Dynamic Picone's identity and its applications

We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.     

The Janson inequalities for general up‐sets

Janson and Janson, ?uczak and Ruciński proved several inequalities for the lower tail of the distribution of the number of events that hold, when all the events are up‐sets (increasing events) of a special form—each event is the intersection of some subset of a single set of independent events (i.e., a principal up‐set). We show that these inequalities in fact hold for arbitrary up‐sets, by modifying existing proofs to use only positive correlation, avoiding the need to assume positive correlation conditioned on one of the events. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 391–395, 2015     

On discrete and continuous wirtinger inequalities

We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.     

On an upper bound of Sanov's inequality for the probability of a large deviation

In this paper a sharp upper bound of Sanov's inequality for the large probability of large deviations under the multinomial distribution is obtained. Examples of computing the upper bounds for the probabilities of misclassification in discriminant analysis are also given.     

Radon不等式的新推广

利用Ho。lder不等式、Young不等式、Chebyshev不等式、幂平均不等式建立Radon不等式的指数推广形式,得到一个具有广泛应用价值的不等式.指出文[7]中给出的关于Radon不等式的推广结果是错误的,并在本文中作了修正.     

A new approach for the derivation of bounds for the Jensen difference

There are many useful applications of Jensen's inequality in several fields of science, and due to this reason, a lot of results are devoted to this inequality in the literature. The main theme of this article is to present a new method of finding estimates of the Jensen difference for differentiable functions. By applying definition of convex function, and integral Jensen's inequality for concave function in the identity pertaining the Jensen difference, we derive bounds for the Jensen difference. We present integral version of the bounds in Riemann sense as well. The sharpness of the proposed bounds through examples are discussed, and we conclude that the proposed bounds are better than some existing bounds even with weaker conditions. Also, we present some new variants of the Hermite–Hadamard and Hölder inequalities and some new inequalities for geometric, quasi-arithmetic, and power means. Finally, we give some applications in information theory.     

Sobolev's inequality for Riesz potentials of functions in Morrey spaces of integral form

Morrey spaces have become a good tool for the study of existence and regularity of solutions of partial differential equations. Our aim in this paper is to give Sobolev's inequality for Riesz potentials of functions in Morrey spaces (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lyapunov type inequalities for a fractional two‐point boundary value problem

In this paper, new Lyapunov‐type inequalities are obtained for the case when one is dealing with a class of fractional two‐point boundary value problems. As an application of this result, we obtain a lower bound for the eigenvalues of corresponding equations. Copyright © 2016 John Wiley & Sons, Ltd.     

On the converse Jensen inequality

We give a survey on the converse Jensen inequality and we show that several recently published inequalities are simple consequences of certain long time known results. We also give a new refinement of the converse Jensen inequality as well as improvements of some related results.     

Generalization of Bernoulli polynomials

The Bernoulli polynomials are generalized and some properties of the resulting generalizations are presented.     

On Pólya–Szegö reverse of Schwarz's inequality

We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.     

An Improved Hardy's Inequality

In this paper we shall extend Hardy's inequality associated with Fourier transform to the strip n(2-p) ≤ σ < n+p(N +1) where N = [n(1/p-1)], the greatest integer not exceeding n(1/p−1).     

An AZ-style identity and Bollobás deficiency

The powerful AZ identity is a sharpening of the famous LYM-inequality. More generally, Ahlswede and Zhang discovered a generalization in which the Bollobás inequality for two set families can be lifted to an identity.In this paper, we show another generalization of the AZ identity. The new identity implies an identity which characterizes the deficiency of the Bollobás inequality for an intersecting Sperner family. We also give some consequences relating to Helly families and LYM-style inequalities.     

Hadamard-type fractional differential equations for the system of integral inequalities on time scales

ABSTRACT

This paper investigates some system of integral inequalities of one independent variable on time scales. The conclusion can be obtained by using Hadamard-type fractional differential equations and Greene's method which bring together and expand some integral inequalities on time scales. The established inequalities give explicit bounds on unknown functions which can be utilized as a key in examining the properties of certain classes of partial dynamic equations and difference equations on time scales. As an application, a system of fractional differential equations is considered to explain the value of our results.     

On Switching to H‐Free Graphs

In this article, we study the problem of deciding if, for a fixed graph H, a given graph is switching equivalent to an H‐free graph. Polynomial‐time algorithms are known for H having at most three vertices or isomorphic to P₄. We show that for H isomorphic to a claw, the problem is polynomial, too. On the other hand, we give infinitely many graphs H such that the problem is NP‐complete, thus solving an open problem [Kratochvíl, Ne?et?il and Zýka, Ann Discrete Math 51 (1992)]. Further, we give a characterization of graphs switching equivalent to a K_{1, 2}‐free graph by ten forbidden‐induced subgraphs, each having five vertices. We also give the forbidden‐induced subgraphs for graphs switching equivalent to a forest of bounded vertex degrees.     

Stochastic flows of diffeomorphisms on an open set in rn

In this paper, we will give sufficient conditions for the solution to a stochastic differential equation (SDE) on an open set D in R" to define a stochastic flow of diffeomorphisms of D onto itself. Since a necessary and sufficient condition for the solution to determine a stochastic flow of diffeomorphisms is that the original SDE and its adjoint SDE are both strictly conservative, we will concentrate our attention on finding sufficient conditions for the SDE to be strictly conservative. It will be etablished that the strict conservativeness follows if the vector fields governing the SDE decay suitably near the boundary dD in the direction transversal to 3D and some additional assumptions are satisfied.     

Carleson measures and conformal self‐mappings in the real unit ball

Bounded and compact Carleson measures in the unit ball B of R ⁿ , n ≥ 2, are characterized by means of global Dirichlet integrals of the conformal self‐map T_a taking a ∈ B to the origin. The same proof applies in the unit ball of C ⁿ . It is also proved that the powers of the Jacobian of T_a satisfy the weak Harnack inequality and even Harnack's inequality with a constant independent of a. As an application of these results it is shown that the two different definitions for Carleson measures in the existing literature are equivalent for a certain range of parameter values. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improved Cotlar's inequality in the context of local Tb theorems

In the context of local Tb theorems with $L^p$ testing conditions we prove an enhanced Cotlar's inequality. This is related to the problem of removing the so called buffer assumption of Hytönen–Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen–Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures μ in ${\mathbb{R}}^d$ satisfying $\mu (B(x,r))\le C{r}^n$, $n\in (0,d]$. The range of exponents in the Cotlar type inequality depend on n. Without assuming buffer we get the full range of exponents $p,q\in (1,2]$ for measures with $n\le 1$, and in general we get $p,q\in [2??(n),2]$, $?(n)>0$. Consequences for (non-homogeneous) local Tb theorems are discussed.     

Bohr's inequality for holomorphic and pluriharmonic mappings with values in complex Hilbert spaces

Let ${\mathbb{B}}_H$ be the unit ball of a complex Hilbert space H. First, we give a Bohr's inequality for the holomorphic mappings with lacunary series with values in complex Hilbert balls. Next, we give several results on Bohr's inequality for pluriharmonic mappings with values in ℓ₂. Note that the Bohr phenomenons that we have obtained are completely different from those in the case with values in $\mathbb{C}$ and are sharp in the case with values in ℓ₂.     

Opial's and wirtinger's type discrete inequalities in two independent variables

We develop discrete Taylor's formula in two independent variables and use it to establish Opial's type inequality involving higher order partial differences. This is followed by several variations of discrete Wirtinger's inequality in two independent variables