# zbMATH — the first resource for mathematics

Two sharp inequalities for power mean, geometric mean, and harmonic mean. (English) Zbl 1187.26013
Summary: For $$p\in\mathbb R$$, the power mean of order $$p$$ of two positive numbers $$a$$ and $$b$$ is defined by $$M_p(a,b)= ((a^p+b^p)/2)^{1/p}$$, $$p\neq 0$$, and $$M_p(a,b)= \sqrt{ab}$$, $$p=0$$. In this paper, we establish two sharp inequalities as follows: $$(2/3)G(a,b)+(1/3)H(a,b)\geq M_{-1/3}(a,b)$$ and $$(1/3)G(a,b)+ (2/3)H(a,b)\geq M_{-2/3}(a,b)$$ for all $$a,b>0$$. Here $$G(a,b)= \sqrt{ab}$$ and $$H(a,b)= 2ab/(a+b)$$ denote the geometric mean and harmonic mean of $$a$$ and $$b$$, respectively.

##### MSC:
 2.6e+61 Means
Full Text:
##### References:
 [1] Wu, SH, Generalization and sharpness of the power means inequality and their applications, Journal of Mathematical Analysis and Applications, 312, 637-652, (2005) · Zbl 1083.26018 [2] Richards, KC, Sharp power mean bounds for the Gaussian hypergeometric function, Journal of Mathematical Analysis and Applications, 308, 303-313, (2005) · Zbl 1065.33005 [3] Wang, WL; Wen, JJ; Shi, HN, Optimal inequalities involving power means, Acta Mathematica Sinica, 47, 1053-1062, (2004) · Zbl 1121.26308 [4] Hästö, PA, Optimal inequalities between Seiffert’s Mean and power means, Mathematical Inequalities & Applications, 7, 47-53, (2004) · Zbl 1049.26006 [5] Alzer, H; Qiu, S-L, Inequalities for means in two variables, Archiv der Mathematik, 80, 201-215, (2003) · Zbl 1020.26011 [6] Alzer, H, A power Mean inequality for the gamma function, Monatshefte für Mathematik, 131, 179-188, (2000) · Zbl 0964.33002 [7] Tarnavas, CD; Tarnavas, DD, An inequality for mixed power means, Mathematical Inequalities & Applications, 2, 175-181, (1999) · Zbl 0938.26010 [8] Bukor, J; Tóth, J; Zsilinszky, L, The logarithmic mean and the power Mean of positive numbers, Octogon Mathematical Magazine, 2, 19-24, (1994) [9] Pečarić, JE, Generalization of the power means and their inequalities, Journal of Mathematical Analysis and Applications, 161, 395-404, (1991) · Zbl 0753.26009 [10] Chen J, Hu B: The identric mean and the power mean inequalities of Ky Fan type.Facta Universitatis 1989, (4):15-18. · Zbl 0698.26008 [11] Imoru, CO, The power mean and the logarithmic Mean, International Journal of Mathematics and Mathematical Sciences, 5, 337-343, (1982) · Zbl 0483.26012 [12] Lin, TP, The power mean and the logarithmic Mean, The American Mathematical Monthly, 81, 879-883, (1974) · Zbl 0292.26015 [13] Alzer, H; Janous, W, Solution of problem 8*, Crux Mathematicorum, 13, 173-178, (1987) [14] Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities, Mathematics and Its Applications (East European Series). Volume 31. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459. [15] Mao, QJ, Power mean, logarithmic mean and heronian dual Mean of two positive number, Journal of Suzhou College of Education, 16, 82-85, (1999)
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.