Showing papers in "Mathematical Inequalities & Applications in 2009"
TL;DR: In this article, the sequence Banach space ψ (Z) is defined for a class of convex functions ψ, and properties of the Kand Jinterpolation spaces (E0,E1)θ,ψ,K and (E 0,E 1) θ ∈ (0,1) are studied.
Abstract: In this paper the sequence Banach space ψ (Z) is defined for a class of convex functions ψ , and properties of the Kand Jinterpolation spaces (E0,E1)θ ,ψ,K and (E0,E1)θ ,ψ,J for a Banach couple E = (E0,E1) and θ ∈ (0,1) are studied. Mathematics subject classification (2000): 46B70.
963 citations
TL;DR: Differential subordination and superordination results for analytic functions in the open unit disk which are associated with the multiplier transformation were obtained by investigating appropriate classes of admissible functions as discussed by the authors.
Abstract: Differential subordination and superordination results are obtained for analytic functions in the open unit disk which are associated with the multiplier transformation These results are obtained by investigating appropriate classes of admissible functions Sandwich-type results are also obtained Mathematics subject classification (2000): 30C80, 30C45
55 citations
TL;DR: In this paper, the authors state and prove weighted Hardy type inequalities with an integral operator A(k) defined by A (k)f(x) := 1/K(x)-integral integral(Omega 2) k(x,y) f(y)d mu(2) (y).
Abstract: We state and prove some new weighted Hardy type inequalities with an integral operator A(k) defined by A(k)f(x) := 1/K(x) integral(Omega 2) k(x,y)f(y)d mu(2) (y), where k : Omega(1) x Omega(2) --> ...
48 citations
TL;DR: In this paper, the complementary incomplete gamma function (G(a,z)$ with complex parameters $a$ and $z$ was shown to be of order (1, 2) for the hyperboloid of one sheet.
Abstract: We prove upper and lower bounds for the complementary incomplete gamma function $\G(a,z)$ with complex parameters $a$ and $z$. Our bounds are refined within the circular hyperboloid of one sheet $\{(a,z):|z|>c|a-1|\}$ with $a$ real and $z$ complex. Our results show that within the hyperboloid, $|\G(a,z)|$ is of order $|z|^{a-1}e^{-\Re(z)}$, and extends an upper estimate of Natalini and Palumbo to complex values of $z$.
43 citations
TL;DR: In this paper, the authors survey some important properties of functions belonging to these classes of functions and prove some new results concerning properties concerning functions from them, and present a survey of the properties of nonnegative quasiconvex functions.
Abstract: In 1985 Godunova and Levin have considered the following class of functions. A function f : I → R is said to belong to the class Q(I) if it is nonnegative and for all x,y ∈ I and t ∈ (0,1) , satisfies the inequality: f ((1− t)x+ ty) f (x) 1− t + f (y) t Here I is an interval of R . It is known that all nonnegative quasiconvex functions belong to this class and this class of functions coincides with the class of Schur functions S (I) , that is, with the class of nonnegative functions that satisfy the inequality ∑ f (x) (x− y) (x− z) 0 for every x,y,z ∈ I The aim of this paper is to survey some important properties of functions belonging to these classes of functions and to prove some new results concerning properties of functions from them. Mathematics subject classification (2000): Primary 26D20, Secondary 26D07, 26A51.
42 citations
29 citations
TL;DR: In this article, the authors give new regularity conditions expressed via epigraphs that assure strong duality between a given primal convex optimization problem and its Lagrange and Fenchel-Lagrange dual problems, respectively, in infinite dimensional spaces.
Abstract: We give new regularity conditions expressed via epigraphs that assure strong duality between a given primal convex optimization problem and its Lagrange and Fenchel-Lagrange dual problems, respectively, in infinite dimensional spaces. Moreover we completely characterize through equivalent statements the so-called stable strong duality between the initial problem and the mentioned duals.
26 citations
TL;DR: In this article, a new operator using the Sălăgean and Ruscheweyh operators is defined, denoted by Sn (δ,α), of analytic functions in the open unit disc.
Abstract: In the present paper we define a new operator using the Sălăgean and Ruscheweyh operators. By Lα we denote the operator given by L n α : A → A, Lα f (z) = (1−α)Rn f (z) + αSn f (z), for z ∈U, where Rn f (z) denotes the Ruscheweyh derivative, Sn f (z) is the Sălăgean operator and An = { f ∈ H (U) : f (z) = z+ an+1z + . . . , z ∈U} is the class of normalized analytic functions with A1 = A. A certain subclass, denoted by Sn (δ ,α) , of analytic functions in the open unit disc is introduced by means of the new operator. By making use of the concept of differential subordination we derive various properties and characteristics of the class Sn (δ ,α) . Also, several differential subordinations are established regarding the operator Lα . Mathematics subject classification (2000): 30C45, 30A20, 34A40.
