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Search: MSC category 26D10 ( Inequalities involving derivatives and differential and integral operators )

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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 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

3. 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

4. CMB 2010 (vol 53 pp. 327)

Luor, Dah-Chin
Multidimensional Exponential Inequalities with Weights
We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$ \Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p}, $$ where 0<$p$, $q$<$\infty$, $\Phi$ is a logarithmically convex function, and $T_k$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin--Cochran-Lee type inequalities.

Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regions
Categories:26D15, 26D10

5. 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

6. CMB 2004 (vol 47 pp. 540)

Jain, Pankaj; Jain, Pawan K.; Gupta, Babita
Compactness of Hardy-Type Operators over Star-Shaped Regions in $\mathbb{R}^N$
We study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in $\RR^N$, and (ii) when the average is taken over all dilations of star-shaped regions in $\RR^N$. These cases include, respectively, the average over annuli and the average over balls centered at origin.

Keywords:Hardy operator, Hardy-Steklov operator, compactness, boundedness, star-shaped regions
Categories:46E35, 26D10

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