We develop in this paper an amelioration of the method given by S. Bobkov and M. Ledoux in [BL00]. We prove by Prékopa-Leindler Theorem an optimal modified logarithmic Sobolev inequality adapted for all log-concave measure on Rn. This inequality implies results proved by Bobkov and Ledoux, the Euclidean Logarithmic Sobolev inequality generalized in the last years and it also implies some convex...

the paper is concerned with robust stability criteria for takagi- sugeno (t-s) fuzzy systems with distributed delays and time delay in the leakage term. by exploiting a model transformation, the system is converted to one of the neutral delay system. global robust stability result is proposed by a new lyapunov-krasovskii functional which takes into account the range of delay and by making use o...

Our aim in this article is to incorporate the notion of "strongly s-convex function" and prove a new integral identity. Some new inequalities of Simpson type for strongly s-convex function utilizing integral identity and Holder's inequality are considered.

Let (sn) be an s-number sequence. We show for each k = 1, 2, . . . and n ≥ k + 1 the inequality

We prove the local Hölder continuity of strong local minimizers of the stored energy functional E(u) = ∫ Ω λ|∇u|2 + h(det∇u) dx subject to a condition of ‘positive twist’. The latter turns out to be equivalent to requiring that u maps circles to suitably star-shaped sets. The convex function h(s) grows logarithmically as s → 0+, linearly as s → +∞, and satisfies h(s) = +∞ if s ≤ 0. These proper...

Bernstein’s inequality for Jacobi polynomials P (α,β) n , established in 1987 by P. Baratella for the region R1/2 = {|α| ≤ 1/2, |β| ≤ 1/2}, and subsequently supplied with an improved constant by Y. Chow, L. Gatteschi, and R. Wong, is analyzed here analytically and, above all, computationally with regard to validity and sharpness, not only in the original region R1/2, but also in larger regions ...

In this paper the extension of Hadamard's type inequality for s– convex function and s–convex functions on the coordinates defined in 2-variables and some applications are given.

We consider variants of van der Corput’s lemma in higher dimensions. 1. The very well-known and extremely useful van der Corput lemma is the following: Van der Corput’s lemma. Let I ⊆ R be an interval and suppose φ : I → R satisfies φ(k) ≥ 1 on I (where k ∈ N). Then, for λ ∈ R, ∣∣∣∣∣ ∫ I e ∣∣∣∣∣ ≤ Ck|λ| 1 k , provided, in addition when k = 1, that φ′ is monotonic on I. An extensive discussion o...

Maybe the most beautiful connection between a discrete object (graph) and a continuous object (eigenvalue) comes from a concept called the Laplacian. It gives a surprisingly simple and algebraic method for splitting a graph into two pieces without cutting too many edges. In a sense, we take an NP-Complete problem of partitioning a graph and relax it to an easy to solve problem of finding an eig...

The aim of this paper is to give effective inequalities in the form of K X ≥?pg(X) −?, where X is a minimal projective 3-fold of general type with at worst Q-factorial terminal singularities. Our method is quite effective in the sense that we have found the optimal form of Noether’s inequality: K X ≥ 4 3 pg(X) − a with a ∈ [ 10 3 , 14 3 ] among smooth canonical models. An interesting applicatio...