Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensen's inequality f(EX) ≤ Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensen's inequality becomes strict are studied. The relation between Jensen's inequality and Fatou's Lemma is examined.     

On Jensen’s inequality for g-expectation and for nonlinear expectation

In this paper, we give a necessary and sufficient condition for g under which Jensen’s inequality holds for g-expectation. In particular, we show that if Jensen’s inequality holds for g-expectation, then g is independent of y and g is superhomogeneous. We also establish a necessary and sufficient condition under which Jensen’s inequality holds for a general filtration-consistent nonlinear expectation. Received: 18 January 2005     

Improved version of Bohr's inequality

In this article, we prove several different improved versions of the classical Bohr's inequality. All the results are proved to be sharp.     

Group-invariant analogues of Hadamard's inequality

A wide class of inequalities for the determinant and other real-valued functions of an n × n complex Hermitian (or real symmetric) matrix H≡(h_jk) may be obtained by generalizing Marshall and Olkin's proof of Hadamard's inequality

$\text{det}\text{H?}\text{∏}\text{j=1}\text{n}\text{h}{}_{\mathrm{jj}}$

for positive definite (pd) H. We shall see that each subgroup G of the group U_n of n x n unitary matrices not only determines an analogue of (1) for det H, but also provides inequalities for a large family of unitarily invariant functions of H (not necessarily pd).     

Wendroff's inequality in n independent variables

The present note obtains some generalizations of Wendroff's integral inequality. These inequalities can be used in the study of the theory of partial differential and integral equations.     

Kato's inequality and the comparison of semigroups

Let A be the generator of a positivity preserving semigroup and let B be another semibounded self-adjoint operator. We give necessary and sufficient conditions in terms of the generators for the inequality $\text{¦ e}{}^{\mathrm{?tB}}\text{u ¦ ? e}{}^{\mathrm{?tA}}\text{¦ u ¦}$ to hold pointwise.     

Another approach to linearized elasticity and Korn's inequality

We describe and analyze an approach to the pure traction problem of three-dimensional linearized elasticity, whose novelty consists in considering the linearized strain tensor as the ‘primary’ unknown, instead of the displacement itself as is customary. This approach leads to a well-posed minimization problem, constrained by a weak form of the St Venant compatibility conditions. It also provides a new proof of Korn's inequality. To cite this article: P.G. Ciarlet, P. Ciarlet Jr., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Convexity of the bounds induced by Markov's inequality

X is a nonnegative random variable such that EX^t < ∞ for 0≤ t < λ ≤ ∞. The (l??) quantile of the distribution of X is bounded above by [?^?1 EX^t]^1?t. We show that there exist positive ?₁ ≥ ?₂ such that for all 0 <?≤?₁ the function g(t) = [?^-1EX^t]^1?t is log-convex in [0, c] and such that for all 0 < ? ≤ ?₂ the function log g(t) is nonincreasing in [0, c].     

Hoeffding’s inequality for supermartingales

We give an extension of Hoeffding’s inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.     

Chebyshev’s inequality for Choquet-like integral

The Chebyshev inequality for the Choquet-like integral is investigated. As an application, a Markov’s inequality for this type of integral is proven. Some previous results obtained by others are generalized.     

Sandor’s inequality for Sugeno integrals

In this paper we prove a fuzzy integral inequality for convex functions. Our results improve recent results that appear in literature. Some examples are given to illustrate our theorems.     

Von Neumann’s inequality for noncommuting contractions

Let T₁,…,T_d be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T₁,…,T_d) is bounded by

     

Young’s inequality and trace

Let φ be a positive linear functional on M_n(C) and f,g mutually conjugate in the sense of Young. In this note we show a necessary and sufficient condition for the inequality

     

Jensen’s inequality for operators without operator convexity

We give Jensen’s inequality for n-tuples of self-adjoint operators, unital n-tuples of positive linear mappings and real valued continuous convex functions with conditions on the bounds of the operators. We also study operator quasi-arithmetic means under the same conditions.     

An asymmetric Kadison’s inequality

Some inequalities for positive linear maps on matrix algebras are given, especially asymmetric extensions of Kadison’s inequality and several operator versions of Chebyshev’s inequality. We also discuss well-known results around the matrix geometric mean and connect it with complex interpolation.     

Kronecker's theorem and Lehmer's problem for polynomials in several variables

Mahler defined the measure of a polynomial in several variables to be the geometric mean of the modulus of the polynomial averaged over the torus. The classical theorem of Kronecker which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk can be regarded as characterizing those polynomials of one variable whose measure is exactly 1. Here this result is generalized to polynomials in several variables. The method employed also gives easy generalizations of recent results of Schinzel and Dobrowolski on Lehmer's problem.