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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 | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [AS] Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications Inc., New York, 1992; reprint of the 1972 edition. MR 1225604 (94b:00012)
  • [Alz03] Horst Alzer, Functional inequalities for the error function, Aequationes Math. 66 (2003), no. 1-2, 119-127. MR 2003459 (2004g:33006), https://doi.org/10.1007/s00010-003-2683-9
  • [Alz09] Horst Alzer, Functional inequalities for the error function. II, Aequationes Math. 78 (2009), no. 1-2, 113-121. MR 2552527 (2010i:33008), https://doi.org/10.1007/s00010-009-2963-0
  • [Alz10] Horst Alzer, Error function inequalities, Adv. Comput. Math. 33 (2010), no. 3, 349-379. MR 2718103 (2011h:33005), https://doi.org/10.1007/s10444-009-9139-2
  • [Bar08a] Árpád Baricz, A functional inequality for the survival function of the gamma distribution, JIPAM. J. Inequal. Pure Appl. Math. 9 (2008), no. 1, Article 13, 5. MR 2391280 (2009c:33006)
  • [Bar08b] Árpád Baricz, Mills' ratio: monotonicity patterns and functional inequalities, J. Math. Anal. Appl. 340 (2008), no. 2, 1362-1370. MR 2390935 (2009i:60031), https://doi.org/10.1016/j.jmaa.2007.09.063
  • [Bec64] E. F. Beckenbach, Superadditivity inequalities, Pacific J. Math. 14 (1964), 421-438. MR 0163996 (29 #1295)
  • [BB65] Edwin F. Beckenbach and Richard Bellman, Inequalities, Second revised printing. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Band 30, Springer-Verlag, New York, Inc., 1965. MR 0192009 (33 #236)
  • [Bru60] Andrew Bruckner, Minimal superadditive extensions of superadditive functions, Pacific J. Math. 10 (1960), 1155-1162. MR 0122943 (23 #A275)
  • [Bru62] A. M. Bruckner, Tests for the superadditivity of functions, Proc. Amer. Math. Soc. 13 (1962), 126-130. MR 0133411 (24 #A3245)
  • [Bru64] A. M. Bruckner, Some relationships between locally superadditive functions and convex functions, Proc. Amer. Math. Soc. 15 (1964), 61-65. MR 0156924 (28 #167)
  • [BO62] A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1962), 1203-1215. MR 0148822 (26 #6326)
  • [Chu55] John T. Chu, On bounds for the normal integral, Biometrika 42 (1955), 263-265. MR 0068148 (16,838f)
  • [HLP52] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge, at the University Press, 1952. MR 0046395 (13,727e)
  • [HP57] Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, rev. ed., American Mathematical Society, Providence, R.I., 1957. MR 0089373 (19,664d)
  • [Mit70] D. S. Mitrinović, Analytic inequalities, in cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York, 1970. MR 0274686 (43 #448)
  • [RS99] Stojan Radenović and Slavko Simić, A note on connection between $ P$-convex and subadditive functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), 59-62. MR 1710969
  • [Ros50] R. A. Rosenbaum, Sub-additive functions, Duke Math. J. 17 (1950), 227-247. MR 0036796 (12,164a)
  • [TWW89] S. Y. Trimble, Jim Wells, and F. T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal. 20 (1989), no. 5, 1255-1259. MR 1009357 (91a:26019), https://doi.org/10.1137/0520082

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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

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