engleski

Multidimensional Hardy-Type Inequalities via Convexity

Let an almost everywhere positive function $\Phi$ be convex for $p > 1$ and $p < 0$, concave for $p \in (0, 1)$, and such that $Ax^p \leq \Phi(x) \leq Bx^p$ holds on $R+$ for some positive constants $A \leq B$. In this paper we derive a class of general integral multidimensional Hardy-type inequalities with power weights, whose left-hand sides involve $\Phi(\int_{; ; ; R^n_+}; ; ; f (t) dt)$ instead of $(\int_{; ; ; R^n_+}; ; ; f (t) dt)^p$, while the corresponding right-hand sides remain as in the classical Hardy&#8217; s inequality and have explicit constants in front of integrals. We also prove the related dual inequalities. The relations obtained are new even for the one-dimensional case and they unify and extend several inequalities of Hardy type known in the literature.

Inequalities; integral inequalities; multidimensional inequalities; Hardy's inequality; weights; power weights; Jensen's inequality; convex functions

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