Showing papers in "Mathematical Inequalities & Applications in 2008"
TL;DR: In this article, the authors introduced the sequence spaces cI(Λ), c0(λk), m I( Λ) and m 0(λ k) associated with the multiplier sequence Λ = (λk), where k is the number of non-zero scalars in the multiplier.
Abstract: In this article we introduce the sequence spaces cI(Λ) , c0(Λ) , m I(Λ) and m0(Λ) associated with the multiplier sequence Λ = (λk) of non-zero scalars. We study the different algebraic and topological properties of these sequence spaces like solidness, symmetricity, sequence algebra, convergence free etc. Also we characterize the multiplier problem and obtain some inclusion relation involving these sequence spaces. Mathematics subject classification (2000): 40A05, 40C05.
71 citations
TL;DR: In this paper, the generalized Hyers-Ulam stability of the Riccati differential equation has been proved under some additional conditions and concrete examples of concrete examples for subject classification.
Abstract: In this paper, we will prove the generalized Hyers-Ulam stability of the Riccati differential equation of the form y′(t)+ g(t)y(t)+ h(t)y(t)2 = k(t) under some additional conditions. Some concrete examples will be introduced. Mathematics subject classification (2000): 26D10, 34A40, 39B82.
43 citations
TL;DR: The concept of I -convergence is a generalization of statistical convergence and it is depended on the notion of the ideal I of subsets of the set N of positive integers as discussed by the authors.
Abstract: The concept of I -convergence is a generalization of statistical convergence and it is depended on the notion of the ideal I of subsets of the set N of positive integers. In this paper for sequences in 2 -normed space the relationship between I -convergence and usual convergence along a filter F (I ) associated with an admissible ideal I with property (AP) is investigated. We introduce the concepts I -Cauchy and I ∗ -Cauchy sequences in 2 -normed spaces and study their certain properties. Mathematics subject classification (2000): 40A05, 46A70, 40A99, 46A99. Keywordsandphrases: Statisticalconvergence, I -convergence, I -Cauchy, I ∗ -Cauchy, 2-normed spaces.
37 citations
TL;DR: In this paper, a new proof of Wilker's inequalities involving trigonometric functions is presented, which is based on the L'Hospital rules for monotonicity and the Wilker-Anglesio Inequality.
Abstract: [1] J. B. WILKER, E3306, The American Mathematical Monthly 96(1989), no.1, 55. [2] J. S. SUMNER, A. A. JAGERS, M. VOWE, AND J. ANGLESIO, Inequalities involving trigonometric functions, The American Mathematical Monthly 98(1991), no.3, 264-267. [3] B. N. GUO, B. M. QIAO, F. QI, AND W. LI, On new proof of Wilker’s inequalities involving trigonometric functions, Mathematical Inequalities and Applications 1(2003), 19-22. [4] I. PINELIS, L’Hospital rules for monotonicity and the Wilker-Anglesio Inequality, The American Mathematical Monthly 111(2004), 905-909.
35 citations
TL;DR: In this paper, it was shown that the van der Waerden's function T (x) can be constructed with the aid of the dyadic expansion of x, which is known as the Takagi function.
Abstract: (where, throughout this paper, R, Z, and N denote the sets of real numbers, integers, and positive integers, respectively, N0 = N∪{0}, and dist(x,Z) = inf{ |x−s| : s ∈ Z }). Functions of this type have been investigated by several authors (e.g. [3], [7], and [10]) as convenient examples for continuous nowhere differentiable functions. In particular, function T is usually cited as “van der Waerden’s function” (e.g. [1], [2]). However, as it was also mentioned by Knopp [7], function T had been constructed earlier by Takagi [9] on the interval [0, 1] in a somewhat different way. Namely, Takagi determined T (x) with the aid of the dyadic expansion of x. It seems therefore historically correct to call T the Takagi function. More historical and mathematical details can be found, for instance, in Kairies’ paper [6]. Recently, Hazy and Pales have discovered that the Takagi function plays a specific role in the theory of approximately convex functions. Namely, in order to extend the celebrated theorem of Bernstein and Doetsch for approximately midconvex functions, they have proved the following result [4, Theorem 4]. Theorem A. Let X be a normed linear space and let D ⊂ X be an open convex set. Suppose that e and δ are nonnegative real numbers, f : D → R fulfils the inequality
33 citations
TL;DR: In this article, the Hilbert's and Hardy-Hilbert integral inequalities in the non-conjugate case were shown to be the best possible, when the parameters satisfy appropriate conditions.
