On Weighted Carleman-Type Inequality (original) (raw)

Note on weighted Carleman-type inequality

International Journal of Mathematics and Mathematical Sciences, 2005

A double inequality involving the constant e is proved by using an inequality between the logarithmic mean and arithmetic mean. As an application, we generalize the weighted Carleman-type inequality.

On Carleman and Knopp's Inequalities

Journal of Approximation Theory, 2002

A sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the "ordinary" sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman-)Knopp's inequality and a new inequality of Hardy-Knopp type is pointed out.

Notes on Inequalities with Doubling Weights

Journal of Approximation Theory, 1999

Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc. inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Sometimes however, like in the weighted Nikolskii inequality, the slightly stronger A∞ condition is used. Throughout their paper the Lp norm is studied under the assumption 1 ≤ p < ∞. In this note we show that their proofs can be modified so that many of their inequalities hold even if 0 < p < 1. The crucial tool is an estimate for quadrature sums for the pth power (0 < p < ∞ is arbitrary) of trigonometric polynomials established by Lubinsky, Máté, and Nevai. For technical reasons we discuss only the trigonometric cases.

Some New Hilbert's Type Inequalities

Journal of Inequalities and Applications, 2009

Some new inequalities similar to Hilbert's type inequality involving series of nonnegative terms are established.

Carleman type inequalities and Hardy type inequalities for monotone functions

This Ph.D. thesis deals with various generalizations of the inequalities by Carleman, Hardy and Pólya-Knopp. In Chapter 1 we give an introduction and overview of the area that serves as a frame for the rest of the thesis. In Chapter 2 we consider Carleman's inequality, which may be regarded as a discrete version of Pólya-Knopp's inequality and also as a natural limiting inequality of the discrete Hardy inequality. We present several simple proofs of and remarks (e.g. historical) about this inequality. In Chapter 3 we give some sharpenings and generalizations of Carleman's inequality. We discuss and comment on these results and put them into the frame presented in the previous chapter. In particular, we present some new proofs and results. In Chapter 4 we prove a multidimensional Sawyer duality formula for radially decreasing functions and with general weights. We also state the corresponding result for radially increasing functions. In particular, these results imply that we can describe mapping properties of operators defined on cones of such monotone functions. Moreover, we point out that these results can also be used to describe mapping properties of operators between some corresponding general weighted multidimensional Lebesgue spaces. In Chapter 5 we give a new weight characterization of the weighted Hardy inequality for decreasing functions and use this result to give a new weight characterization of the weighted Pólya-Knopp inequality for decreasing functions and we also give a new scale of weightconditions for characterizing the embedding Λ p (v) → Γ q (u) for the case 1 < p ≤ q < ∞. In Chapter 6 we make a unified approach to Hardy type inequalitits for decreasing functions and prove a result which covers both the Sinnamon result with one condition and Sawyer's result with two independent conditions for the case when one weight is nondecreasing. In all cases we point out that this condition is not unique and can even be chosen among some (infinite) scales of conditions. In Chapter 7 v vi Abstract we prove a weight characterization of L p ν [0, ∞)−L q μ [0, ∞

On weighted Hardy-type inequalities

Mathematical Inequalities & Applications, 2020

We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.