关于一种用计算机证明不等式的方法的注记

A note on certain inequalities for bivariate means

We obtain simple proofs of certain inequalites for bivariate means.

A note on the Coifman-Fefferman and Fefferman-Stein inequalities

A condition on a Banach function space $X$ is given under which the Coifman-Fefferman and Fefferman-Stein inequalities on $X$ are equivalent.

A Note on Hadamard Inequalities for the Product of the Convex Functions

The main aim of the present note is to prove new Hadamard like integral inequalities for the product of the convex functions.

A note on Khabibullin's conjecture for integral inequalities

An integral transformation relating two inequalities in Khabibullin's conjecture is found. Another proof of this conjecture for some special values of its numeric parameters is suggested.

A Note On Mixed Mean Inequalities

We give a simpler proof of a result of Holland concerning a mixed arithmetic-geometric mean inequality. We also prove a result of mixed mean inequality involving the symmetric means.

A Note on Sharp Multivariate Bernstein- and Markov-Type Inequalities

Let $V$ be a symmetric convex body in $\R^m$. We prove sharp Bernstein-type inequalities for entire functions of exponential type with the spectrum in $V$ and discuss certain properties of the extremal functions. Markov-type inequalities with sharp constants for algebraic polynomials on $V$ and certain non-symmetric convex bodies are proved as well.

A note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalities

The main result of this paper supports a conjecture by C. Pérez and E. Rela about a very recent result of theirs on self-improving theory. Also, we extend the conclusions of their theorem to the range $p<1$. As an application of our result, we give a unified vision of weighted improved Poincaré-type inequalities in the Euclidean setting, which gathers both weighted improved classical and fractional Poincaré inequalities within an approach which avoids any representation formula. We improve some already known results. Finally, we also explore analog inequalities in the context of metric spaces …

A Note on the Weighted Harmonic-Geometric-Arithmetic Means Inequalities

In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of a symmetric positive definite matrix and an inequality related to the coefficients of polynomials with positive roots.

Note On Certain Inequalities for Neuman Means

In this paper, we give the explicit formulas for the Neuman means $N_{AH}$, $N_{HA}$, $N_{AC}$ and $N_{CA}$, and present the best possible upper and lower bounds for theses means in terms of the combinations of harmonic mean $H$, arithmetic mean $A$ and contraharmonic mean $C$.

Note on constants appearing in refined Young inequalities

In this short note, we give the refined Young inequality with Specht's ratio by only elementary and direct calculations. The obtained inequality is better than one previously shown by the author in 2012. In addition, we give a new property of Specht's ratio. These imply an alternative proof of the refined Young ineqaulity shown by author in 2012. We also give a remark on the relation to Kantorovich constant. Finally, we give a proposition for a corresponding general function.

The natural algorithmic approach of mixed trigonometric-polynomial problems

The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities by reducing to polynomial inequalities. Finally, we show the great applicability of this algorithm and as examples, we use it to analyze some new rational (Pade) approximations of the function $\cos^2(x)$, and to improve a class of inequalities by Z.-H. Yang. The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities.

关于傅里叶不等式的注记

Note on Fourier inequalities

本文证明变指数勒贝格空间中豪斯多夫--杨不等式成立当且仅当指数为常数，并在函数单调性条件下给出经典与变指数情形下 Pitt 型加权傅里叶不等式的完整刻画。

Two weight inequalities for individual Haar multipliers and other well localized operators

In this paper we are proving that Sawyer type condition for boundedness work for the two weight estimates of individual Haar multipliers, as well as for the Haar shift and other "well localized" operators.

Turán type inequalities for classical and generalized Mittag-Leffler functions

In this paper some Turán type inequalities for classical and generalized Mittag-Leffler functions are considered. The method is based on proving monotonicity for special ratio of sections for series of Mittag-Leffler functions. Some applications are considered to Lazarević and Wilker-type inequalities for classical and generalized Mittag-Leffler functions. At the end some open problems are formulated.

Two weight norm inequalities for fractional integral operators and commutators

In these lecture notes we describe some recent work on two weight norm inequalities for fractional integral operators, also known as Riesz potentials, and for commutators of fractional integrals. These notes are based on three lectures delivered at the 6th International Course of Mathematical Analysis in Andalucia, held in Antequera, Spain, September 8-12, 2014. They are, however, greatly expanded to include both new results and many details that I did not present in my lectures due to time constraints.

Weighted norm inequalities for fractional Bergman operators

We prove in this note one weight norm inequalities for some positive Bergman-type operators.

On Turán type inequalities for modified Bessel functions

In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note a conjecture…

三维情形下的Evans-Lewis不等式

Evans-Lewis inequalities in dimension three

本文针对三维情形下的Evans-Lewis不等式，提出了一种鲁棒的统一证明方法，实现了对该不等式的严格改进，并清晰解释了最佳常数的出现原因。

Ky Fan inequalities for bivariate means

In this note we give sufficient conditions for bivariate, homogeneous, symmetric means $M$ and $N$ to satisfy Ky Fan inequalities $$\frac{M}{M'}<\frac{N}{N'}\qquad\text{ and }\quad \frac{1}{M}-\frac{1}{M'}<\frac{1}{N}-\frac{1}{N'}.$$

On functional inequalities for the psi function

In this note we study the monotonicity of the function $x\mapsto ψ(1 +bx)^a/ψ(1 + ax)^b$. We also give the several inequalities involving the psi function, whic is the logarithmic derivative of the gamma function.

Inequalities related to the error function

In this note we consider inequalities involving the error function $ϕ$. Our methodes give new proofs of some known inequalities of Komatsu, and of Szarek and Werner, and also produce two families of inequalities that give upper and lower bounds for $ϕ$. Moreover the continued fractions expansion of $ϕ$ is obtained.