Applications of de Rham theorem in approximate midconvexity

作者：Murenko Anna^*; Tabor Jacek; Tabor Jozef

来源：Journal of Difference Equations and Applications, 2012, 18(3): 335-344.

DOI：10.1080/10236190903156283

• approximately midconvex function
• Takagi function
• Jensen difference

Let X be a normed space, V be an open convex subset of X and let Theta : [0; infinity) -%26gt; [0, infinity) be a given function. A function f : V -%26gt; R is called Theta-midconvex if %26lt;br%26gt;f(x + y)/2) %26lt;= f(x) + f(y)/2 + Theta(parallel to x - y parallel to) for x, y is an element of V. %26lt;br%26gt;By the result of Tabor and Tabor, we know that under respective conditions on Theta, if f is Theta-midconvex and locally bounded above at a point then there exists a continuous function phi : [0, 1] -%26gt; R such that %26lt;br%26gt;f(tx + (1 - t)y) %26lt;= tf(x) + (1- t)f(y) + phi(t)Theta(parallel to x - y vertical bar parallel to) for x, y is an element of V, %26lt;br%26gt;t is an element of [0, 1]. %26lt;br%26gt;In this paper we determine the smallest function phi satisfying the above inequality. The required conditions on Q are such that the functions phi(t) = t(p), p is an element of [1, 2] satisfy them. As the main tool we use de Rham theorem.

• 出版日期2012