• Title, Summary, Keyword: Jensen integral inequality

Search Result 6, Processing Time 0.029 seconds

A note on Jensen type inequality for Choquet integrals

  • Jang, Lee-Chae
    • International Journal of Fuzzy Logic and Intelligent Systems
    • /
    • v.9 no.2
    • /
    • pp.71-75
    • /
    • 2009
  • The purpose of this paper is to prove a Jensen type inequality for Choquet integrals with respect to a non-additive measure which was introduced by Choquet [1] and Sugeno [20]; $$\Phi((C)\;{\int}\;fd{\mu})\;{\leq}\;(C)\;\int\;\Phi(f)d{\mu},$$ where f is Choquet integrable, ${\Phi}\;:\;[0,\;\infty)\;\rightarrow\;[0,\;\infty)$ is convex, $\Phi(\alpha)\;\leq\;\alpha$ for all $\alpha\;{\in}\;[0,\;{\infty})$ and ${\mu}_f(\alpha)\;{\leq}\;{\mu}_{\Phi(f)}(\alpha)$ for all ${\alpha}\;{\in}\;[0,\;{\infty})$. Furthermore, we give some examples assuring both satisfaction and dissatisfaction of Jensen type inequality for the Choquet integral.

TIME SCALES INTEGRAL INEQUALITIES FOR SUPERQUADRATIC FUNCTIONS

  • Baric, Josipa;Bibi, Rabia;Bohner, Martin;Pecaric, Josip
    • Journal of the Korean Mathematical Society
    • /
    • v.50 no.3
    • /
    • pp.465-477
    • /
    • 2013
  • In this paper, two different methods of proving Jensen's inequality on time scales for superquadratic functions are demonstrated. Some refinements of classical inequalities on time scales are obtained using properties of superquadratic functions and some known results for isotonic linear functionals.

Stability of Time-delayed Linear Systems with New Integral Inequality Proportional to Integration Interval (새로운 적분구간 비례 적분 부등식을 이용한 시간지연 선형시스템의 안정성)

  • Kim, Jin-Hoon
    • The Transactions of The Korean Institute of Electrical Engineers
    • /
    • v.65 no.3
    • /
    • pp.457-462
    • /
    • 2016
  • In this paper, we consider the stability of time-delayed linear systems. To derive an LMI form of result, the integral inequality is essential, and Jensen's integral inequality was the best in the last two decades until Seuret's integral inequality is appeared recently. However, these two are proportional to the inverse of integration interval, so another integral inequality is needed to make it in the form of LMI. In this paper, we derive an integral inequality which is proportional to the integration interval which can be easily converted into LMI form without any other inequality. Also, it is shown that Seuret's integral inequality is a special case of our result. Next, based on this new integral inequality, we derive a stability condition in the form of LMI. Finally, we show, by well-known two examples, that our result is less conservative than the recent results.

Stability on Time Delay Systems: A Survey (시간지연시스템의 안정성에 관한 연구동향)

  • Park, PooGyeon;Lee, Won Il;Lee, Seok Young
    • Journal of Institute of Control, Robotics and Systems
    • /
    • v.20 no.3
    • /
    • pp.289-297
    • /
    • 2014
  • This article surveys the control theoretic study on time delay systems. Since time delay systems are infinite dimensional, there are not analytic but numerical solutions on almost analysis and synthesis problems, which implies that there are a tremendous number of approximated solutions. To show how to find such solutions, several results are summarized in terms of two different axes: 1) theoretic tools like integral inequality associated with the derivative of delay terms, Jensen inequality, lower bound lemma for reciprocal convexity, and Wirtinger-based inequality and 2) various candidates for Laypunov-Krasovskii functionals.

APPROXIMATE CONVEXITY WITH RESPECT TO INTEGRAL ARITHMETIC MEAN

  • Zoldak, Marek
    • Bulletin of the Korean Mathematical Society
    • /
    • v.51 no.6
    • /
    • pp.1829-1839
    • /
    • 2014
  • Let (${\Omega}$, $\mathcal{S}$, ${\mu}$) be a probabilistic measure space, ${\varepsilon}{\in}\mathbb{R}$, ${\delta}{\geq}0$, p > 0 be given numbers and let $P{\subset}\mathbb{R}$ be an open interval. We consider a class of functions $f:P{\rightarrow}\mathbb{R}$, satisfying the inequality $$f(EX){\leq}E(f{\circ}X)+{\varepsilon}E({\mid}X-EX{\mid}^p)+{\delta}$$ for each $\mathcal{S}$-measurable simple function $X:{\Omega}{\rightarrow}P$. We show that if additionally the set of values of ${\mu}$ is equal to [0, 1] then $f:P{\rightarrow}\mathbb{R}$ satisfies the above condition if and only if $$f(tx+(1-t)y){\leq}tf(x)+(1-t)f(y)+{\varepsilon}[(1-t)^pt+t^p(1-t)]{\mid}x-y{\mid}^p+{\delta}$$ for $x,y{\in}P$, $t{\in}[0,1]$. We also prove some basic properties of such functions, e.g. the existence of subdifferentials, Hermite-Hadamard inequality.