jensen s inequality revisited and refined

... first 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 ... λc, s1 , s2 (x) = s2 λc,ϕ,ψ (x) + − s2 (s1 = 0) (3) and λc, s1 , s2 (x) = s2 s − s1 − s1 s1 λc,ϕ,ψ (x) − s1 −1 s1 (s1 = 0) (4) for each x Î I\{c} If s1 = 0, then it is trivial by (3) that λc, s1 ... (8) by Jensen s inequality This also means M  (f) ≤ M ψ (f) because ψ is monotone decreasing on I For the strict case, since g is a non-constant function on Ω, we obtain the desired results from...

Ngày tải lên: 21/06/2014, 00:20

... Inequalities and Applications holds for every x, y ∈ I It is clear that every convex function is Jensen- convex To see that the class of convex functions is a proper subclass of Jensen- convex functions, ... proof, see 2, page 71 or 1, page 53 A class of functions which is between the class of convex functions and the class of Jensen- convex functions is the class of Wright-convex functions A function ... weights pi , i 1, , n The most important result in this direction is the JensenSteffensen inequality see, e.g., 1, page 57 which states that 1.1 holds also if x1 ≤ x2 ≤ k · · · ≤ xn and ≤ Pk...

Ngày tải lên: 22/06/2014, 02:20

... 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 lemma gives a simple sufficient ... Equation 1.11 is true if and only if θ1 ≤ 1−f t x1 /f x − θ1 σ − q tx1 1.12 The right-hand side The cases q or t not pose problems because the right-hand side is still finite is an increasing function ... Theorem 1.2 Suppose that f x is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval I ⊆ R, and suppose that f x is strictly convex on I Suppose that...

Ngày tải lên: 22/06/2014, 03:20

... this last expression is the value of the European derivative security Of course, the LHS cannot be strictly less than the RHS above, since stopping at time n is always allowed, and we conclude ... g S  = IE 1 + r,n g Sn  : 94 S2 (HH) = 16 S (H) = S2 (HT) = S =4 S2 (TH) = S1 (T) = S2 (TT) = Figure 7.3: A three period binomial model 7.3 Stopped Martingales Let fYk gn=0 be a stochastic ... American call Assume that r  Consider the American derivative security with payoff g Sk  in period k The value of this security is the same as the value of the simple European derivative security...

Ngày tải lên: 18/10/2013, 03:20

... Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993 S S Dragomir, J Peˇ ari´ , and L E Persson, “Properties of some functionals related to Jensen s c c inequality, ” ... 5.1 Since this inequality is homogeneous in and ak , and also in Qi and Rk , without loss of and Qi Using the notations ak x2 and Rk p, where generality, assume that x ≥ and p > 0, the inequality ... Rk The proposition is proved ak 5.13 12 Journal of Inequalities and Applications References D S Mitrinovi´ , J E Peˇ ari´ , and A M Fink, Classical and New Inequalities in Analysis, vol 61 of...

Ngày tải lên: 21/06/2014, 07:20

... s ♦α s − a b h s ♦α s f b a a b b h s f g s ♦α s − a h s g s ♦α s b a h s ♦α s h s ♦α s f t a b hs f g s − f t ♦α s a ≥ at 3.8 b hs g s −t ♦α s a b at a b at b h s g s ♦α s − t h s ♦α s a h s ... thesis see also 8, In Section 3, we present our main results which are generalizations of Jensen s inequality on time scales Some examples and applications are given in Section Preliminaries ... Equations on Time Scales, Birkh¨ user, Boston, Mass, USA, 2001 a 16 M Bohner and A Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkh¨ user, Boston, a Mass, USA, 2003 17 J W Rogers...

Ngày tải lên: 22/06/2014, 03:20

Ngày tải lên: 16/03/2014, 03:20

... f(x) + g(x) is constant or 1/p = (1 /s + 1/t)/2 On the other hand, in view of Minkowski s inequality for the cases of < s

Ngày tải lên: 20/06/2014, 22:20

... 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 ... inequality [31] and 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 ... (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

... Fuzzy Sets Syst 159, 730–738 (2008) doi:10.1016/j.fss.2007.07.011 Mirmostafee, AK, Moslehian, MS: Fuzzy versions of Hyers-Ulam-Rassias theorem Fuzzy Sets Syst 159, 720–729 (2008) doi:10.1016/j.fss.2007.09.016 ... Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality Math Inequal Appl 4, 93–118 (2001) Jung, S- M: Hyers-Ulam-Rassias stability of Jensen s equation and its application Proc Am Math Soc ... Trans Am Math Soc 347, 3111–3119 (1995) doi:10.2307/2154775 Hyers, DH, Isac, G, Rassias, ThM: Stability of Functional Equations in Several Variables Birkhäuser, Basel (1998) Hyers, DH, Isac,...

