Abstract
We present various inequalities for the error function. One of our theorems states: Let α ≥ 1. For all x,y > 0 we have
with the best possible bounds
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Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965)
Alzer, H.: Functional inequalities for the error function. Aequationes Math. 66, 119–127 (2003)
Alzer, H.: Functional inequalities for the error function, II. Aequationes Math. (in print)
Alzer, H., Berg, C.: Some classes of completely monotonic functions. Ann. Acad. Sci. Fenn. 27, 445–460 (2002)
Alzer, H., Ruscheweyh, S.: A subadditive property of the gamma function. J. Math. Anal. Appl. 285, 564–577 (2003)
Baricz, Á.: Mills’ ratio: monotonicity patterns and functional inequalities. J. Math. Anal. Appl. 340, 1362–1370 (2008)
Baricz, Á.: A functional inequality for the survival function of the gamma distribution. J. Inequal. Pure Appl. Math. 9(1), Article 13, 5 pp. (2008)
Bochner, S.: Harmonic Analysis and the Theory of Probability. Univ. of California Press, Berkeley (1960)
Chu, J.T.: On bounds for the normal integral. Biometrika 42, 263–265 (1955)
Dhombres, J.G.: Solutions générale sur un groupe abélian de l’équation fonctionelle f(x*f(y)) = f(f(x)*y). Aequationes Math. 15, 173–193 (1977)
Dhombres, J.G.: Relations de dépendance entre les l’équations fonctionelles de Cauchy. Aequationes Math. 35, 186–212 (1988)
Erf, Wolfram Research: http://mathworld.wolfram.com/Erf.html (2009)
Frenzen, C.L.: Error bounds for asymptotic expansions of the ratio ot two gamma functions. SIAM J. Math. Anal. 18, 890–896 (1987)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 5th edn. In: Jeffrey, A. (ed.). Academic Press, Boston (1994)
Luke, Y.L.: Mathematical Functions and their Approximations. Academic Press, New York (1975)
Miller, K.S., Samko, S.G.: Completely monotonic functions. Integr. Transf. Spec. Funct. 12, 389–402 (2001)
Mitrinović, D.S.: Problem 5555. Amer. Math. Monthly 75, 84, 1129–1130 (1968)
Mitrinović, D.S.: Analytic Inequalities. Springer, New York (1970)
Niculescu, C.P.: Convexity according to the geometric mean. Math. Inequal. Appl. 3, 155–167 (2000)
Ross, D.K., Mahajan, A.: On enveloping series for some of the special functions, and on integral inequalities involving them. In: General Inequalities 2, pp. 161–175. Birkhäuser, Basel (1980)
Widder, D.V.: The Laplace Transform. Princeton Univ. Press, Princeton (1941)
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Communicated by Juan Manuel Pena.
To my father—for light in deep darkness.
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Alzer, H. Error function inequalities. Adv Comput Math 33, 349–379 (2010). https://doi.org/10.1007/s10444-009-9139-2
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DOI: https://doi.org/10.1007/s10444-009-9139-2
Keywords
- Error function
- Functional inequalities
- Differential inequalities
- Mitrinović-type inequalities
- Dhombres-type inequalities
- Hölder and Minkowski-type inequalities
- Mean value inequalities
- Enveloping series
- Convex
- Sub- and superadditive
- Completely monotonic