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1. CMB Online first

Atıcı, Ferhan M.; Yaldız, Hatice
Convex Functions on Discrete Time Domains
In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f \geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept, we state and prove discrete Hermite-Hadamard inequality using the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$). Second, we state and prove the discrete fractional Hermite-Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.

Keywords:discrete calculus, discrete fractional calculus, convex functions, discrete Hermite-Hadamard inequality
Categories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15

2. CMB Online first

Lin, Minghua
A determinantal inequality involving partial traces
Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes \mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved an inequality for Schatten $p$-norms: \[ 1+\|\mathbf{A}\|_p\ge \|\tr_1 \mathbf{A}\|_p+\|\tr_2 \mathbf{A}\|_p, \] where $\tr_1, \tr_2$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality \[ 1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n. \]

Keywords:determinantal inequality, partial trace, block matrix
Categories:47B65, 15A45, 15A60

3. CMB 2015 (vol 59 pp. 204)

Spektor, Susanna
Restricted Khinchine Inequality
We prove a Khintchine type inequality under the assumption that the sum of Rademacher random variables equals zero. We also show a new tail-bound for a hypergeometric random variable.

Keywords:Khintchine inequality, Kahane inequality, Rademacher random variables, hypergeometric distribution.
Categories:46B06, 60E15, 52A23, 46B09

4. CMB 2015 (vol 58 pp. 486)

Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong
Inequalities for Partial Derivatives and their Applications
We present various weighted integral inequalities for partial derivatives acting on products and compositions of functions which are applied to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations.

Keywords:inequality for integral, Opial-type inequality, Hölder's inequality, partial differential operator, partial differential equation
Categories:26D10, 35A23

5. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators
Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic Schrödinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse Hölder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained.

Keywords:Musielak-Orlicz-Hardy space, magnetic Schrödinger operator, atom, second-order Riesz transform, maximal inequality
Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

6. CMB 2014 (vol 58 pp. 561)

Martinez-Maure, Yves
Plane Lorentzian and Fuchsian Hedgehogs
Parts of the Brunn-Minkowski theory can be extended to hedgehogs, which are envelopes of families of affine hyperplanes parametrized by their Gauss map. F. Fillastre introduced Fuchsian convex bodies, which are the closed convex sets of Lorentz-Minkowski space that are globally invariant under the action of a Fuchsian group. In this paper, we undertake a study of plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the Fuchsian analogues of classical geometrical inequalities (analogues which are reversed as compared to classical ones).

Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequality
Categories:52A40, 52A55, 53A04, 53B30

7. CMB 2011 (vol 55 pp. 555)

Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang
Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications
In this paper we prove weighted norm inequalities with weights in the $A_p$ classes, for pseudodifferential operators with symbols in the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the scope of Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield $L^{p}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\mathrm{OP}S^{m}_{\rho, \delta}$.

Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates
Categories:42B20, 42B25, 35S05, 47G30

8. CMB 2011 (vol 55 pp. 708)

Demeter, Ciprian
Improved Range in the Return Times Theorem
We prove that the Return Times Theorem holds true for pairs of $L^p-L^q$ functions, whenever $\frac{1}{p}+\frac{1}{q}<\frac{3}{2}$.

Keywords:Return Times Theorem, maximal multiplier, maximal inequality
Categories:42B25, 37A45

9. CMB 2011 (vol 55 pp. 597)

Osękowski, Adam
Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
We determine the best constants $C_{p,\infty}$ and $C_{1,p}$, $1 < p < \infty$, for which the following holds. If $u$, $v$ are orthogonal harmonic functions on a Euclidean domain such that $v$ is differentially subordinate to $u$, then $$ \|v\|_p \leq C_{p,\infty} \|u\|_\infty,\quad \|v\|_1 \leq C_{1,p} \|u\|_p. $$ In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of $\mathbb R^2$. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.

Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem
Categories:31B05, 60G44, 60G40

10. CMB 2011 (vol 55 pp. 537)

Kang, Dongsheng
Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents
In this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy-Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated.

Keywords:elliptic problem, solution, Hardy-Sobolev inequality, singularity, Moser iteration
Categories:35B33, 35B40, 35J60

11. CMB 2011 (vol 55 pp. 88)

Ghanbari, K.; Shekarbeigi, B.
Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a connected bounded domain in $\mathbb{R}^n$. Let $0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues of the following Dirichlet problem: $$ \begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in D u|_{\partial D}=\frac{\partial u}{\partial n}|_{\partial D}=0, \end{cases} $$ where $V(x)$ is a nonnegative potential, and $\rho(x)\in C(\bar{D})$ is positive. We prove the following inequalities: $$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}} {\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times \frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2}, $$ $$\frac{n^2k^2}{8(n+2)}\leq \Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr] \times\sum_{i=1}^k\mu_i^{1/2}. $$

Keywords:biharmonic operator, eigenvalue, eigenvector, inequality

12. CMB 2011 (vol 54 pp. 630)

Fiorenza, Alberto; Gupta, Babita; Jain, Pankaj
Mixed Norm Type Hardy Inequalities
Higher dimensional mixed norm type inequalities involving certain integral operators are characterized in terms of the corresponding lower dimensional inequalities.

