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Ngày tải lên: 19/06/2014, 15:20

... than those issued by FASAB Closed 50 07-4 The Secretary of the Treasury Treasury s checklist was revised in fiscal year 2008 to address should direct the Fiscal these concerns Assistant Secretary ... Treasury analysis, ensure direct linking of the CFS and agency note disclosures Open Treasury has showed progress by demonstrating that amounts in the Statement of Social Insurance were consistent ... information was disclosed in Note 2, Cash and Other Monetary Assets, in the fiscal year 2008 CFS Closed 36 04-20 The Secretary of the Treasury should direct the Fiscal Assistant Secretary to ensure that...

Ngày tải lên: 20/06/2014, 08:20

... mechanism for monitoring and assessing the effectiveness of internal control over the preparation of the CFS See status of recommendation No 04 -6 56 07 -10 The Secretary of the Treasury should ... processes for monitoring and assessing the effectiveness of internal control over the processes used to prepare the CFS Treasury is currently reviewing its Open internal control procedures to ... Appendix I: Status of Treasury s and OMB s Progress in Addressing GAO s Prior Year Recommendations for Preparing the CFS Status of recommendation Count No Recommendation Per Treasury and...

Ngày tải lên: 20/06/2014, 08:20

... Beckenbach-type inequality that is due to Wang [32] The rest of this paper is organized as follows In Section 2, we present extensions of (3) and (4) and establish their corresponding reversed versions In Section ... purpose of this work is to give extensions of inequalities (3) and (4) and establish their corresponding reversed versions Moreover, the obtained results will be applied to improve Hao Z-C inequality ... (27) The classical arithmetic-geometric mean inequality is one of the most important inequalities in analysis This classical inequality has been widely studied by many authors, and it has motivated...

Ngày tải lên: 20/06/2014, 22:20