×

zbMATH — the first resource for mathematics

Optimal power mean bounds for Yang mean. (English) Zbl 1339.26086
Summary: In this paper, we prove that the double inequality \(M_p(a,b)<U(a,b)<M_q(a,b)\) holds for all \(a,b>0\) with \(a\neq b\) if and only if \(p\leq 2\log 2/(2\log\pi-\log 2)=0.8684\ldots\) and \(q\geq 4/3\), where \(U(a,b)\) and \(M_r(a,b)\) are the Yang and \(r\)th power means of \(a\) and \(b\), respectively.

MSC:
26E60 Means
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities. Reidel, Dordrecht; 1988. · Zbl 0687.26005
[2] Seiffert, HJ, Aufgabe \(β\)16, Ginkgo-Wurzel, 29, 221-222, (1995)
[3] Jagers, AA, Solution of problem 887, Nieuw Arch. Wiskd, 12, 230-231, (1994)
[4] Hästö PA: A monotonicity property of ratios of symmetric homogeneous means.JIPAM. J. Inequal. Pure Appl. Math. 2002.,3(5): Article ID 71 · Zbl 0292.26015
[5] Hästö, PA, Optimal inequalities between seiffert’s mean and power Mean, Math. Inequal. Appl, 7, 47-53, (2004) · Zbl 1049.26006
[6] Witkowski, A, Interpolations of Schwab-Borchardt Mean, Math. Inequal. Appl, 16, 193-206, (2013) · Zbl 1261.26027
[7] Costin, I; Toader, G, Optimal evaluations of some Seiffert-type means by power means, Appl. Math. Comput, 219, 4745-4754, (2013) · Zbl 06447280
[8] Chu, Y-M; Long, B-Y, Bounds of the neuman-Sándor Mean using power and identric means, No. 2013, (2013) · Zbl 1264.26038
[9] Alzer, H, Ungleichungen für mittelwerte, Arch. Math, 47, 422-426, (1986) · Zbl 0585.26014
[10] Alzer, H, Ungleichungen für [inlineequation not available: see fulltext.], Elem. Math, 40, 120-123, (1985) · Zbl 0596.26014
[11] Burk, F, The geometric, logarithmic, and arithmetic Mean inequality, Am. Math. Mon, 94, 527-528, (1987) · Zbl 0632.26008
[12] Lin, TP, The power mean and the logarithmic Mean, Am. Math. Mon, 81, 879-883, (1974) · Zbl 0292.26015
[13] Pittenger, AO, Inequalities between arithmetic and logarithmic means, Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz, 678, 15-18, (1980) · Zbl 0469.26009
[14] Pittenger, AO, The symmetric, logarithmic and power means, Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz, 678, 19-23, (1980) · Zbl 0469.26010
[15] Stolarsky, KB, The power and generalized logarithmic means, Am. Math. Mon, 87, 545-548, (1980) · Zbl 0455.26008
[16] Alzer, H; Qiu, S-L, Inequalities for means in two variables, Arch. Math, 80, 201-215, (2003) · Zbl 1020.26011
[17] Yang, Z-H, Three families of two-parameter means constructed by trigonometric functions, No. 2013, (2013) · Zbl 1297.26071
[18] Yang, Z-H; Chu, Y-M; Song, Y-Q; Li, Y-M, A sharp double inequality for trigonometric functions and its applications, No. 2014, (2014)
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.