A Functional Inequality (original) (raw)

Dominated extensions of functionals and V-convex functions of cancellative cones

Bulletin of the Australian Mathematical Society, 2003

Let C be a cancellative cone and consider a subcone Co of C. We study the natural problem of obtaining conditions on a non negative homogeneous function : C-> R + so that for each linear functional / defined in Co which is bounded by , there exists a linear extension to C In order to do this we assume several geometric conditions for cones related to the existence of special algebraic basis of the linear span of these cones.

On Functional Inequalities Associated with

2016

In the paper, the equivalence of the functional inequality 2f (x) + f (y) + f (−y) − f (x − y) ≤ f (x + y) (x, y ∈ G) and the Drygas functional equation f (x + y) + f (x − y) = 2f (x) + f (y) + f (−y) (x, y ∈ G) is proved for functions f : G → E where (G, +) is an abelian group, (E, < •, • >) is an inner product space, and the norm is derived from the inner product in the usual way.

Quasilinearity of Some Composite Functionals with Applications

Bulletin of the Australian Mathematical Society, 2010

The quasilinearity of certain composite functionals defined on convex cones in linear spaces is investigated. Applications in refining the Jensen, Hölder, Minkowski and Schwarz inequalities are given.

Convex decompositions and the valence of some functions

Journal of Nonlinear and Variational Analysis, 2020

The aim of this paper is twofold. On one hand, we provide examples of R n-valued functions on some open subsets D of R n whose restrictions to the convex subsets of D are all injective. Such applications are shortly called CIP functions. On the other hand, we provide alternative descriptions of the maximal convex subsets of the convex open sets with compact convex subsets removed. The maximal convex subsets of R n with convex sets removed were characterized before by Martínez-Legaz and Singer [Compatible Preorders and Linear Operators on R n , Linear Algebra Appl. 153 (1991), 53-66] as being the convex subsets of R n , shortly called hemispaces, whose complements are convex too. The two topics merge together as the smallest number k of convex subsets, of the considered open set, needed to cover it, is an upper bound for the valence of every CIP function on that open set.

Form convex functions and the wydl and other inequalities

Letters in Mathematical Physics, 1978

We show that the functional calculus of sesquilinear forms derived in [3] leads to a general theory of the Wigner, Yanase, Dyson-Lieb-type inequalities. In particular, we obtain the joint convexity and the monotonicity of the relative entropy functional. 1. THE GENERAL THEORY Let 91 be a complex vector space. A real function a defined on 91 is called a quadratic form if it is homogeneous of degree 2 (i.e. a(M) = IX 12 a(a) for all a E 91 and X E r and ,,atisfies the parallelogram equality: a(a + b) + a(a-b) = 2a(a) + 2or(b) a, b E 9.I. (1) The set of all quadratic forms on 9~ will be denoted by o~-(91). A quadratic form is called positive if it takes a non-negative value. The corresponding set will be denoted by o~' § Let 911 be another complex vector space and 7r: 91.I 1 ~ 91 be a linear map. Then, for any E ~'(9~), the inverse image 7r*a defined by n*a(a) = a(n(a)), a C 911, is an element of ~(9 1 1): rr*a E ~' (911). Moreover, ~r*a is positive if a is positive. The well-known polarization formula gives us a one-to-one correspondence between ~'(gff) and the set of all Hermitian sesquillnear forms on 91 (cf. [2]). Therefore, the functional calculus of sesquilinear forms derived in [3] can be applied to quadratic forms via the correspondence mentioned above. Let us remind ourselves of the basic definitions. Let a,/3 E ~'+(91) a n d f b e a homogeneous measurable, locally bounded function on IR 2 = fir, s) E IR 2" r ~ 0 and s i> 0). We recall that the function is homogeneous if f(Xr, Xs) = ~f(r, s), A, r, s >~ O. The function is locally bounded if it is bounded on each compact set. The measurability is understood with respect to the o-algebra of Borel subsets of l~2+. To def'me f(a,/3), we consider any tetrad (H, h, A, B) where H i s a Hilbert space, h is a linear map of 9.I onto a dense subset of H, A and B are commuting positive bounded operators acting on H such that