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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

##### MSC:
 39B72 Systems of functional equations and inequalities
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