Generalized Young and Cauchy--Bunyakowsky Inequalities with Applications: a survey

On the Hölder and Cauchy--Schwarz inequalities

A generalization of the Hölder inequality is considered. Its relations with a previously obtained improvement of the Cauchy--Schwarz inequality are discussed.

加强的Cauchy-Schwarz和Hölder不等式

Strengthened Cauchy-Schwarz and Hölder inequalities

本文通过推导恒等式，给出了复内积空间中加强Cauchy-Schwarz不等式基本结果的新证明，并推广到加强Hölder不等式。

An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind

In the paper, the author establishes an integral representation for Cauchy numbers of the second kind, finds the complete monotonicity, minimality, and logarithmic convexity of Cauchy numbers of the second kind, and presents some inequalities for determinants and products of Cauchy numbers of the second kind.

Two Weight Inequalities for the Cauchy Transform from $ \mathbb{R}$ to $ \mathbb{C}_+$

We characterize those pairs of weights $ σ$ on $ \mathbb{R}$ and $ τ$ on $ \mathbb{C}_+$ for which the Cauchy transform $\mathsf{C}_σ f (z) \equiv \int_{\mathbb{R}} \frac {f(x)} {x-z} \; σ(dx)$, $ z\in \mathbb{C}_+$, is bounded from $L ^2(\mathbb{R};σ)$ to $L ^{2}(\mathbb{C}_+; τ)$. The characterization is in terms of an $A_2$ condition on the pair of weights and testing conditions for the transform, extending the recent solution of the two weight inequality for the Hilbert transform. As corollaries of this result we derive (1) a characterization of embedding measures for the model space $K_\v…

通过柯西中值定理的若干Ostrowski型不等式

Some Ostrowski Type Inequalites via Cauchy's Mean Value Theorem

本文利用柯西中值定理推导出若干Ostrowski型不等式，并将其应用于特定函数实例，拓展了该类不等式的适用范围。

Refinements of Some Reverses of Schwarz's Inequality in 2-Inner Product Spaces and Applications for Integrals

Refinements of some recent reverse inequalities for the celebrated Cauchy-Bunyakovsky-Schwarz inequality in 2-inner product spaces are given. Using this framework, applications for determinantal integral inequalities are also provided.