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Journal of Mathematical Analysis and Applications, 1999 The following result is proved: Let \(0< b_{n+1}\leq b_n\), \(B_n= \sum^n_{k= 1}b_k\), \(a_n\geq 0\), \(0< \sum^\infty_{n=1} b_na_n< \infty\). Then \[ \sum^\infty_{n= 1} b_{n+ 1}(a^{b_1}_1\cdots a^{b_n}_n)^{1/B_n}< e \sum^\infty_{n= 1} \Biggl[1- {b_n\over 2(B_n+ b_n)}\Biggr] b_na_n.
Bicheng, Yang
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On Bicheng-Debnath's generalizations of Hardy's integral inequality [PDF]
International Journal of Mathematics and Mathematical Sciences, 2001 We consider Hardy's integral inequality and we obtain some new generalizations of Bicheng-Debnath's recent results. We derive two distinguished classes of inequalities covering all admissible choices of parameter k from Hardy's original relation ...
Aleksandra Cižmešija, Josip Pecaric
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An Equality Underlying Hardy’s Inequality [PDF]
The American Mathematical Monthly, 2022 A classical inequality of G. H. Hardy states that Cx≤2x for x in l2, where C is the Cesàro (alias averaging) operator. This inequality has been strengthened to (C−I)x≤x. It has also been shown that CTx≤Cx for x in l2. We present equalities that imply these inequalities, together with the reverse inequalities (C−I)x≥(1/√2)x and Cx≤√2CTx. We also present
Jameson, Graham
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Hardy’s inequalities and integral operators on Herz-Morrey spaces
Open Mathematics, 2020 We obtain some estimates for the operator norms of the dilation operators on Herz-Morrey spaces. These results give us the Hardy’s inequalities and the mapping properties of the integral operators on Herz-Morrey spaces.
Yee Tat-Leung, Ho Kwok-Pun
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An Improved Discrete Hardy Inequality [PDF]
The American Mathematical Monthly, 2018 We improve the classical discrete Hardy inequality \begin{equation*}\label{1} \sum _{n=1}^{\infty }a_{n}^{2}\geq \left({\frac {1}{2}}\right)^{2} \sum _{n=1}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{2}, \end{equation*} where $\{a_n\}_{n=1}^\infty$ is any sequence of non-negative real numbers.
Matthias Keller +2 more
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Some new scales of characterization of Hardy’s inequality; pp. 7–18 [PDF]
Proceedings of the Estonian Academy of Sciences, 2010 Let 1 lt; p ⤠q lt; â. Inspired by some recent results concerning Hardy-type inequalities where the equivalence of four scales of integral conditions was proved, we use related ideas to find ten new equivalence scales of integral conditions.
Amiran Gogatishvili +2 more
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Some Hardy's inequalities on conformable fractional calculus
Demonstratio MathematicaIn this article, we will demonstrate some Hardy’s inequalities by utilizing Hölder inequality, integration by parts, and chain rule of the conformable fractional calculus.
AlNemer Ghada +5 more
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A logarithmic Hardy inequality
Journal of Functional Analysis, 2010 We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither ...
Del Pino, Manuel +3 more
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Semilinear Parabolic Equations on the Heisenberg Group with a Singular Potential
Abstract and Applied Analysis, 2012 We discuss the asymptotic behavior of solutions for semilinear parabolic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy's inequality, and the nonlinearity is controlled by Sobolev's inequality. We also
Houda Mokrani, Fatimetou Mint Aghrabatt
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Ground state solutions for Kirchhoff-type equations with general nonlinearity in low dimension
Boundary Value Problems, 2021 This paper is dedicated to studying the following Kirchhoff-type problem: { − m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) Δ u + V ( x ) u = f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , $$ \textstyle\begin{cases} -m ( \Vert \nabla u \Vert ^{2}_{L^{2}(\mathbb{R} ^{N})} )\Delta u+
Jing Chen, Yiqing Li
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