zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An optimal double inequality between geometric and identric means. (English) Zbl 1247.26040
Summary: We find the greatest value p and least value q in (0,1/2) such that the double inequality G(pa+(1-p)b,pb+(1-p)a)<I(a,b)<G(qa+(1-q)b,qb+(1-q)a) holds for all a,b>0 with ab. Here, G(a,b), and I(a,b) denote the geometric, and identric means of two positive numbers a and b, respectively.
MSC:
26D15Inequalities for sums, series and integrals of real functions
26E60Means
References:
[1]Chu, Y. -M.; Xia, W. -F.: Two sharp inequalities for power mean, geometric mean, and harmonic mean, J. inequal. Appl., 6 (2009) · Zbl 1187.26013 · doi:10.1155/2009/741923
[2]Chu, Y. -M.; Xia, W. -F.: Inequalities for generalized logarithmic means, J. inequal. Appl., 7 (2009) · Zbl 1187.26014 · doi:10.1155/2009/763252
[3]Long, B. -Y.; Chu, Y. -M.: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means, J. inequal. Appl., 10 (2010) · Zbl 1187.26015 · doi:10.1155/2010/806825
[4]Chu, Y. -M.; Long, B. -Y.: Best possible inequalities between generalized logarithmic mean and classical means, Abstr. appl. Anal., 13 (2010) · Zbl 1185.26064 · doi:10.1155/2010/303286
[5]Chu, Y. -M.; Long, B. -Y.: Sharp inequalities between means, Math. inequal. Appl. 14, No. 3, 647-655 (2011) · Zbl 1223.26049 · doi:http://files.ele-math.com/abstracts/mia-14-55-abs.pdf
[6]Chu, Y. -M.; Zong, C.; Wang, G. D.: Optimal convex combination bounds of seiffert and geometric means for the arithmetic mean, J. math. Inequal. 5, No. 3, 429-434 (2011) · Zbl 1225.26061 · doi:http://files.ele-math.com/articles/jmi-05-37.pdf
[7]Alzer, H.: Inequalities for arithmetic, geometric and harmonic means, Bull. lond. Math. soc. 22, No. 4, 362-366 (1990) · Zbl 0707.26014 · doi:10.1112/blms/22.4.362
[8]Seiffert, H. -J.: Eine integralungleichung für streng monotonc funktionen mit logarithmisch konvexer umkehrfunktion, Elem. math. 44, No. 1, 16-18 (1989) · Zbl 0721.26010
[9]Chen, C. -P.; Qi, F.: Monotonicity properties and inequalities of functions related to means, Rocky mountain J. Math. 36, No. 3, 857-865 (2006) · Zbl 1131.26007 · doi:10.1216/rmjm/1181069432 · doi:euclid:rmjm/1181069432
[10]Zhu, L.: Some new inequalities for means in two variables, Math. inequal. Appl. 11, No. 3, 443-448 (2008)
[11]Neuman, E.; Sándor, J.: Companion inequalities for certain bivariate means, Appl. anal. Discrete math. 3, No. 1, 46-51 (2009) · Zbl 1199.26087 · doi:10.2298/AADM0901046N
[12]Du, H. -X.: Some inequalities for bivariate means, Commu. korean math. Soc. 24, No. 4, 553-559 (2009)
[13]Kouba, O.: New bounds for the identric mean of two arguments, JIPAM. J. Inequal. pure appl. Math. 9, No. 3, 6 (2008) · Zbl 1172.26314 · doi:emis:journals/JIPAM/article1008.html?sid=1008
[14]Trif, T.: On certain inequalities involving the identric mean in n variables, Studia univ. Babeş–bolyai math. 46, No. 4, 105-114 (2001) · Zbl 1027.26024
[15]Vamanamurthy, M. K.; Vuorinen, M.: Inequalities for means, J. math. Anal. appl. 183, No. 1, 155-166 (1994) · Zbl 0802.26009 · doi:10.1006/jmaa.1994.1137
[16]Sándor, J.: On the identric and logarithmic means, Aequationes math. 40, No. 2–3, 261-270 (1990) · Zbl 0717.26014 · doi:10.1007/BF02112299
[17]Sándor, J.: A note on some inequalities for means, Arch. math. 56, No. 5, 471-473 (1991) · Zbl 0693.26005 · doi:10.1007/BF01200091
[18]Sándor, J.: On refinements of certain inequalities for means, Arch. math. Brno 31, No. 4, 279-282 (1995) · Zbl 0847.26015
[19]Sándor, J.: Two inequalities for means, Internat. J. Math. math. Sci. 18, No. 3, 621-623 (1995) · Zbl 0827.26016 · doi:10.1155/S0161171295000792 · doi:http://www.hindawi.com/journals/ijmms/volume-18/issue-3.html
[20]Alzer, H.: Ungleichungen für mittelwerte, Arch. math. 47, No. 5, 422-426 (1986) · Zbl 0585.26014 · doi:10.1007/BF01189983
[21]Qiu, Y. -F.; Wang, M. -K.; Chu, Y. -M.; Wang, G. -D.: Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean, J. math. Inequal. 5, No. 3, 301-306 (2011) · Zbl 1226.26019 · doi:http://files.ele-math.com/articles/jmi-05-27.pdf
[22]Alzer, H.: Bestmögliche abschätzungen für spezielle mittelwerte, Zb. rad. Prir.-mat. Fak. ser. Mat. 23, No. 1, 331-346 (1993) · Zbl 0815.26014
[23]Alzer, H.; Qiu, S. -L.: Inequalities for mean in two variables, Arch. math. 80, No. 2, 201-215 (2003) · Zbl 1020.26011 · doi:10.1007/s00013-003-0456-2
[24]Chu, Y. -M.; Qiu, Y. -F.; Wang, M. -K.: Sharp power mean bounds for the combination of seiffert and geometric means, Abstr. appl. Anal., 12 (2010) · Zbl 1197.26054 · doi:10.1155/2010/108920
[25]Shi, M. -Y.; Chu, Y. -M.; Jiang, Y. -P.: Optimal inequalities among various means of two arguments, Abstr. appl. Anal., 10 (2009) · Zbl 1187.26017 · doi:10.1155/2009/694394
[26]Long, B. -Y.; Chu, Y. -M.: Optimal power mean bounds for the weighted geometric mean of classical means, J. inequal. Appl., 6 (2010) · Zbl 1187.26016 · doi:10.1155/2010/905679