Abstract: By introducing some parameters and a norm |x|α , x ∈ R+ , we give higherdimensional Hilbert’s and Hardy-Hilbert’s integral inequalities in non-conjugate case. Further, we prove that their constant factors are the best possible, in the conjugate case, when the parameters satisfy appropriate conditions. We also compare our results with some known results. Mathematics subject classification (2000): 26D15.
33 citations
TL;DR: In this article, the Hyers-Ulam stability of the Pexider functional equation was investigated, and it was shown that f 1 (x + y) + f 2(x + σ(y)) = f 3(x)+ f 4(y), x, y ∈ E, where E is a normed space and σ : E −→ E is an involution.
Abstract: In this paper we investigate the Hyers-Ulam stability of the Pexider functional equation f 1(x + y) + f 2(x + σ(y)) = f 3(x) + f 4(y), x, y ∈ E, where E is a normed space and σ : E −→ E is an involution. Mathematics subject classification (2000): 39B52, 39B82,.
28 citations
TL;DR: Characterization of Lvp[0, ∞) -L μq[O, √ ∞] boundedness of the general Hardy operator (Hsf) for monotone functions f ≥ 0 for 0 is given in this article.
Abstract: Characterization of Lvp[0, ∞) - L μq[O, ∞) boundedness of the general Hardy operator (Hsf)(x) =(∫[0,x] fsudλ) 1/s restricted to monotone functions f ≥ 0 for 0
28 citations
TL;DR: In this article, the second Kershaw double inequality concerning the γρερδεratio of two gamma functions is refined, extended, and generalized elegantly.
Abstract: In the paper, the second Kershaw’s double inequality concerning
ratio of two gamma functions is refined, extended and generalized elegantly.
26 citations
TL;DR: Refinements of Jensen-Steffensen's inequality, Slater-Pecaric's inequality and majorization theorems for superquadratic functions are presented in this paper.
Abstract: Refinements of Jensen-Steffensen's inequality, Slater-Pecaric's inequality and majorization theorems for superquadratic functions are presented.
23 citations
TL;DR: In this paper, higher-order convexity properties of real functions are introduced and surveyed based on J. L. Jensen's concept of convex functions as well on its generalization by E. V. Wright and related to T. M. Popoviciu's convexness notions.
Abstract: Based on J. L. W. V. Jensen’s concept of convex functions as well on its generalization byE. M. Wright and related to T. Popoviciu’s convexity notions, higher-order convexity properties of real functions are introduced and surveyed. Mathematics subject classification (2000): 26A51, 26A48, 39B62.
TL;DR: In this article, the main aim of the present note is to establish two new weighted Grüss type integral inequalities by using a fairly elementary analysis, which is the same as the one presented in this paper.
Abstract: The main aim of the present note is to establish two new weighted Grüss type integral inequalities by using a fairly elementary analysis. Mathematics subject classification (2000): 26D15, 26D20.
TL;DR: In this article, the validity of the inequalities for non-negative Borel measurable functions g on the interval (a, b)⊆R, where 0 < p 1, 0 < q +∞, λ, μ and ν are non-positive Borel measures on (b, c), and u is a weight function on (c, d), is characterized.