Keywords:Hardy inequality, reverse Hardy inequality, mixed norm, Hardy-Steklov operator
Categories:26D10, 26D15

13. CMB 2010 (vol 54 pp. 159)

Sababheh, Mohammad
Hardy Inequalities on the Real Line
We prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

Keywords:Hardy's inequality, inequalities including the Fourier transform and Hardy spaces
Categories:42A05, 42A99

14. CMB 2009 (vol 53 pp. 153)

Niu, Pengcheng; Ou, Yafei; Han, Junqiang
Several Hardy Type Inequalities with Weights Related to Generalized Greiner Operator
In this paper, we establish several weighted $L^p (1\lt p \lt \infty)$ Hardy type inequalities related to the generalized Greiner operator by improving the method of Kombe. Then the best constants in inequalities are discussed by introducing new polar coordinates.

Keywords:generalized Greiner operator, polar coordinates, Hardy inequality
Categories:35B05, 35H99

15. CMB 2008 (vol 51 pp. 161)

Agarwal, Ravi P.; Otero-Espinar, Victoria; Perera, Kanishka; Vivero, Dolores R.
Wirtinger's Inequalities on Time Scales
This paper is devoted to the study of Wirtinger-type inequalities for the Lebesgue $\Delta$-integral on an arbitrary time scale $\T$. We prove a general inequality for a class of absolutely continuous functions on closed subintervals of an adequate subset of $\T$. By using this expression and by assuming that $\T$ is bounded, we deduce that a general inequality is valid for every absolutely continuous function on $\T$ such that its $\Delta$-derivative belongs to $L_\Delta^2([a,b)\cap\T)$ and at most it vanishes on the boundary of $\T$.

Keywords:time scales calculus, $\Delta$-integral, Wirtinger's inequality

16. CMB 2006 (vol 49 pp. 256)

Neelon, Tejinder
A Bernstein--Walsh Type Inequality and Applications
A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak--Siciak theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a double power series $F(x,y)$\ converges on a set of lines of positive capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero.

Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series
Categories:32A05, 26E05

17. CMB 2006 (vol 49 pp. 185)

Averkov, Gennadiy
On the Inequality for Volume and Minkowskian Thickness
Given a centrally symmetric convex body $B$ in $\E^d,$ we denote by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex body in $\M^d(B).$ The relationship between volume $V(K)$ and the Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can naturally be given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple corollary of the Rogers--Shephard inequality we obtain that $\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach--Mazur distance to the regular hexagon.

Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratio
Categories:52A40, 46B20

18. CMB 2006 (vol 49 pp. 82)

Gogatishvili, Amiran; Pick, Luboš
Embeddings and Duality Theorem for Weak Classical Lorentz Spaces
We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$ \left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t), $$ where $$ f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds. $$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$ \|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t), $$ where $$ f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds. $$

Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality
Categories:26D10, 46E20

19. CMB 2004 (vol 47 pp. 206)

Hurri-Syrjänen, Ritva
The Poincaré Inequality and Reverse Doubling Weights
We show that Poincar\'e inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains.

Keywords:reverse doubling weights, Poincaré inequality, John domains

20. CMB 2001 (vol 44 pp. 376)

Zhang, Xi
A Note on $p$-Harmonic $1$-Forms on Complete Manifolds
In this paper we prove that there is no nontrivial $L^{q}$-integrably $p$-harmonic $1$-form on a complete manifold with nonnegatively Ricci curvature $(0
Keywords:$p$-harmonic, $1$-form, complete manifold, Sobolev inequality
Categories:58E20, 53C21

21. CMB 1999 (vol 42 pp. 478)

Pruss, Alexander R.
A Remark On the Moser-Aubin Inequality For Axially Symmetric Functions On the Sphere
Let $\scr S_r$ be the collection of all axially symmetric functions $f$ in the Sobolev space $H^1(\Sph^2)$ such that $\int_{\Sph^2} x_ie^{2f(\mathbf{x})} \, d\omega(\mathbf{x})$ vanishes for $i=1,2,3$. We prove that $$ \inf_{f\in \scr S_r} \frac12 \int_{\Sph^2} |\nabla f|^2 \, d\omega + 2\int_{\Sph^2} f \, d\omega- \log \int_{\Sph^2} e^{2f} \, d\omega > -\oo, $$ and that this infimum is attained. This complements recent work of Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang concerning the Moser-Aubin inequality.

Keywords:Moser inequality, borderline Sobolev inequalities, axially symmetric functions
Categories:26D15, 58G30

22. CMB 1999 (vol 42 pp. 87)

Kittaneh, Fuad
Some norm inequalities for operators
Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ which satisfy the relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then $$ \Biglvert \,\Bigl|\sum^n_{i=1} A^*_i X_i B_i \Bigr|^r \,\Bigrvert^2 \leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert $$ for every $r>0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.

Keywords:Unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequality
Categories:47A30, 47B10, 47B15, 47B20

23. CMB 1997 (vol 40 pp. 169)

Cruz-Uribe, David
The class $A^{+}_{\infty}(\lowercase{g})$ and the one-sided reverse Hölder inequality
We give a direct proof that $w$ is an $A^{+}_{\infty}(g)$ weight if and only if $w$ satisfies a one-sided, weighted reverse H\"older inequality.

Keywords:one-sided maximal operator, one-sided $(A_\infty)$, one-sided, reverse Hölder inequality

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