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.

Certain results of starlike and convex functions in some conditions

Thermal Science

The theory of geometric functions was first introduced by Bernard Riemann in 1851. In 1916, with the concept of normalized function revealed by Bieberbach, univalent function concept has found application area. Assume f(z)=z+??, n?(anzn) converges for all complex numbers z with |z|<1 and f(z)is one-to-one on the set of such z. Convex and starlike functions f(z) and g(z) are discussed with the help of subordination. The f(z) and g(z) are analytic in unit disc and f(0)=f'(0)=1, and g(0)=0, g'(0)-1=0. A single valued function f(z) is said to be univalent (or schlict or one-to-one) in domain D?C never gets the same value twice; that is, if f(z1)-f(z2)?0 for all z1 and z2 with z1 ? z2. Let A be the class of analytic functions in the unit disk U={z:|z|<1} that are normalized with f(0)=F'(0)=1. In this paper we give the some necessary conditions for f(z) ? S* [a, a2] and 0?a2?a?1 f'(z)(2r-1)[1-f'(z)]+zf?(z / 2r[f'(z)]2. This condition means that convexity and ...

Mazur's type problem for convexity of higher orders

Publicationes Mathematicae Debrecen, 2016

I. Labuda and R. D. Mauldin [4] have solved in affirmative the following S. Mazur’s problem posed about 1935 (see [6]): “In a space E of type (B), there is given an additive functional F (x) with the following property: if x(t) is a continuous function in 0 ≤ t ≤ 1 with values in E, then F (x(t)) is a measurable function. Is F (x) continuous?” In [1], we showed that the same remains true in the case where F is a Jensen-convex functional on an open and convex subdomain of a real Banach space. In the present paper, we study the possibilities of an extension of this result to convexity of higher orders. Some eighty years ago Stanis law Mazur asked the following question (see “Problem 24” from the famous “Scottish Book” [6]): “In a space E of type (B), there is given an additive functional F (x) with the following property: if x(t) is a continuous function in 0 ≤ t ≤ 1 with values in E, then F (x(t)) is a measurable function. Is F (x) continuous?” The solution, in the affirmative, was given by I. Labuda and R. D. Mauldin [4]. As a matter of fact, they have proved a more general theorem: instead of functionals, they considered additive operators from a Banach space into a Hausdorff topological vector space. Fairly soon afterwards, this result was generalized by Z. Lipecki [5] to the case where the domain and range of the additive transformations considered are suitable Abelian topological groups. Mathematics Subject Classification: 26B25, 39B62, 46A19.

Some results on abstract convexity of functions

Mathematica Slovaca, 2018

In this paper, we study abstract convexity topical and sub-topical functions. We obtain some results on abstract convexity such as support set and subdifferential in view of new elementary functions. Indeed, first we show that the ICR and IR functions are dense in IPH functions then the topical function and sub-topical function on topological vector space by another elementary function is studied. These elementary functions lead to obtain similar results with easier proofs.