Abstract: In this paper we characterize the validity of the inequalities ‖g‖p,(a,b),λ c ∥∥∥∥u(x) ∫ (a,x) g(y) dμ ∥∥∥∥ q,(a,b),ν and ‖g‖p,(a,b),λ c ∥∥∥∥u(x) ∫ (x,b) g(y) dμ ∥∥∥∥ q,(a,b),ν for non-negative Borel measurable functions g on the interval (a, b)⊆R , where 0 < p 1 , 0 < q +∞ , λ , μ and ν are non-negative Borel measures on (a, b) , and u is a weight function on (a, b) . Mathematics subject classification (2000): 26D10, 26D15, 46E30.
TL;DR: For any Banach space X the n-th James constants J(n)(X) and K-p,q(n) are investigated and discussed in this article, where new properties of these constants are presented.
Abstract: For any Banach space X the n-th James constants J(n)(X) and the n-th Khintchine constants K-p,q(n)(X) are investigated and discussed. Some new properties of these constants are presented. The main ...
TL;DR: In this paper, the inequalities in convolutions in weighted Lp(R,ρ) spaces and their important applications to partial differential equations and integral transforms are discussed, and the importance of these inequalities is discussed.
Abstract: In this paper, we give the inequalities in convolutions in weighted Lp(R,ρ) spaces and their important applications to partial differential equations and integral transforms. Mathematics subject classification (2000): 44A35, 35A22, 26D20. Keywordsandphrases: Convolution, inequality,weighted Lp norm,Green function, integral transform, partial differential equations.
TL;DR: The construction of grand Furuta inequality (GFI) is improved in order to give a constructive proof of Uchiyama’s result (2003) and the generalization of UChiyama's result is obtained by mathematical induction which implies Furuta's recent results.
Abstract: As a continuation of our previous work with the same title, the construction of grand Furuta inequality (GFI) is improved in order to give a constructive proof of Uchiyama’s result (2003). Afterwards the generalization of Uchiyama’s result is obtained by mathematical induction which implies Furuta’s recent results. Mathematics subject classification (2010): 47A63, 47B15, 47B65.
TL;DR: In this article, some new bounds for Lp(x, y) and Ip(p, r) in terms of Ap(x and r) and Gp(r, r), respectively, were established.
Abstract: In this paper, some new bounds for Lp(x, y) and Ip(x, y) in terms of Ap(x, y) and Gp(x, y) are established. Mathematics subject classification (2000): 26E60, 26D07.
TL;DR: In this paper, the Hyers-Ulam-Rassias stability of linear mappings in quasi-Banach modules associated to the Cauchy functional equation and a generalized Jensen functional equation was proved.
Abstract: A quasi norm is a non-negative function ‖.‖ on a linear space X satisfying the same axioms as a norm except for the triangle inequality, which is replaced by the weaker condition that “there is a constant K 1 such that ‖x + y‖ K(‖x‖ + ‖y‖) for all x, y ∈ X ”. In this paper, we prove the Hyers–Ulam–Rassias stability of linear mappings in quasi-Banach modules associated to the Cauchy functional equation and a generalized Jensen functional equation. Mathematics subject classification (2000): 39B82, 39B52, 46L05, 47Jxx.
TL;DR: In this article, some bounds for I(x, y) in terms of A(x and y) and L(x x, y), and for L(X, y, y).
Abstract: In this paper, some bounds for I(x, y) in terms of A(x, y) and L(x, y) , and L(x, y) in terms of G(x, y) and I(x, y) are established. Mathematics subject classification (2000): 26E60, 26D07.
TL;DR: In this article, a comparision theorem for quasi-arithmetic means of Mercer's type for operators was proved, which is a generalization of Jensen's inequality for operator convex functions.
Abstract: Refinements of Jensen's inequality for operator convex functions, which are generalizations of Mercer's result, are proved. Obtained results are used to refine monotonicity properties for power means of Mercer's type, and a comparision theorem for quasi-arithmetic means of Mercer's type for operators.
TL;DR: In this paper, the Frobenius number and the number of gaps of a numerical semigroup generated by three positive integers are computed using an algorithm that allows us to calculate the smallest positive integer that is solution of an inequality of this type.
