Characterization Of Convex Functions Research Articles

Characterizations of convex functions by level sets

It is shown that a lsc function is convex if and only if the minimal set of any linear perturbation of the function is convex. That fact also yields that the convexity of a function is equivalent to the quasiconvexity of its all linear perturbations.

Convex Inequalities and Functional Space with the Convex Semi-Norm

In this article, we establish new characterizations of convex functions, prove some connected convex type integral inequality; consider the pair of convex functions as the dual semi-norms in functional space. The properties of the integral operators are considered in the scales of the convex semi-norm under the standard conditions on singular kernels.

A Characterization of Convex Functions

Let f be a real-valued radially lower semicontinuous function defined on a convex subset D of a real vector space. It is shown that f is convex if and only if, for all there exists such that .

Properties and Characterizations of Convex Functions on Time Scales

Abstract In this research we deal with algebraic properties and characterizations of convex functions in the context of a time scale; this notion of convexity has been studied for some other authors but the setting of properties are establish here. Moreover, characterizations, a separation theorem and an inequality of Jensen type for this class of functions are shown as well.

Some properties of convex functions

We treat two problems on convex functions of one real variable. The first one is concerned with properties of tangent lines to the graph of a convex function and essentially is related to the questions on the first derivative (if it exists). The second problem is related to Schwarz?s derivative, in fact its upper limit modification. It gives an interesting characterization of convex functions. Let us recall the definition of a convex functions.

Generalizations of Sherman’s inequality

The concept of majorization is a powerful and useful tool which arises frequently in many different areas of research. Together with the concept of Schur-convexity it gives an important characterization of convex functions. The well known Majorization theorem plays a very important role in majorization theory—it gives a relation between one-dimensional convex functions and n-dimensional Schur-convex functions. A more general result was obtained by S. Sherman. In this paper, we get generalizations of these results for n-convex functions using Taylor’s interpolating polynomial and the Cebysev functional. We apply the exponentially convex method in order to interpret our results in the form of exponentially, and in the special case logarithmically convex functions. The outcome is some new classes of two-parameter Cauchy-type means.

Characterization of Lower Semicontinuous Convex Functions on Riemannian Manifolds

In this paper, an upper subderivative of a lower semicontinuous function on a Riemannian manifold is introduced. Then, an approximate mean value theorem for the upper subderivative on a Hadamard manifold is presented. Moreover, the results are used for characterization of convex functions on Riemannian manifolds.

Convex separation by regular pairs

It is known that the existence of a convex (resp., concave) separator between two given functions can be characterized via a simple inequality. The notion of convexity can be generalized applying regular pairs (in other words, two dimensional Chebyshev systems). The aim of the present note is to extend the above mentioned result to this setting. In the proof, a modified version of the classical Caratheodory’s theorem and the characterization of convex functions play the key role.

Characterizations of r-Convex Functions

This paper discusses some properties of r-convexity and its relations with some other types of convexity. A characterization of convex functions in terms of r-convexity is given without assuming differentiability. The concept of strict r-convexity is introduced. For a twice continuously differentiable function f, it is shown that the strict r-convexity of f is equivalent to a certain condition on ∇ 2 f. Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.

Some characterizations of convex functions

The main result in this paper is to establish some new characterizations of convex functions, in which we also simplify the proof of the characterizations given by Bessenyei and Páles.

Characterization of convex functions

There are many inequalities which in the class of continuous functions are equivalent to convexity (for example the Jensen inequality and the Hermite–Hadamard inequalities). We show that this is not a coincidence: every nontrivial linear inequality

Mollified Derivatives and Second-order Optimality Conditions

The class of strongly semicontinuous functions is considered. For these functions the notion of mollified derivatives, introduced by Ermoliev, Norkin and Wets [8], is extended to the second order. By means of a generalized Taylor's formula, second order necessary and sufficient conditions are proved for both unconstrained and constrained optimization. Finally a characterization of convex functions is given.

Almost everywhere convex functions on R n and weak derivatives

The classical subdifferential calculus is a useful tool in order to establish characterizations of convex functions and optimality conditions; but it becomes useless when one thinks to study almost everywhere convex functions. In this paper by using Sobolev space theory we give some characterizations of this class of functions.

A.e. Convex Functions on R nth power

The classical subdifferential calculus is a useful tool in order to estabilish characterizations of convex functions and optimality conditions; but it becomes usless when one thinks to study perturbed convex functions. In this paper by using Sobolev space theory we give some characterizations of this class of functions and we show optimality conditions based on the notion of the weak derivative.

Generalized second-order characterizations of convex functions

We introduce a second-order upper Dini-directional derivative and obtain a second-order characterization for a continuously Gâteaux differentiable function to be a convex function.

Sub differential Monotonicity as Characterization of Convex Functions

We prove that a lower semicontinuous function defined on a Banach space is convex if and only if its subdifferential ismonotone.

A characterization of convex functions

A characterization of convex functions

A note on an inequality for the gamma function

Some inequalities for the Wallis functions are proved. The results of this paper are consequences of some characterization of convex functions. A generalization of a result of Boyd (1) and an extentlon of an inequality of Gantschi (3) are obtained.