22 citations
TL;DR: In this article, an inequality for curvature-like tensor fields was given for the case of Lagrangian submanifolds of complex space forms and centroaffine hypersurfaces.
Abstract: We give an inequality for curvature-like tensor fields and apply this to Lagrangian submanifolds of complex space forms and to centroaffine hypersurfaces. In both settings we investigate the equality case and give a classification theorem if equality is attained at every point of the submanifold. We also provide an example showing that this inequality is best possible, in a sense explained in the paper. Mathematics subject classification (2000): 53A15, 53B25.
21 citations
TL;DR: In this paper, discrete moment problems with given finite supports and unimodal distributions with known mode are formulated and used to obtain sharp lower and upper bounds for expectations of higher order convex functions of discrete random variables as well as probabilities of the union of events.
Abstract: Discrete moment problems with given finite supports and unimodal distributions with known mode, are formulated and used to obtain sharp lower and upper bounds for expectations of higher order convex functions of discrete random variables as well as probabilities of the union of events. The bounds are based on the knowledge of some of the power moments of the random variables involved, or the binomial moments of the number of events which occur. The bounding problems are formulated as LP’s and dual feasible basis structure theorems as well as the application of the dual method of linear programming provide us with the results. Applications in PERT and reliability are presented. Mathematics subject classification (2000): 60E15, 90B15, 90B25, 90C05. Keywordsandphrases: Discretemomentproblem, linear programming, bounding probabilities, discrete unimodality.
18 citations
TL;DR: In this paper, the reverse Cauchy-Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C*-algebra were proved.
Abstract: We prove two new reverse Cauchy-Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C*-algebra. We apply our results to ...
TL;DR: In this article, new refined lower and upper bound forms of Jordan's inequality are proved, and the lower bound form is shown to improve L. Yang's inequality that plays a pivotal role in the theory of distribution of values of functions.
Abstract: New refined lower and upper bound forms of Jordan’s inequality are proved. As an application, the lower bound form is shown to improve L. Yang’s inequality that plays a pivotal role in the theory of distribution of values of functions. Some numerical results are included. Mathematics subject classification (2000): 26D15.
TL;DR: In this article, the authors give some further considerations about logarithmic convexity for differences of power for positive linear functionals as well as some related results, and give a classification of mathematics subject classification.
Abstract: We give some further considerations about logarithmic convexity for differences of power Means for positive linear functionals as well as some related results. Mathematics subject classification (2000): 26A51, 26A46, 26A48.
TL;DR: In this article, the authors obtained the inequalities for the iterated convolution and their applications to physical problems and also obtained the inequality ∥ ∥∥∦∥ ∦∦ ∦ √ ∑ m ⎛ ⎝ r ∏ j=1 ∗(Fm,jρm, j) ⎞ ⎠ √ r ∈ j= 1 ∗|ρm,j|⎠ 1 p−1 √∥
Abstract: In this paper, we obtain the inequalities for the iterated convolution and their applications to physical problems. We also get the inequality ∥∥∥∥∥∥ ∑ m ⎛ ⎝ r ∏ j=1 ∗(Fm,jρm,j) ⎞ ⎠ ⎛ ⎝ r ∏ j=1 ∗|ρm,j| ⎞ ⎠ 1 p−1 ∥∥∥∥∥∥ Lp(Rn) ∑ m r ∏ j=1 ‖Fm,j‖Lp(Rn,|ρm,j|) and its applications in Lp(R, |ρ|) space. Mathematics subject classification (2000): 44A35, 35A22, 26D20.
TL;DR: In this paper, a Chebyshev type sharp upper bound is presented for the tail probability of scale mixtures of the zero mean multivariate normal distribution, only in terms of the variance.
Abstract: In this short note a Chebyshev type sharp upper bound is presented for the tail probability of scale mixtures of the zero mean multivariate normal distribution, only in terms of the variance. Similar estimation is proved for the probability content of an arbitrary ellipsoid containing the origin. Mathematics subject classification (2000): 60E15.
TL;DR: Assuming differentiability of these additive generators, equivalent sufficient conditions that can be expressed as inequalities involving derivatives of the additive generators are proposed, avoiding the need of composing them.
Abstract: Dominance between triangular norms (t-norms) is a versatile relationship. For continuous Archimedean t-norms, dominance can be verified by checking one of many sufficient conditions derived from a generalization of the Mulholland inequality. These conditions pertain to various convexity properties of compositions of additive generators and their inverses. In this paper, assuming differentiability of these additive generators, we propose equivalent sufficient conditions that can be expressed as inequalities involving derivatives of the additive generators, avoiding the need of composing them. We demonstrate the powerfulness of the results by the straightforward rediscovery of dominance relationships in the Schweizer-Sklar t-norm family, as well as by unveiling some formerly unknown dominance relationships in the Sugeno-Weber t-norm family. Finally, we illustrate that the results can also be applied to members of different parametric families of t-norm.
