In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for R n i j with 1 ≤ i < j < m n = m . We also discuss other limit theorems such as the central limit theorem, the law of iterated...

In this paper, we present new Poisson-type inequalities for Poisson integrals with continuous data on the boundary. The obtained inequalities are used to obtain growth properties at infinity of positive superharmonic functions in a smooth cone.

We study the log-convexity of the extended beta functions. As a consequence, we establish Turán-type inequalities. The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the inequalities on extended beta functions. The particular cases of those results also give the Turán-type inequalities for extended confluent and extended...

Convex risk minimization is a commonly used setting in learning theory. In this paper, we firstly give a perturbation analysis for such algorithms, and then we apply this result to differential private learning algorithms. Our analysis needs the objective functions to be strongly convex. This leads to an extension of our previous analysis to the non-differentiable loss functions...

In this paper, we study the Fourier series related to higher-order Bernoulli functions and give new identities for higher-order Bernoulli functions which are derived from the Fourier series of them. MSC: 11B68, 42A16.

We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set...

In this paper, we consider the compact differences of weighted composition operators on the standard weighted Bergman spaces. Some necessary and sufficient conditions for the differences of weighted composition operators to be compact are given, which extends Moorhouse’s results in (J. Funct. Anal. 219:70-92, 2005 ). MSC: 47B33, 30D55, 46E15.

Sharp remainder terms are explicitly given on the standard Hardy inequalities in L p ( R n ) with 1 < p < n . Those remainder terms provide a direct and exact understanding of Hardy type inequalities in the framework of equalities as well as of the nonexistence of nontrivial extremals. MSC: 26D10, 26D15, 46E35.

In this paper, we first introduce some new Morrey-type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators I α in these new Morrey-type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators...

In this study, the simplex whose vertices are barycenters of the given simplex facets plays an essential role. The article provides an extension of the Hermite-Hadamard inequality from the simplex barycenter to any point of the inscribed simplex except its vertices. A two-sided refinement of the generalized inequality is obtained in completion of this work. MSC: 26B25, 52A40.

We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi’s inequality. The obtained results are then applied for...

In many applications, the available data come from a sampling scheme that causes loss of information in terms of left truncation. In some cases, in addition to left truncation, the data are weakly dependent. In this paper we are interested in deriving the asymptotic normality as well as a Berry-Esseen type bound for the kernel density estimator of left truncated and weakly...

This paper formulates an infected predator-prey model with Beddington-DeAngelis functional response from a classical deterministic framework to a stochastic differential equation (SDE). First, we provide a global analysis including the global positive solution, stochastically ultimate boundedness, the persistence in mean, and extinction of the SDE system by using the technique of...

In this article, some new oscillation criterion for the second order Emden-Fowler functional differential equation of neutral type ( r ( t ) | z ′ ( t ) | α − 1 z ′ ( t ) ) ′ + q ( t ) | x ( σ ( t ) ) | β − 1 x ( σ ( t ) ) = 0 , where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) , α > 0 and β > 0 are established. Our results improve some well-known results which were published...

The uniqueness of the solution for the definite problem of a parabolic variational inequality is proved. The problem comes from the study of the optimal exercise strategies for the perpetual executive stock options with unrestricted exercise in financial market. Because the variational inequality is degenerate and the obstacle condition contains the partial derivative of an...

This paper is purported to investigate a food chain reaction-diffusion predator-prey system with nonlocal delays in a bounded domain with no flux boundary condition. We investigate the global stability and find the sufficient conditions of global stability of the unique positive equilibrium for this system. The derived results show that delays often restrain stability. Using the...

Conjugate gradient methods play an important role in many fields of application due to their simplicity, low memory requirements, and global convergence properties. In this paper, we propose an efficient three-term conjugate gradient method by utilizing the DFP update for the inverse Hessian approximation which satisfies both the sufficient descent and the conjugacy conditions...

We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out. MSC: 26A33, 26D10, 47G10.

In the article, we provide the necessary and sufficient conditions for the parameters α and β such that the generalized Wilker-type inequality 2 β α + 2 β ( sin x x ) α + α α + 2 β ( tan x x ) β − 1 > ( < ) 0 holds for all x ∈ ( 0 , π / 2 ) . MSC: 26D05, 33B10.

Based on the Padé approximation method, we present new inequalities for Gauss lemniscate functions. We also solve a conjecture on inequalities for Gauss lemniscate functions proposed by Sun and Chen. MSC: 26D07, 41A60.

We study the asymptotic properties of the solutions of a class of even-order damped differential equations with p-Laplacian like operators, delayed and advanced arguments. We present new theorems that improve and complement related contributions reported in the literature. Several examples are provided to illustrate the practicability, maneuverability, and efficiency of the...

In this paper, we give the notion of locally convex probabilistic seminormed spaces and discuss some property of locally convex probabilistic seminormed spaces. MSC: 4E70, 46S50.

In this paper we introduce q-Szász-Mirakjan-Kantorovich operators generated by a Dunkl generalization of the exponential function and we propose two different modifications of the q-Szász-Mirakjan-Kantorovich operators which preserve some test functions. We obtain some approximation results with the help of the well-known Korovkin theorem and the weighted Korovkin theorem for...

In this paper, we obtain an extended Halanay inequality with unbounded coefficient functions on time scales, which extends an earlier result in Wen et al. (J. Math. Anal. Appl. 347:169-178, 2008 ). Two illustrative examples are also given.

We consider the extremal problem of maximizing functions u in the class of real-valued biconvex functions satisfying a boundary condition ψ on a product of the unit ball with itself, with the ℓ p -norm. In 1986, Burkholder explicitly found the maximal function for p = 2 . In this paper, we find some characterizations of such extremal functions. We establish that sufficiently...