Higher Order Convex Research Articles

Convex solutions to polynomial-like iterative equations on open intervals

The paper deals with the polynomial-like iterative functional equation $$\lambda_1 f(x)+\lambda_2 f^2(x)+\cdots+\lambda_n f^n(x)=F(x).$$ By using Schauder’s fixed point theorem and a version of the uniform boundedness principle for families of convex (respectively higher order convex) functions as basic tools, the existence of nondecreasing convex (respectively higher order convex) solutions to this equation on open (possibly unbounded) intervals is investigated. The results of the paper complement similar ones established by other authors, concerning the existence of monotonic or convex solutions to the above equation on compact intervals. Some examples illustrating their applicability are provided.

On some extremalities in the approximate integration

Some extremalities for quadrature operators are proved for convex functions of higher order. Such results are known in the numerical analysis, however they are often proved under suitable differentiability assumptions. In our considerations we do not use any other assumptions apart from higher order convexity itself. The obtained inequalities refine the inequalities of Hadamard type. They are applied to give error bounds of quadrature operators under the assumptions weaker from the commonly used.

Higher order duality in multiobjective programming with cone constraints

We consider the nonlinear multiobjective programming problems involving cone constraints. For this program, we construct higher order dual problems and establish weak, strong and converse duality theorems for an efficient solution by using higher order generalized invexity conditions due to Zhang (Higher order convexity and duality in multiobjective programming, in “Progress in Optimization”, Contributions from Australasia, Applied Optimization, Vol. 30, eds. A. Eberhard, R. Hill, D. Ralph and B.M. Glover, Kluwer Academic Publishers, Dordrecht, 1999, 101–116). As special cases of our duality relations, we give some known duality results.

Decomposition of higher-order Wright-convex functions

In this paper we consider higher-order Wright-convex functions and prove that they are representable as the sum of a continuous higher-order convex function and a polynomial function.

On convex functions of higher order

Based on J. L. W. V. Jensen’s concept of convex functions as well on its generalization byE. M. Wright and related to T. Popoviciu’s convexity notions, higher-order convexity properties of real functions are introduced and surveyed. Mathematics subject classification (2000): 26A51, 26A48, 39B62.

Some properties of generalized higher-order convexity

Some properties of generalized higher-order convexity

Higher-Order Convex Approximations of Young Measures in Optimal Control

The general theory of approximation of (possibly generalized) Young measures is presented, and concrete cases are investigated. An adjoint-operator approach, combined with quasi-interpolation of test integrands, is systematically used. Applicability is demonstrated on an optimal control problem for an elliptic system, together with one-dimensional illustrative calculations of various options.

On Dinghas-Type Derivatives and Convex Functions of Higher Order

In this paper higher-order convexity properties of real functions are characterized in terms of a Dinghas-type derivative. The main tool used is a mean value inequality for Dinghas-type derivatives.

Gerber's inequality and some related inequalities

In this paper simple proofs of various extensions of Bernoulli's inequality and then higher order convexity is used to give more precise forms of these generalizations.

On a Korovkin-type theorem for simultaneous approximation

Several years ago, Knoop and Pottinger [9] (see also 1201) proved a quantitative Korovkin-type theorem for some positive linear operators acting on the Banach space C’(K) of real-valued and r-times continuously differentiable functions on a compact interval K of the real axis. Recently, the same authors proved [lo] an analogous quantitative theorem for functions defined on an unbounded interval of the real axis. The purpose of the present paper is to generalize their first non-quantitative assertion for compact intervals 16, Korollar 2.23 by replacing the so-called almostconvexity property by the condition of convexity-preserving of two types of higher-order convex functions and by adding another condition :in terms of CebySev norms and derivatives of order Y. These two types of higher-order convex functions are some spline functions and some monosplines. Using a result of Bojanic and Roulier [2] (for a history of this result see the next section), one may give a system of Y (types of) higher-order convex functions which may be used in Knoop-Pottinger’s result but only when the operators L,, are applicabe also to functions in c’~ '(K), say. However, if L,, , n 3 1, act only on c“ ‘(K), just two more simpler types of higher-order convex functions will be indicated. These generalizations are motivated by the fact that even classical instances of positive linear operators such as those of Meyer-Konig and Zeller have the property of simultaneous approximation but do not entirely satisfy the conditions of Knoop’s and Pottinger’s result (see [ 13, 91). The present paper is organized in the following manner. The next section contains the notations and the basic concepts used in the sequel. In the third section we review the previous results on test function theorems on simultaneous approximation while in Section 4 the main results are stated and proved. In the last section we show that the main results, are more general than the result of Knoop and Pottinger, among other remarks. 223 002 l-9045/90 $3.00

Smooth rate of weak convergence of convex type positive finite measures

Let μ be a positive finite measure of mass c 0 on [ a, b]⊂R. For a fixed x 0 ϵ [ a, b] and τ(X) = ¦x − x 0¦ , let the probability measure ϱ = c 0 −1 μ ○ τ −1. Assume that the corresponding to distribution function fulfills certain higher-order convexity conditions. By the use of convex moment methods, upper bounds for |∫ [a,b](ƒ(x)− Σ k=0 n ƒ (k)(x 0) k! (x−x 0) k)·μ(dx)| and ¦∝ [a, b]ƒ dμ − ƒ(x 0)¦, ƒ ϵ C n([a, b]), n ⩾ 1 are obtained involving a power moment of μ and the first modulus of continuity of ƒ (n) . These produce sharp inequalities that are attained. The established estimates improve the corresponding ones in the literature. Applications to probabilistic distributions are given at the end.

A geometric characterization ofnth order convex functions

A new geometric characterization is presented for a function convex of order n on an open interval, distinct from the whole of R. We shall prove that if /: (a, b)~^R with b < + oo, if a is an arbitrarily fixed number, a ^ 6, and if F(x) denotes the of the point of intersection in the x, y-plsme between the vertical line x = a and the osculating parabola of order n to the graph of / at the point (x, /(#)), then / is convex of order n on {a, b) iff F is increasing thereon. l Introduction* Let / be a real-valued convex function <yi the interval (α, 6) where a ^ o: it is stated in [1; p. I. 51, Exercise 7], and is indeed elementary to prove, that the function F(x) = f(x) — xfuix) is decreasing on (α, 6), fή denoting the right derivative of /. F(x) is none other than the ordinate at the origin of the right tangent line to the graph of / at the point (x, f(x)). Besides proving the converse of this proposition in this paper we shall extend the result to nth order convex functions, the role of the tangent line being played by the osculating parabola of order n. The result we present provides a meaningful geometric characterization of such a class of functions (Theorem 2.1 below). The notion of higher-order convexity is classical: its systematic study essentially began with a paper by Popoviciu [4] and was continued in many other works by the same author. The entire theory is surveyed in his monograph [5]. Other properties can be found in books [2; Chp. 4 §3 and Chp. 3 §2] and [3; Chp. XI] in the context of generalized convex functions. Many characterizations of nth order convex functions can be obtained from these references; in order to establish our main result we shall only need a few of such characterizations and shall state them here for the sake of convenience.