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An optimal double inequality between power-type Heron and Seiffert means. (English) Zbl 1210.26021
Purpose of this paper is to present the optimal upper and lower power-type Heron mean bounds for the Seiffert mean \(T(a,b)\).
For \(k\in[0;+\infty),\) the power-type Heron mean \(H_k(a,b)\) and the Seiffert mean \(T(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by:
\[ H_k(a,b)=\begin{cases} \bigl((a^k+(a\,b)^{k/2}+b^k)/3\bigr)^{1/k}, &k\neq 0,\\ \sqrt{a\,b}, &k=0, \end{cases} \] and
\[ T(a,b)=\begin{cases} (a-b)/2\arctan\bigl((a-b)/(a+b)\bigr), &a\neq b,\\ a,&a=b, \end{cases} \] respectively.
It is proved that for all \(a,b>0\), with \(a\neq b\), one has:
\[ H_{\log 3/ \log(\pi/2)}(a,b)<T(a,b)<H_{5/2}(a,b) \] and both \(H\)’s are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean \(T(a,b),\) respectively.

26D15 Inequalities for sums, series and integrals
26E60 Means
Full Text: DOI EuDML
[1] Wang, M-K; Chu, Y-M; Qiu, Y-F, Some comparison inequalities for generalized muirhead and identric means, No. 2010, 10, (2010) · Zbl 1187.26018
[2] Long, B-Y; Chu, Y-M, Optimal inequalities for generalized logarithmic, arithmetic, and geometric means, No. 2010, 10, (2010) · Zbl 1187.26015
[3] Long, B-Y; Chu, Y-M, Optimal power mean bounds for the weighted geometric Mean of classical means, No. 2010, 6, (2010) · Zbl 1187.26016
[4] Chu, Y-M; Long, B-Y, Best possible inequalities between generalized logarithmic Mean and classical means, No. 2010, 13, (2010) · Zbl 1185.26064
[5] Chu, Y-M; Qiu, Y-F; Wang, M-K; Wang, G-D, The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert’s Mean, No. 2010, 7, (2010) · Zbl 1209.26018
[6] Long, B; Xia, W; Chu, Y, An optimal inequality for power mean, geometric mean and harmonic Mean, International Journal of Modern Mathematics, 5, 149-155, (2010) · Zbl 1201.26009
[7] Chu, Y-M; Xia, W-f, Two optimal double inequalities between power mean and logarithmic Mean, Computers & Mathematics with Applications, 60, 83-89, (2010) · Zbl 1205.26041
[8] Zhang, X-M; Xi, B-Y; Chu, Y-M, A new method to prove and find analytic inequalities, No. 2010, 19, (2010) · Zbl 1185.26058
[9] Zhang, X-M; Chu, Y-M, A new method to study analytic inequalities, No. 2010, 13, (2010) · Zbl 1185.26057
[10] Chu, Y-M; Xia, W-f, Inequalities for generalized logarithmic means, No. 2009, 7, (2009) · Zbl 1187.26014
[11] Shi, M-y; Chu, Y-M; Jiang, Y-p, Optimal inequalities among various means of two arguments, No. 2009, 10, (2009) · Zbl 1187.26017
[12] Chu, Y-M; Xia, W-f, Two sharp inequalities for power mean, geometric mean, and harmonic Mean, 6, (2009) · Zbl 1187.26013
[13] Chu, Y; Xia, W, Solution of an open problem for Schur convexity or concavity of the gini Mean values, Science in China A, 52, 2099-2106, (2009) · Zbl 1179.26068
[14] Chu, Y; Zhang, X, Necessary and sufficient conditions such that extended Mean values are Schur-convex or Schur-concave, Journal of Mathematics of Kyoto University, 48, 229-238, (2008) · Zbl 1153.26307
[15] Chu, Y; Zhang, X; Wang, G, The Schur geometrical convexity of the extended Mean values, Journal of Convex Analysis, 15, 707-718, (2008) · Zbl 1163.26004
[16] Jia, G; Cao, J, A new upper bound of the logarithmic Mean, (2003) · Zbl 1048.26015
[17] Zhang, Z-H; Lokesha, V; Wu, Y-d, The new bounds of the logarithmic Mean, Advanced Studies in Contemporary Mathematics, 11, 185-191, (2005) · Zbl 1079.26021
[18] Zhang, Z-H; Wu, Y-d, The generalized Heron Mean and its dual form, Applied Mathematics E-Notes, 5, 16-23, (2005) · Zbl 1087.26021
[19] Seiffert, J, Aufgabe [inlineequation not available: see fulltext.]16, Die Wurzel, 29, 221-222, (1995)
[20] Hästö, PA, A monotonicity property of ratios of symmetric homogeneous means, (2002) · Zbl 1026.26025
[21] Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities. D. Reidel, , he Netherlands; 1988:xx+459. · Zbl 0687.26005
[22] Sándor, J, A note on some inequalities for means, Archiv der Mathematik, 56, 471-473, (1991) · Zbl 0693.26005
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