TL;DR: In this paper, the authors prove Lp boundedness of certain singular integral operators associated with a variable surface of revolution (SOMR) under the assumption that related lower dimensional maximal operators are bounded.
Abstract: We prove Lp boundedness of certain singular integral operators associated with a variable surface of revolution assuming a boundedness of related lower dimensional maximal operators. The singular integrals are defined by rough kernels satisfying certain size and cancellation conditions. Mathematics subject classification (2000): Primary 42B20.
TL;DR: In this paper, three different criteria for Lp -Lq boundedness of Volterra integral operator with locally integrable weight functions w,v and a non-negative kernel k(x,y) satisfying Oinarov's condition are given.
Abstract: Three different criteria for Lp -Lq boundedness of Volterra integral operator with locally integrable weight functions w,v and a non-negative kernel k(x,y) satisfying Oinarov’s condition are given. Relations between components of the boundedness constants are described.
TL;DR: In this paper, the value distribution of the differential polynomials f nf (k)−1 where n( 2), k are positive integers was considered and some estimates were obtained only by the reduced counting function.
Abstract: In this paper, we consider the value distribution of the differential polynomials f nf (k)− 1 where n( 2), k are positive integers, and obtain some estimates only by the reduced counting function. Mathematics subject classification (2000): 30D35, 26D10.
TL;DR: In this paper, the authors established a generalisation of the Pecaric-Rajic inequality by providing upper and lower bounds for the norm of the linear combination ∑n j=1 αjxj where αj ∈ K and xj∈ X for j ∈ {1,..., n} with n ≥ 2.
Abstract: In this paper we establish a generalisation of the recent PecaricRajic inequality by providing upper and lower bounds for the norm of the linear combination ∑n j=1 αjxj where αj ∈ K and xj ∈ X for j ∈ {1, . . . , n} with n ≥ 2. Applications for two vectors that are related to the Massera-Schaffer, Dunkl-Williams and Maligranda-Mercer inequalities are given. Some bounds for the quantity ‖x/ ‖y‖ − y/ ‖x‖‖ with x, y ∈ X {0}, are also provided.
TL;DR: In this article, a generalized generalized system for nonlinear variational inequalities in Hilbert spaces is presented, which is based on the convergence of projection methods and can be applied to variational problems.
Abstract: In this paper, we introduce a new algorithm for a generalized system for a relaxed cocoercive nonlinear inequality and an asymptotically nonexpansive mapping in Hilbert spaces by the convergence of projection methods. Our results include the previous results as special cases extend and improve the main results of [R.U. Verma, General convergence analysis for two-step projection methods and application to variational problems. Appl. Math. Lett. 18 (11) (2005), 1286-1292], [R.U. Verma, Generalized system for relaxed cocoercive variational inequalities and its projection methods, J. Optim. Theory Appl. 121 (1) (2004), 203-210], [R.U. Verma, Generalized class of partial relaxed monotonicity and its connections, Adv. Nonlinear Var. Inequal. 7 (2) (2004), 155-164], [N.H. Xiu, J.Z. Zhang, Local convergence analysis of projection type algorithms: Unified approach, J. Optim. Theory Appl. 115 (2002) 211-230], [N.H. Nie, Z. Liu, K.H. Kim, S.M. Kang, A system of nonlinear variational inequalities involving strong monotone and pseudocontractive mappings, Adv. Nonlinear Var. Inequal. 6 (2) (2003), 91-99], [S.S. Chang, H.W. Joseph Lee, C.K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Math. Lett. 20 (3) (2007), 329-334] and many others. Mathematics subject classification (2000): 47J05; 47J25.
TL;DR: In this paper, the authors prove coefficient bounds and distortion inequalities, and associated inclusion relations for the (n, δ) -neighborhoods of a class of p -valently analytic functions with negative coefficients, defined by means of a certain nonhomogeneous Cauchy-Euler differential equation.
Abstract: In this present paper, by making use of the familiar concept of neighborhoods of p valent functions, the author prove coefficient bounds and distortion inequalities, and associated inclusion relations for the (n, δ) -neighborhoods of a class of p -valently analytic functions with negative coefficients, which is defined by means of a certain non-homogeneous Cauchy-Euler differential equation. Relevant connections of some of the results obtained in this paper with those in earlier works are also provided. Mathematics subject classification (2000): 30C45.
