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A subadditive property of the error function


Authors: Horst Alzer and Man Kam Kwong
Journal: Proc. Amer. Math. Soc. 142 (2014), 2697-2704
MSC (2010): Primary 33B20, 26D15
DOI: https://doi.org/10.1090/S0002-9939-2014-11996-4
Published electronically: April 21, 2014
MathSciNet review: 3209325
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Abstract: We prove the following subadditive property of the error function:

$\displaystyle \mbox {erf}\,(x)=\frac {2}{\sqrt {\pi }}\int _0^x e^{-t^2}dt \quad {(x\in \mathbf {R})}. $

Let $ a$ and $ b$ be real numbers. The inequality

$\displaystyle \mbox {erf}\,\bigl ((x+y)^a\bigr )^b< \mbox {erf}\,(x^a)^b + \mbox {erf}\,(y^a)^b $

holds for all positive real numbers $ x$ and $ y$ if and only if $ ab\leq 1$.

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Additional Information

Horst Alzer
Affiliation: Morsbacher Str. 10, 51545 Waldbröl, Germany
Email: H.Alzer@gmx.de

Man Kam Kwong
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong
Email: mankwong@polyu.edu.hk

DOI: https://doi.org/10.1090/S0002-9939-2014-11996-4
Keywords: Error function, complementary error function, sub- and superadditive, inequalities, convex, concave
Received by editor(s): March 4, 2012
Received by editor(s) in revised form: August 22, 2012
Published electronically: April 21, 2014
Additional Notes: The research of the second author was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 5012/10P)
Communicated by: Walter Van Assche
Article copyright: © Copyright 2014 American Mathematical Society