Alsina, Claudi; Ger, Roman On some inequalities and stability results related to the exponential function. (English) Zbl 0918.39009 J. Inequal. Appl. 2, No. 4, 373-380 (1998). 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 Cited in 2 ReviewsCited in 157 Documents MSC: 39B72 Systems of functional equations and inequalities Keywords:inequalities; exponential function; Hyers-Ulam stability; functional equation; Jensen convex function Citations:Zbl 0907.39025 PDFBibTeX XMLCite \textit{C. Alsina} and \textit{R. Ger}, J. Inequal. Appl. 2, No. 4, 373--380 (1998; Zbl 0918.39009) Full Text: DOI EuDML