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Integrals Refining Convex Inequalities
Bulletin of the Malaysian Mathematical Sciences Society, 2019The authors prove that, if \(\Phi:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{K})\) is a normalized positive linear map, \(A\in\mathcal{B}(\mathcal{H})\) is a self-adjoint operator with the spectrum in \(J\), and \(f:J\rightarrow\mathbb{R}\) is a convex function, then \begin{align*} f\left(\frac{\langle\Phi(A)x, x\rangle+\langle\phi(A)y, y ...
Mohammad Sababheh +2 more
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Symmetrization and Integral Inequalities
Mathematical Notes, 2023This paper offers a comprehensive investigation into Steiner symmetrizations applied to anisotropic integral functionals within the multivariate calculus of variations, with a specific focus on functions belonging to the Sobolev class and characterized by compact support.
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Integral inequalities resembling Copson's inequality
Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 1990AbstractThe present paper deals with two inequalities which resemble Copson's integral inequalities. From our theorems, we obtain two interesting corollaries.
Mohapatra, R. N., Vajravelu, K.
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2000
Some integral inequalities of the following type are proved: Suppose \(f\) has a continuous \(n\)th order derivative on \([a,b]\); \(f^{(i)}(a)\geq 0\) and \(f^{(n)}(x)\geq n!\) for all \(x\in [a,b]\) and \(0\leq i\leq n-1\). Then \[ \int^b_a [f(x)]^{n+2} dx\geq \Biggl[\int^b_a f(x) dx\Biggr]^{n+ 1}. \] An open problem is also stated.
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Some integral inequalities of the following type are proved: Suppose \(f\) has a continuous \(n\)th order derivative on \([a,b]\); \(f^{(i)}(a)\geq 0\) and \(f^{(n)}(x)\geq n!\) for all \(x\in [a,b]\) and \(0\leq i\leq n-1\). Then \[ \int^b_a [f(x)]^{n+2} dx\geq \Biggl[\int^b_a f(x) dx\Biggr]^{n+ 1}. \] An open problem is also stated.
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