On An Application of Hayashi's Inequality for Differentiable Functions

Jackson-Stechkin type inequalities for differentiable functions in weighted Orlicz spaces

In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt's $A_{p}$ condition. To obtain refined version of the Jackson type inequality we prove an extrapolation theorem, Marcinkiewicz multiplier theorem and Littlewood Paley type results. As a consequence refined inverse Marchaud type inequalities are obtained. By means of a realization result we find an equivalence between the fractional order weighted modulus of smoothness and the classical weighted Peetre's K-functional.

Brouwer不动点定理的一个应用：梯度范数为常数的连续可微凸函数

An application of Brouwer's fixed-point theorem: continuously differentiable convex functions with gradient of constant norm

本文利用Brouwer不动点定理和Cauchy--Bunyakovsky--Schwarz不等式证明梯度范数为常数的连续可微凸函数必为仿射映射，并讨论了对Hilbert空间函数的可能推广。

Integral inequalities for mappings whose derivatives are s-convex in the second sense and applications to special means for positive real numbers

In this paper, the authors establish a new type integral inequalities for differentiable s-convex functions in the second sense. By the well-known Hölder inequality and power mean inequality, they obtain some integral inequalities related to the s-convex functions and apply these inequalities to special means for positive real numbers.

A class of nowhere differentiable functions satisfying some concavity type estimate

In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition seems to be independent of nowhere differentiable character, it turns out that each function in P is nowhere differentiable. The class P naturally appear from both a geometrical viewpoint and an analytic viewpoint. In fact, we prove that a function belongs to P if and only if some geometrical inequality holds for a family of parabolas with vertexes on this fun…

On some integral inequalities for s-geometrically convex functions and their applications

In this paper, we establish three inequalities for differentiable s-geometrically and geometrically convex functions which are connected with the famous Hermite-Hadamard inequality holding for convex functions. Some applications to special means of pozitive real numbers are given.

On new inequalities of Hermite-Hadamard-Fejer type for convex functions via fractional integrals

In this paper, we establish some weighted fractional inequalities for differentiable mappings whose derivatives in absolute value are convex. These results are connected with the celebrated Hermite-Hadamard-Fejer type integral inequality. The results presented here would provide extensions of those given in earlier works.

On the growth of real functions and their derivatives

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $α>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot \log_q x \cdot f^{(k)}(x)}{1+|f(x)|^α}\le 0 $$ holds.

受控导数的可微函数的定量驯服性质

Quantitative tame properties of differentiable functions with controlled derivatives

本文证明定义在凸体上、导数受控于给定序列的可微函数具有多项式的定量性质，包括零点集大小、Sobolev 参数化及 Remez 型不等式，并由此导出子水平集与范数比较等推论。

Weakly differentiable functions on varifolds

The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. In this class the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincaré type embeddings, embeddings into spaces of continuous and sometimes Hölder continuous functions, pointwise differentiability results both of approximate a…

关于带交替符号的Weinberger不等式推广

On the Generalization of Weinberger's Inequality with Alternating Signs

本文给出了带交替符号的 Weinberger 不等式推广的严格数学证明，阐明了条件 f(0)≤0 的重要性，并证明了适用于奇数项和的 Szegö 不等式。

一维Pólya-Szegő原理的一个变体

A variation on the Pólya-Segő principle in one dimension

证明了Riesz $(p,\alpha)$-变差的Pólya-Szegő原理，该变差在有界$p$-变差和Sobolev空间$W^{1,p}$之间插值，结果对某些具有分数光滑性的不可微函数也成立。