×

zbMATH — the first resource for mathematics

Optimal lower and upper bounds for the geometric convex combination of the error function. (English) Zbl 1332.33003
Summary: For \(x\in\mathbb R\), the error function \(\operatorname{erf}(x)\) is defined as \[ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt. \] In this paper, we answer the question: what are the greatest value \(p\) and the least value \(q\), such that the double inequality \(\operatorname {erf}(M_{p}(x,y;\lambda))\leq G(\operatorname{erf}(x),\operatorname {erf}(y);\lambda)\leq\operatorname{erf}(M_{q}(x,y;\lambda))\) holds for all \(x,y\geq1\) (or \(0< x,y<1\)) and \(\lambda\in(0,1)\)? Here, \(M_{r}(x,y;\lambda)=(\lambda x^{r}+(1-\lambda)y^{r})^{1/r}\) (\(r\neq0\)), \(M_{0}(x,y;\lambda)=x^{\lambda}y^{1-\lambda}\) and \(G(x,y;\lambda )=x^{\lambda}y^{1-\lambda}\) are the weighted power and the weighted geometric mean, respectively.

MSC:
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
26D15 Inequalities for sums, series and integrals
Software:
DLMF; Equator
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abramowitz, M, Stegun, I (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965) · Zbl 0515.33001
[2] Oldham, K, Myland, J, Spanier, J: An Atlas of Functions: With Equator, the Atlas Function Calculator, 2nd edn. Springer, New York (2009) · Zbl 1167.65001
[3] Clendenin, W, Rational approximations for the error function and for similar functions, Commun. ACM, 4, 354-355, (1961) · Zbl 0102.05203
[4] Hart, RG, A close approximation related to the error function, Math. Comput., 20, 600-602, (1966) · Zbl 0196.48504
[5] Cody, WJ, Rational Chebyshev approximations for the error function, Math. Comput., 23, 631-637, (1969) · Zbl 0182.49403
[6] Matta, F; Reichel, A, Uniform computation of the error function and other related functions, Math. Comput., 25, 339-344, (1971) · Zbl 0218.33002
[7] Bajić, B, On the computation of the inverse of the error function by means of the power expansion, Bull. Math. Soc. Sci. Math. Roum., 17, 115-121, (1973) · Zbl 0299.33001
[8] Blair, JM; Edwards, CA; Johnson, JH, Rational Chebyshev approximations for the inverse of the error function, Math. Comput., 30, 7-68, (1976)
[9] Bhaduri, RK; Jennings, BK, Note on the error function, Am. J. Phys., 44, 590-592, (1976) · Zbl 0341.68063
[10] Zimmerman, IH, Extending menzel’s closed-form approximation for the error function, Am. J. Phys., 44, 592-593, (1976)
[11] Elbert, Á; Laforgia, A, An inequality for the product of two integrals relating to the incomplete gamma function, J. Inequal. Appl., 5, 39-51, (2000) · Zbl 0947.33002
[12] Gawronski, W; Müller, J; Reinhard, M, Reduced cancellation in the evaluation of entire functions and applications to the error function, SIAM J. Numer. Anal., 45, 2564-2576, (2007) · Zbl 1155.65317
[13] Baricz, Á, Mills’ ratio: monotonicity patterns and functional inequalities, J. Math. Anal. Appl., 340, 1362-1370, (2008) · Zbl 1138.60022
[14] Alzer, H, Functional inequalities for the error function. II, Aequ. Math., 78, 113-121, (2009) · Zbl 1208.33003
[15] Dominici, D, Some properties of the inverse error function, Contemp. Math., 457, 191-203, (2008) · Zbl 1173.33302
[16] Temme, NM, Error functions, dawson’s and fresnel integrals, 159-171, (2010), Washington
[17] Kharin, SN, A generalization of the error function and its application in heat conduction problems, No. 176, 51-59, (1981)
[18] Chaudhry, MA; Qadir, A; Zubair, SM, Generalized error functions with applications to probability and heat conduction, Int. J. Appl. Math., 9, 259-278, (2002) · Zbl 1034.33002
[19] Aumann, G: Konvexe Funktionen und die Induktion bei Ungleichungen zwischen Mittelwerten. Münchner Sitzungsber. 109, 403-415 (1933) · JFM 59.0962.05
[20] Anderson, GD; Vamanamurthy, MK; Vuorinen, M, Generalized convexity and inequalities, J. Math. Anal. Appl., 335, 1294-1308, (2007) · Zbl 1125.26017
[21] Gronau, D, Selected topics on functional equations, Dubrovnik, 1993, Aarhus · Zbl 0826.39005
[22] Gronau, D; Matkowski, J, Geometrical convexity and generalization of the Bohr-mollerup theorem on the gamma function, Math. Pannon., 4, 153-160, (1993) · Zbl 0791.33002
[23] Gronau, D; Matkowski, J, Geometrically convex solutions of certain difference equations and generalized Bohr-mollerup type theorems, Results Math., 26, 290-297, (1994) · Zbl 0876.39003
[24] Matkowski, J, \(\mathbf{L}^{p}\)-like paranorms, Proceedings of the Austrian-Polish Seminar, Graz, 1991, Graz · Zbl 0786.46031
[25] Niculescu, CP, Convexity according to the geometric Mean, Math. Inequal. Appl., 3, 155-167, (2000) · Zbl 0952.26006
[26] Alzer, H, Error function inequalities, Adv. Comput. Math., 33, 349-379, (2010) · Zbl 1211.33002
[27] Xia, W; Chu, Y, Optimal inequalities for the convex combination of error function, J. Math. Inequal., 9, 85-99, (2015) · Zbl 1314.33003
[28] Chu, Y; Li, Y; Xia, W; Zhang, X, Best possible inequalities for the harmonic Mean of error function, J. Inequal. Appl., 2014, (2014) · Zbl 1372.33003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.