Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0918.39009
Alsina, Claudi; Ger, Roman
On some inequalities and stability results related to the exponential function. (English)
[J] J. Inequal. Appl. 2, No.4, 373-380 (1998). ISSN 1029-242X/e

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.
[P.Sahoo]

MSC 2000:: *39B72 Functional inequalities involving unknown functions

Keywords: inequalities; exponential function; Hyers-Ulam stability; functional equation; Jensen convex function

Citations: Zbl 0907.39025

Cited in: Zbl 1118.39014 Zbl 1116.39019

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]