On a local form of Jensen’s functional equation. (English) Zbl 0594.39006

Kostyrko, Pavel

Aequationes Math. 30, 65-69 (1986).

A function f:(a,b)\(\rightharpoonup R\) is said to be locally Jensen at \(x\in (a,b)\) if there exists \(\delta >0\) such that (1) \(1/2(f(x+h)+f(x- h))=f(x)\) holds for \(0<h<\delta\). f is locally Jensen at x for each \(x\in (a,b)\) and f is Jensen function if (1) holds for each \(x,x+h,x-h\in (a,b)\). A closed and countable set M is said to be an s-set if for every \(x\in M\) there exists \(\delta >0\) such that for each \(0<h<\delta\) \(x+h\in M\) iff x-h\(\in M\). The author proves the following result. A function f:(a,b)\(\to R\) is locally Jensen iff there exists a Jensen function g, an s-set M with the contiguous intervals \(\{J_ n\}\) and a sequence \(\{a_ n\}\) such that \(f_{| J_ n}=g_{| J_ n}+a_ n\) holds and f is locally Jensen at each \(x\in M\).

Reviewer: M.C.Zdun

MSC:

39B99	Functional equations and inequalities
26A51	Convexity of real functions in one variable, generalizations

Keywords:

locally Jensen functions; Jensen functional equation

References:

[1]	Davies, Roy O.,Symmetric sets are measurable. Real Anal. Exchange4 (1978–79), 87–89. · Zbl 0398.28001
[2]	Thomson, B. S.,On full covering properties. Real Anal. Exchange6 (1980–81), 77–93.