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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 \(\varepsilon > 0\) and let \(f : I \to {\mathbb R}\) (the set of reals) be a differential function. Then \(| f' (x) - f(x) | \leq \varepsilon\) holds for all \(x\) in an interval \(I\) if and only if \(f\) can be represented in the form \(f(x) = \varepsilon + e^x \ell (e^{-x})\) where \(\ell\) is an arbitrary differentiable function defined on the interval \(J = \{ e^{-x}\mid x \in I\}\), nonincreasing and \(2\varepsilon\)-Lipschitz. They also prove that given an \(\varepsilon >0\), a nondecreasing Jensen convex function \(f: I \to {\mathbb R}\) satisfying \(f(x) \geq -\varepsilon\) for all \(x \in I\), is a solution of the inequality \({{f(y)-f(x)} \over {y-x}} - \varepsilon \leq f( {{x+y} \over 2})\) if and only if \(f(x) = d(x) e^x - \varepsilon\) where \(d: I \to {\mathbb R}^+\) is nonincreasing and \(I \owns x\mapsto d(x) e^x\) is Jensen concave.
Reviewer: P.Sahoo

39B72 Systems of functional equations and inequalities
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