Abstract: AproportionallymodularDiophantineinequalityisanexpressionoftheform axmodb cx ,w herea, b and c are positive integers. In this paper we present an algorithm that allows us to calculate the smallest positive integer that is solution of an inequality of this type. We also obtain an algorithm that computes the Frobenius number and the number of gaps of a numerical semigroup generated by three positive integers.
TL;DR: In this paper, a sharpened version of the classical fundamental triangle inequality is presented, where φ = min 1 i
Abstract: In this note, we show a sharpened version of the classical fundamental triangle inequality, as follows 2R2+10Rr−r2−2(R−2r) √ R2 − 2Rr cos φ s 2R2+10Rr−r2+2(R−2r) √ R2 − 2Rr cos φ, where φ = min 1 i
TL;DR: In this paper, the approximation properties of a Kantorovich variant of the Bleimann, Butzer and Hahn operators for locally bounded functions are studied. And the authors estimate their rate of convergence by some techniques of probability theory and analysis methods.
Abstract: We study the approximation properties of a Kantorovich variant of the Bleimann, Butzer and Hahn operators for locally bounded functions, and estimate their rate of convergence by some techniques of probability theory and analysis methods. Mathematics subject classification (2000): 41A36, 41A25, 41A10.
TL;DR: In this paper, the generalized Hyers-Ulam stability of mappings on normed spaces for the Pexiderized Cauchy-Jensen additive mapping was proved.
Abstract: We prove the generalized Hyers–Ulam stability of mappings on normed spaces for the Pexiderized Cauchy–Jensen additive mapping
TL;DR: In this article, the well-known Young's inequality is refined by a double inequality, and the double inequality is used for mathematics subject classification. But it is not used in this paper.
Abstract: In this short note, the well-known Young’s inequality is refined by a double inequality. Mathematics subject classification (2000): 26D15.
TL;DR: In this article, the authors present new existence results for the singular boundary value problem −u = g(t,u )+ λh(t,u), t ∈ (0, 1) u(0 )= 0 = u(1).
Abstract: This paper presents new existence results for the singular boundary value problem −u = g(t,u )+ λh(t,u), t ∈ (0,1) u(0 )= 0 = u(1). In particular our nonlinearity may be singular at t = 0, 1a ndu = 0 and is allowed to change sign. Existence in this paper will be established by obtaining a sequence of upper and lower solutions which in turn will generate a sequence of approximate solutions.
TL;DR: In this paper, a series of new one-dimensional and multidimensional integral and discrete inequalities of the Hilbert and the Hardy-Hilbert type with non-conjugate exponents are derived.
Abstract: In this paper we derive a series of new one-dimensional and multidimensional integral and discrete inequalities of the Hilbert and the Hardy-Hilbert type, with non-conjugate exponents. First, prove and discuss two equivalent general inequalities of such type, as well as their corresponding reverse inequalities. The obtained results are then applied to various settings considering homogeneous functions of a negative real degree. In particular, we prove generalizations and refinements of some recent results of Rassias et al, related to the Hilbert-type inequalities with conjugate exponents, and some new multidimensional inequalities of the Godunova type. Mathematics subject classification (2000): 26D10, 26D15.
TL;DR: In this paper, the matrix norms of a GCD related matrix were investigated for multiplicative arithmetical functions f, and upper and lower bounds for these infinite prime products were obtained by using particular norm inequalities.
Abstract: In this paper we investigate the matrix norms of a GCD related matrix, i.e., (Sf ) = ( f (i, j)/(irjr) ) for multiplicative arithmetical functions f . In particular, we obtain upper bounds for the p norms of (Sf ) for f = φ , σα , and ψ in terms of infinite prime products. Furthermore, we give lower and upper bounds for these infinite prime products by using particular norm inequalities. Mathematics subject classification (2000): 11C20, 15A36, 15A60, 11A25.