On a Refinement of Hardy’s Inequalities via Superquadratic And Subquadratic Functions
DOI:
https://doi.org/10.5644/SJM.07.2.03Keywords:
Kernels, measures, Hardy type operators, superquadratic function, subquadratic function, integral identitiesAbstract
Let $A_k$ be an integral operator defined by
$$
A_kf(x):=\frac{1}{K(x)} \iom2 k(x,y)f(y)d\oy,
$$
where $k:\Omega_1\times \Omega_2 \to \R$ is a general nonnegative kernel,and
$(\Omega_1,\Sigma_1,\mu_1)$, $(\Omega_2,\Sigma_2,\mu_2)$ are measure spaces with $\sigma$-finite measures and
$$
K(x):=\iom2 k(x,y)d\oy, \quad x \in \Omega_1.
$$
In this paper we define a functional as a difference between the right-hand side and the left-hand side of the refined Hardy type inequality with general measures and kernels using the notation of superquadratic and subquadratic functions inequality and study its properties, such as exponential and logarithmic convexity.
2000 Mathematics Subject Classification. Primary 26D10, Secondary 26D15
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