An optimal double inequality between logarithmic and generalized logarithmic means

• Autorzy: Jiang Y. , Long B. , Chu Y.

Identyfikatory

DOI

10.1515/JAA-2013-016

• Języki publikacji: EN

Abstrakty

EN

In this paper, we answer the question: for any q > 0 with q ≠ 1, what are the greatest value p₁= p₁(q) and the least value p₂= p₂(q) such that the double inequality Lp₁(a, b) < [L(a^q, b^q)]^1/q < Lp₂(a, b) holds for all a, b > 0 with a ≠ b? Here L(a, b) and Lp(a, b) are the logarithmic and pth generalized logarithmic means of a and b, respectively.

Słowa kluczowe

EN

logarithmic mean; generalized logarithmic mean; power mean

PL

uogólniona średnia logarytmiczna; średnia zgeneralizowana

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2013
• Tom: Vol. 19, nr 2
• Strony: 271--182
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Jiang Y.

• School of Information & Engineering, Huzhou Teachers College, Huzhou 313000, P.R. China

autor

Long B.

• School of Mathematics Science, Anhui University, Hefei 230039, P.R. China

autor

Chu Y.

• Department of Mathematics, Huzhou Teachers College, Huzhou 313000, P.R. China

Bibliografia

• [1] H. Alzer and S.-L. Qiu, Inequalities for means in two variables, Arch. Math. 80 (2003), 201-215.
• [2] P. S. Bullen, D. S. Mitrinović and P. M. Vasić, Means and Their Inequalities, Kluwer, Dordrecht, 2003.
• [3] B. C. Carlson, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly 78 (1971), 496-505.
• [4] B. C. Carlson, The logarithmic mean, Amer. Math. Monthly 79 (1972), 615-618.
• [5] B. C. Carlson and J. L. Gustafson, Total positive of mean values and hypergeometric functions, SIAM J. Math. Anal. 14 (1983), 389-395.
• [6] Y.-M. Chu and B.-Y. Long, Best possible inequalities between generalized logarithmic mean and classical means, Abstr. Appl. Anal. (2010), article ID 303286.
• [7] Y.-M. Chu and W. F. Xia, Inequalities for generalized logarithmic means, J. Inequal. Appl (2009), article ID 763252.
• [8] C. O. Imoru, The power mean and the logarithmic mean, Internat. J. Math. Math. Sci. 5 (1982), 337-343.
• [9] P. Kahlig and J. Matkowski, Functional equations involving the logarithmic mean, Z Angew. Math. Mech. 76 (1996), 385-390.
• [10] T. P. Lin, The power mean and the logarithmic mean, Amer. Math. Monthly 81 (1974), 879-883.
• [11] B.-Y. Long and Y.-M. Chu, Optimal inequalities for generalized logarithmic, arithmetic, and geometric means, J. Inequal. Appl. (2010), article ID 806825.
• [12] A. O. Pittenger, Inequalities between arithmetic and logarithmic means, Univ. Beograd Publ. Elektrotehn. Fok. Ser. Mat. Fiz. 678-715 (1980), 15-18.
• [13] A. O. Pittenger, The logarithmic mean in n variables, Amer. Math. Monthly 92 (1985), 99-104.
• [14] G. Pólya and G. Szegó, Isoperimetric Inequalities in Mathematical Physics, Prince-ton University Press, Princeton, 1951.