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On some inequalities and stability results related to the exponential function. (English) Zbl 0918.39009
The authors examine the Hyers-Ulam stability [see D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhäuser, Boston (1998; Zbl 0907.39025)] of the differential equation f ' =f and prove the following result: Given an ε>0 and let f:I (the set of reals) be a differential function. Then |f ' (x)-f(x)|ε holds for all x in an interval I if and only if f can be represented in the form f(x)=ε+e x (e -x ) where is an arbitrary differentiable function defined on the interval J={e -x xI}, nonincreasing and 2ε-Lipschitz. They also prove that given an ε>0, a nondecreasing Jensen convex function f:I satisfying f(x)-ε for all xI, is a solution of the inequality f(y)-f(x) y-x-εf(x+y 2) if and only if f(x)=d(x)e x -ε where d:I + is nonincreasing and Ixd(x)e x is Jensen concave.
Reviewer: P.Sahoo

MSC:
39B72Systems of functional equations and inequalities