Ngày tải lên: 20/06/2014, 22:20

... Proschan, and Y L Tong, Convex Functions, Partial Orderings, and Statistical Applications, c c vol 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992 S Simi´ ... ari´ , and Y Seo, “Converses of Jensen s operator inequality, ” Operators and Matrices, vol cc c c 4, no 3, pp 385–403, 2010 c c S Iveli´ and J Peˇ ari´ , “Generalizations of converse Jensen s inequality ... D S Mitrinovi´ , J E Peˇ ari´ , and A M Fink, Classical and New Inequalities in Analysis, vol 61 of c c c Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands,...

Ngày tải lên: 21/06/2014, 05:20

... have special case of s1 s2 /2 if s1 < s2 and for t 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 ... special case of s1 s2 /2 if s1 < s2 and for 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 and for r s2 , s s1 , and t Acknowledgments The research of the first and second authors was funded by Higher Education Commission, Pakistan The research...

Ngày tải lên: 21/06/2014, 17:20

... sequel we will give some examples to demonstrate the fruitfulness of the assertions from Theorem 2.1 Since all bounds will be given as a combination of means from the Stolarsky class, here is ... The expression Sf a, b represents the best possible global upper bound for Jensen s inequality written in the form 1.4 Journal of Inequalities and Applications 3 Proofs We will give proofs of the ... ⎪ s xs − y s ⎪ ⎨ ⎪ xs − y s ⎪ ⎪ ⎪ ⎪ ⎪ s log x − log y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪√ ⎪ ⎪ xy, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x, rs r − s / 0, r s / 0, 4.3 1 /s , s / 0, r 0, r s x y > 0, 0, and this form is introduced by Stolarsky...

Ngày tải lên: 22/06/2014, 03:20

... B max {x, y } and give some new generalizations of Hilbert s inequality As applications, we also consider some particular results Yongjin Li et al Main results and applications Lemma 2.1 Define ... 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...

Ngày tải lên: 22/06/2014, 06:20

... B max {x, y } and give some new generalizations of Hilbert s inequality As applications, we also consider some particular results Yongjin Li et al Main results and applications Lemma 2.1 Define ... 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...

Ngày tải lên: 22/06/2014, 06:20

... 2002 [12] M Johansson, L.-E Persson, and A Wedestig, “Carleman s inequality- history, proofs and some new generalizations,” Journal of Inequalities in Pure and Applied Mathematics, vol 4, no 3, ... Mathematical Analysis and Applications, vol 234, no 2, pp 717–722, 1999 [3] P Yan and G.-Z Sun, “A strengthened Carleman s inequality, ” Journal of Mathematical Analysis and Applications, vol 240, no ... Mathematical Analysis and Applications, vol 253, no 2, pp 691–694, 2001 [8] X.-J Yang, “Approximations for constant e and their applications,” Journal of Mathematical Analysis and Applications, vol 262,...

Ngày tải lên: 22/06/2014, 06:20

... 2002 [12] M Johansson, L.-E Persson, and A Wedestig, “Carleman s inequality- history, proofs and some new generalizations,” Journal of Inequalities in Pure and Applied Mathematics, vol 4, no 3, ... Mathematical Analysis and Applications, vol 234, no 2, pp 717–722, 1999 [3] P Yan and G.-Z Sun, “A strengthened Carleman s inequality, ” Journal of Mathematical Analysis and Applications, vol 240, no ... Mathematical Analysis and Applications, vol 253, no 2, pp 691–694, 2001 [8] X.-J Yang, “Approximations for constant e and their applications,” Journal of Mathematical Analysis and Applications, vol 262,...

Ngày tải lên: 22/06/2014, 06:20

... version of Popoviciu s inequality, which is a unified improvement of Acz´ l s inequality and Popoviciu s inequality In e Section 4, the obtained result will be used to establish an integral inequality ... [14], Wu established a sharp and generalized version of Popoviciu s inequality as follows Theorem 1.3 Let p > 0, q > 0, 1/ p + 1/q ≥ 1, and let , bi (i = 1,2, ,n) be positive p p q q numbers such ... matical Analysis and Applications, vol 245, no 2, pp 393–403, 2000 [4] L Losonczi and Z P´ les, “Inequalities for indefinite forms,” Journal of Mathematical Analysis a and Applications, vol 205,...

Ngày tải lên: 22/06/2014, 18:20

... two of the numbers to be less than or equal to (neither can all the numbers be less than 1) Because if a number is less than and two are greater than the inequality is obviously true (the product ... 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 − ... ≤ − 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...

Ngày tải lên: 15/08/2014, 17:07

... organization s business strategies Because market demand is changing, business strategies should be ready to change to satisfy customers’ needs and wants However, any changes of strategies should be ... solutions for their businesses Moreover, these lessons will lower risks when implementing business strategies and losses resulting from bad decisions Finally, manager should be alerted and ready ... reserve stock As a result, it is more necessary for Vinalink s warehouses to make good use of space to stock as much goods as possible Having many wide aisles becomes a disadvantage in this case because...

Ngày tải lên: 20/04/2013, 11:13