×

zbMATH — the first resource for mathematics

The optimal convex combination bounds for Seiffert’s mean. (English) Zbl 1221.26037
The authors prove the following optimal bounds for the Seiffert mean \(P(a,b)=(a-b)/[2\arcsin ((a-b)/(a+b))]\) by convex combinations of contraharmonic mean \(C(a,b)=(a^{2}+b^{2})/(a+b)\) and geometric mean \(G(a,b)= \sqrt{ab}\), respectively, harmonic mean \(H(a,b)=2ab/(a+b)\).
1) The double inequality \(\alpha _{1}C(a,b)+(1-\alpha _{1})G(a,b)<P(a,b)<\beta _{1}C(a,b)+(1-\beta _{1})G(a,b)\) holds for all \( a,b>0\) with \(a\neq b\) if and only if \(\alpha _{1}\leq 2/9\) and \(\beta _{1}\geq 1/\pi\).
2) The double inequality \(\alpha _{2}C(a,b)+(1-\alpha _{2})H(a,b)<P(a,b)<\beta _{2}C(a,b)+(1-\beta _{2})H(a,b)\) holds for all \( a,b>0\) with \(a\neq b\) if and only if \(\alpha _{2}\leq 1/\pi \) and \(\beta _{2}\geq 5/12\).

MSC:
26E60 Means
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Seiffert, H-J, Problem 887, Nieuw Archief voor Wiskunde, 11, 176, (1993)
[2] Seiffert, H-J, Aufgabe [inlineequation not available: see fulltext.] 16, Die Wurzel, 29, 221-222, (1995)
[3] Hästö, PA, Optimal inequalities between Seiffert’s Mean and power means, Mathematical Inequalities & Applications, 7, 47-53, (2004) · Zbl 1049.26006
[4] Neuman E, Sándor J: On certain means of two arguments and their extensions.International Journal of Mathematics and Mathematical Sciences 2003, (16):981-993. · Zbl 1040.26015
[5] Neuman, E; Sándor, J, On the Schwab-Borchardt Mean, Mathematica Pannonica, 14, 253-266, (2003) · Zbl 1053.26015
[6] Hästö, PA, A monotonicity property of ratios of symmetric homogeneous means, Journal of Inequalities in Pure and Applied Mathematics, 3, 1-54, (2002)
[7] Seiffert, H-J, Ungleichungen für einen bestimmten mittelwert, Nieuw Archief voor Wiskunde, 13, 195-198, (1995) · Zbl 0830.26008
[8] 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, 7, (2010) · Zbl 1209.26018
[9] 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
[10] Wang, M-K; Qiu, Y-F; Chu, Y-M, Sharp bounds for Seiffert means in terms of Lehmer means, Journal of Mathematical Inequalities, 4, 581-586, (2010) · Zbl 1204.26053
[11] Wang, S; Chu, Y, The best bounds of the combination of arithmetic and harmonic means for the Seiffert’s Mean, International Journal of Mathematical Analysis, 4, 1079-1084, (2010) · Zbl 1207.26033
[12] Zong, C; Chu, Y, An inequality among identric, geometric and Seiffert’s means, International Mathematical Forum, 5, 1297-1302, (2010) · Zbl 1206.26034
[13] Long, B-Y; Chu, Y-M, Optimal inequalities for generalized logarithmic, arithmetic, and geometric means, No. 2010, 10, (2010) · Zbl 1187.26015
[14] 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
[15] Xia, W-F; Chu, Y-M; Wang, G-D, The optimal upper and lower power Mean bounds for a convex combination of the arithmetic and logarithmic means, No. 2010, 9, (2010) · Zbl 1190.26038
[16] Chu, Y-M; Long, B-Y, Best possible inequalities between generalized logarithmic Mean and classical means, No. 2010, 13, (2010) · Zbl 1185.26064
[17] Shi, M-Y; Chu, Y-M; Jiang, Y-P, Optimal inequalities among various means of two arguments, No. 2009, 10, (2009) · Zbl 1187.26017
[18] Chu, Y-M; Xia, W-F, Two sharp inequalities for power mean, geometric mean, and harmonic Mean, No. 2009, 6, (2009) · Zbl 1187.26013
[19] Chu, Y-M; Xia, W-F, Inequalities for generalized logarithmic means, No. 2009, 7, (2009) · Zbl 1187.26014
[20] Wen, J; Wang, W-L, The optimization for the inequalities of power means, No. 2006, 25, (2006) · Zbl 1133.26324
[21] Hara, T; Uchiyama, M; Takahasi, S-E, A refinement of various Mean inequalities, Journal of Inequalities and Applications, 2, 387-395, (1998) · Zbl 0917.26017
[22] Neuman, E; Sándor, J, On the Schwab-Borchardt Mean, Mathematica Pannonica, 17, 49-59, (2006) · Zbl 1100.26011
[23] Jagers, AA, Solution of problem 887, Nieuw Archief voor Wiskunde, 12, 230-231, (1994)
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.