TL;DR: In this paper, upper and lower bounds for the p-angular distance in normed linear spaces are given. But none of the obtained upper bounds are better than the corresponding results due to L. Maligranda recently established in the paper======[simple norm inequalities, Amer. Math. Monthly, 113(2006), 256-260].
Abstract: New upper and lower bounds for the p-angular distance in normed
linear spaces are given. Some of the obtained upper bounds are better than the
corresponding results due to L. Maligranda recently established in the paper
[Simple norm inequalities, Amer. Math. Monthly, 113(2006), 256-260].
TL;DR: In this article, a general form of Hardy-Hilbert's inequality with perturbed Hilbert's kernel with the best possible estimation in the case of conjugate exponents is obtained.
Abstract: A general form of recently obtained Hardy-Hilbert’s inequalitywith perturbed Hilbert’s kernel with the best possible estimation in the case of conjugate exponents is obtained. The multidimensional case is also considered. The case of non-conjugate exponents is briefly given. Mathematics subject classification (2000): 26D15.
TL;DR: In this article, the authors obtained an absolute summability factor theorem for lower triangular matrices, which is the first one known for lower triangulation matrices and the first for lower quadratic matrices.
Abstract: In this paper we obtain an absolute summability factor theorem for lower triangular matrices.
TL;DR: In this paper, the strong convergence of Browder type iteration for multivalued nonexpansive non-self-mapping T satisfying the weakly inwardness condition in a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm was established.
Abstract: In this paper, we established the strong convergence of Browder type iteration {xt} for the multivalued nonexpansive nonself-mapping T satisfying the weakly inwardness condition in a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm or in a reflexive Banach space with weakly sequentially continuous duality mapping. Furthermore, we also obtained the strong convergent results for the Halpern type iteration {xn} for multivalued nonexpansive nonself-mapping T . Mathematics subject classification (2000): 47H05, 47H10, 47H17.
TL;DR: In this paper, the authors considered the inequalities of type MJn(f,x,qq) Jn(m,x,pp) mJn(k,k,q) for convex functions on the real line and gave an alternative proof of the Jensen-Steffensen inequalities.
Abstract: We consider the inequalities of type MJn(f ,x,qq) Jn(f ,x,pp) mJn(f ,x,qq), where f is a convex function and Jn(f ,xx,pp )= n=1 pif (xi) � f n=1 pixi , recently introduced by S.S. Dragomir. We give an alternative proof of such inequalities and prove another similar result for the case when f is a convex function on an interval in the real line, while p and q satisfy the conditions for Jensen-Steffensen inequality. We show that our result improves the result of Dragomir in this special case. We also prove the integral versions of all our results, including those related to Boas' generalization of Jensen-Steffensen integral inequality.
TL;DR: In this paper, the authors consider elliptic equations in non-chicco form with discontinuous coefficients and obtain local and non-local a priori bounds for solutions of Dirichlet problem and study dependence of the constants in the estimates.
Abstract: In the present paper we consider Morrey spaces in unbounded domains and study elliptic equations in nondivergence form with discontinuous coefficients when the class of discontinuities is of Chicco type. In particular we state some local and non local a priori bounds for solutions of Dirichlet problem and study the dependence of the constants in the estimates. The idea is to approximate the principal coefficients by functions with derivatives which belong locally to the space Ls , 2 < s n , while the coefficients of lower terms in the differential operator belong to Morrey spaces. Our results are based on embedding theorems which allow us to require a summability lower than n for the coefficients of the operator L . Mathematics subject classification (2000): 35J25, 46E35.
TL;DR: The comparison theorem for R(u, v; r, s; x, y) = (E(r, s, xv, yv) E(r, s; Xu, yy) )1/(v−u), u = v, where E is the Stolarsky mean, is proved as discussed by the authors.
Abstract: The comparison theorem for R(u, v; r, s; x, y) = ( E(r, s; xv, yv) E(r, s; xu, yy) )1/(v−u) , u = v, where E is the Stolarsky mean, is proved. This generalises the results of Leach, Sholander and Páles. Mathematics subject classification (2000): 26D15.
TL;DR: In this article, it was shown that for all nonnegative μ-integrable simple functions x,y :Ω → R (where Ω(x) stands for the sup- port of x), there exists a real p > 1 such that ϕ1(t)
Abstract: Let (Ω,Σ,μ) be a measure space such that 0 0). We prove that if Ω xydμ ψ 1 Ω(x) ϕ 1 ◦| x|dμ ψ2 Ω(y) ϕ 2 ◦| y|dμ for all nonnegative μ-integrable simple functions x,y :Ω → R (where Ω(x) stands for the sup- port of x), then then there exists a real p > 1 such that ϕ1(t)