# 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
##### Keywords:
error function; power mean; functional inequalities
DLMF; Equator
Full Text:
##### References:
  Abramowitz, M, Stegun, I (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1965) · Zbl 0515.33001  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  Clendenin, W, Rational approximations for the error function and for similar functions, Commun. ACM, 4, 354-355, (1961) · Zbl 0102.05203  Hart, RG, A close approximation related to the error function, Math. Comput., 20, 600-602, (1966) · Zbl 0196.48504  Cody, WJ, Rational Chebyshev approximations for the error function, Math. Comput., 23, 631-637, (1969) · Zbl 0182.49403  Matta, F; Reichel, A, Uniform computation of the error function and other related functions, Math. Comput., 25, 339-344, (1971) · Zbl 0218.33002  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  Blair, JM; Edwards, CA; Johnson, JH, Rational Chebyshev approximations for the inverse of the error function, Math. Comput., 30, 7-68, (1976)  Bhaduri, RK; Jennings, BK, Note on the error function, Am. J. Phys., 44, 590-592, (1976) · Zbl 0341.68063  Zimmerman, IH, Extending menzel’s closed-form approximation for the error function, Am. J. Phys., 44, 592-593, (1976)  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  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  Baricz, Á, Mills’ ratio: monotonicity patterns and functional inequalities, J. Math. Anal. Appl., 340, 1362-1370, (2008) · Zbl 1138.60022  Alzer, H, Functional inequalities for the error function. II, Aequ. Math., 78, 113-121, (2009) · Zbl 1208.33003  Dominici, D, Some properties of the inverse error function, Contemp. Math., 457, 191-203, (2008) · Zbl 1173.33302  Temme, NM, Error functions, dawson’s and fresnel integrals, 159-171, (2010), Washington  Kharin, SN, A generalization of the error function and its application in heat conduction problems, No. 176, 51-59, (1981)  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  Aumann, G: Konvexe Funktionen und die Induktion bei Ungleichungen zwischen Mittelwerten. Münchner Sitzungsber. 109, 403-415 (1933) · JFM 59.0962.05  Anderson, GD; Vamanamurthy, MK; Vuorinen, M, Generalized convexity and inequalities, J. Math. Anal. Appl., 335, 1294-1308, (2007) · Zbl 1125.26017  Gronau, D, Selected topics on functional equations, Dubrovnik, 1993, Aarhus · Zbl 0826.39005  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  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  Matkowski, J, $$\mathbf{L}^{p}$$-like paranorms, Proceedings of the Austrian-Polish Seminar, Graz, 1991, Graz · Zbl 0786.46031  Niculescu, CP, Convexity according to the geometric Mean, Math. Inequal. Appl., 3, 155-167, (2000) · Zbl 0952.26006  Alzer, H, Error function inequalities, Adv. Comput. Math., 33, 349-379, (2010) · Zbl 1211.33002  Xia, W; Chu, Y, Optimal inequalities for the convex combination of error function, J. Math. Inequal., 9, 85-99, (2015) · Zbl 1314.33003  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.