A subadditive property of the error function
HTML articles powered by AMS MathViewer
- by Horst Alzer and Man Kam Kwong PDF
- Proc. Amer. Math. Soc. 142 (2014), 2697-2704 Request permission
Abstract:
We prove the following subadditive property of the error function: \[ \mbox {erf} (x)=\frac {2}{\sqrt {\pi }}\int _0^x e^{-t^2}dt \quad {(x\in \mathbf {R})}. \] Let $a$ and $b$ be real numbers. The inequality \[ \mbox {erf} \bigl ((x+y)^a\bigr )^b< \mbox {erf} (x^a)^b + \mbox {erf} (y^a)^b \] holds for all positive real numbers $x$ and $y$ if and only if $ab\leq 1$.References
- 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
- Horst Alzer, Functional inequalities for the error function, Aequationes Math. 66 (2003), no. 1-2, 119–127. MR 2003459, DOI 10.1007/s00010-003-2683-9
- Horst Alzer, Functional inequalities for the error function. II, Aequationes Math. 78 (2009), no. 1-2, 113–121. MR 2552527, DOI 10.1007/s00010-009-2963-0
- Horst Alzer, Error function inequalities, Adv. Comput. Math. 33 (2010), no. 3, 349–379. MR 2718103, DOI 10.1007/s10444-009-9139-2
- 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
- Árpád Baricz, Mills’ ratio: monotonicity patterns and functional inequalities, J. Math. Anal. Appl. 340 (2008), no. 2, 1362–1370. MR 2390935, DOI 10.1016/j.jmaa.2007.09.063
- E. F. Beckenbach, Superadditivity inequalities, Pacific J. Math. 14 (1964), 421–438. MR 163996, DOI 10.2140/pjm.1964.14.421
- Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 30, Springer-Verlag, Inc., New York, 1965. Second revised printing. MR 0192009, DOI 10.1007/978-3-662-35199-4
- Andrew Bruckner, Minimal superadditive extensions of superadditive functions, Pacific J. Math. 10 (1960), 1155–1162. MR 122943, DOI 10.2140/pjm.1960.10.1155
- A. M. Bruckner, Tests for superadditivity of functions, Proc. Amer. Math. Soc. 13 (1962), 126–130. MR 133411, DOI 10.1090/S0002-9939-1962-0133411-9
- A. M. Bruckner, Some relationships between locally superadditive functions and convex functions, Proc. Amer. Math. Soc. 15 (1964), 61–65. MR 156924, DOI 10.1090/S0002-9939-1964-0156924-4
- A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1962), 1203–1215. MR 148822, DOI 10.2140/pjm.1962.12.1203
- John T. Chu, On bounds for the normal integral, Biometrika 42 (1955), 263–265. MR 68148, DOI 10.2307/2333443
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686, DOI 10.1007/978-3-642-99970-3
- 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
- R. A. Rosenbaum, Sub-additive functions, Duke Math. J. 17 (1950), 227–247. MR 36796, DOI 10.1215/S0012-7094-50-01721-2
- 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, DOI 10.1137/0520082
Additional Information
- Horst Alzer
- Affiliation: Morsbacher Str. 10, 51545 Waldbröl, Germany
- MR Author ID: 238846
- Email: H.Alzer@gmx.de
- Man Kam Kwong
- Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung- hom, Hong Kong
- MR Author ID: 108745
- ORCID: 0000-0003-0808-0925
- Email: mankwong@polyu.edu.hk
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3209325