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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 {\it D. H. Hyers, G. Isac} and {\it 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 $f : I \to {\Bbb R}$ (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 $f: I \to {\Bbb R}$ 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 $d: I \to {\Bbb R}^+$ is nonincreasing and $I \owns x\mapsto d(x) e^x$ is Jensen concave.
Reviewer: P.Sahoo

39B72Systems of functional equations and inequalities
Full Text: DOI EuDML