Moreover we know exactly when the equality holds: if and only if D is a disc. For any optimal geometric inequality it is important to have a complete understanding of the equality case. Sometimes this can be easily achieved by checking the proof of the inequality. Take as an example the following elegant theorem due to Lichenerowicz [L]: let (M; g) be a compact Riemannian manifold of dimension ...

Let φ be an arbitrary bijection of R+. We prove that if the two-place function φ−1[φ(s) + φ(t)] is subadditive in R+ then φ must be a convex homeomorphism of R+. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of R+ are also given. We apply the above results to obtain several converses of Minkowski’s inequality. Introduction. Throughout this ...

W ( ; ) := inf 2 ( ; ) sZZ 1 2 d(x; y)2 d (x; y); where d(x; y) = kx yk2 and ( ; ) denotes the set of all probability measures on R R with marginals and , i.e., ( R) = and (R ) = . The transportation cost inequality (TCI) obtained by M. Talagrand [15] is W ( ; ) p S( ; ) ; where is the standard Gaussian measure and is any probability measure on R. Recently Talagrand's inequality and its counter...

This paper focuses on the problem of finite-time boundedness and finite-time passivity of discrete-time T-S fuzzy neural networks with time-varying delays. A suitable Lyapunov--Krasovskii functional(LKF) is established to derive sufficient condition for finite-time passivity of discrete-time T-S fuzzy neural networks. The dynamical system is transformed into a T-S fuzzy model with uncertain par...

The superadditivity and monotonicity properties of some functionals associated with convex functions and the Hermite-Hadamard inequality in the general setting of linear spaces are investigated. Applications for norms and convex functions of a real variable are given. Some inequalities for arithmetic, geometric, harmonic, logarithmic and identric means are improved. 1. Introduction For any conv...

We will demonstrate in this paper that Bell′s theorem (Bell′s inequality) does not really conflict with quantum mechanics, the controversy between them originates from the different definitions for the expectation value using the probability distribution in Bell′s inequality and the expectation value in quantum mechanics. We can not use quantum mechanical expectation value measured in experimen...

Refining some results of S. S. Dragomir, several new reverses of the triangle inequality in inner product spaces are obtained.

The classical Hermite-Hadamard inequality characterizes the continuous convex functions of one real variable. The aim of the present paper is to give an analogous characterization for functions of a vector variable. 1. The Hermite-Hadamard inequality In a letter sent on November 22, 1881, to the journal Mathesis (and published there two years later), Ch. Hermite [10] noted that every convex fun...

When a quantum system is divided into subsystems, their entanglement entropies are subject to an inequality known as strong subadditivity. For a field theory this inequality can be stated as follows: given any two regions of space A and B, S(A)+S(B) ≥ S(A∪B)+ S(A ∩ B). Recently, a method has been found for computing entanglement entropies in any field theory for which there is a holographically...

Refining some results of Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if a is a unit vector in a real or complex inner product space (H ;〈·,·〉), r,s > 0, p ∈ (0,s], D = {x ∈ H ,‖rx− sa‖ ≤ p}, x1,x2 ∈D−{0}, and αr,s = min{(r2‖x...