A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Here a compound set means a set obtained by varying each argument of a function of several variables over a set associated with that argument, wh...

In this paper, a class of hybrid impulsive systems with time delays and stochastic effects are considered. We obtain some criteria on the global exponential stability in mean square for the impulsive stochastic delayed systems. To do this, differential inequalities and L-operator inequalities are developed. An example is given to illustrate the effectiveness of our results.

The purpose of this paper is to investigate some nonlinear integral inequalities and their discrete analogues. The inequalities given here can be used as handy tools in the qualitative theory of certain differential equations, integral equations and difference equations. Mathematics subject classification (2010): 26D10, 26D15.

*Correspondence: [email protected] 1School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, P.R. China Full list of author information is available at the end of the article Abstract In this paper, some new generalized retarded inequalities for discontinuous functions are discussed, which are effective in dealing with the qualitative theory of some impulsive differential ...

Keywords: Analytic functions Starlike and convex functions Multivalent functions Hadamard product (or convolution) Coefficient bounds Distortion inequalities Neighborhood properties Non-homogeneous Cauchy–Euler differential equations a b s t r a c t In this paper, by making use of the familiar concept of neighborhoods of p-valently analytic functions, we prove coefficient bounds, distortion ine...

Differential and integral inequalities play a fundamental role in global existence,uniqueness, stability, and other properties of the solutions of various nonlinear differential equations; see 1–4 . A great deal of attention has been given to differential and integral inequalities; see 1, 2, 5–8 and the references given therein. Motivated by the results in 1, 5, 7 , the main purpose of this pap...

The attracting set of dynamical systems have been extensively studied over the past few decades and various results are reported. It is well known that one of the most researching tools is inequality technique. With the establishment of various differential and integral inequalities 1–12 , the sufficient conditions on the attracting sets for different differential systems are obtained 12–16 . H...

In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the coarsest level, these are often mysterious-looking inequalities that hold for ‘positive’ solutions of some parabolic PDE, and can be verified quickly by grinding ou...

In this paper, we generalize some integral inequalities to more general situations. These on the one hand generalize and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of integral equations and differential equations. Applications are given to illustrate the usefulness of the inequalities.

We prove various pointwise estimates for solutions of linear elliptic difference inequalities with random coefficients. These estimates include discrete versions of the maximum principle of Aleksandrov and Harnack inequalities and Holder estimates of Krylov and Safonov for elliptic differential operators with bounded coefficients.