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Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. (English) Zbl 1187.26015
Summary: For $$p\in\mathbb R$$, the generalized logarithmic mean $$L_p(a,b)$$, arithmetic mean $$A(a,b)$$, and geometric mean $$G(a,b)$$ of two positive numbers $$a$$ and $$b$$ are defined by $$L_p(a,b)=a$$, for $$a=b$$, $$L_p(a,b)= [(b^{p+1}-a^{p+1})/((p+1)(b-a))]^{1/p}$$, for $$p\neq 0$$, $$p\neq-1$$, and $$a\neq b$$, $$L_p(a,b)= (1/e)(b^b/a^a)^{1/(b-a)}$$, for $$p=0$$, and $$a\neq b$$, $$L_p(a,b)= (b-a)/(\log b-\log a)$$, for $$p=-1$$, and $$a\neq b$$, $$A(a,b)= (a+b)/2$$, and $$G(a,b)= \sqrt{ab}$$, respectively. In this paper, we find the greatest value $$p$$ (or least value $$q$$, resp.) such that the inequality $$L_p(a,b)<\alpha A(a,b)+ (1-\alpha) G(a,b)$$ (or $$\alpha A(a,b)+ (1-\alpha)G(a,b)< L_q(a,b)$$, resp.) holds for $$\alpha\in(0,1/2)$$ (or $$\alpha\in(1/2,1)$$, resp.) and all $$a,b>0$$ with $$a\neq b$$.

MSC:
 2.6e+61 Means
Full Text:
References:
 [1] Alzer, H, Ungleichungen für mittelwerte, Archiv der Mathematik, 47, 422-426, (1986) · Zbl 0585.26014 [2] Alzer, H; Qiu, S-L, Inequalities for means in two variables, Archiv der Mathematik, 80, 201-215, (2003) · Zbl 1020.26011 [3] Burk, F, The geometric, logarithmic and arithmetic Mean inequality, The American Mathematical Monthly, 94, 527-528, (1987) · Zbl 0632.26008 [4] Janous, W, A note on generalized Heronian means, Mathematical Inequalities & Applications, 4, 369-375, (2001) · Zbl 1128.26302 [5] Leach, EB; Sholander, MC, Extended Mean values. II, Journal of Mathematical Analysis and Applications, 92, 207-223, (1983) · Zbl 0517.26007 [6] Sándor, J, On certain inequalities for means, Journal of Mathematical Analysis and Applications, 189, 602-606, (1995) · Zbl 0822.26014 [7] Sándor, J, On certain inequalities for means. II, Journal of Mathematical Analysis and Applications, 199, 629-635, (1996) · Zbl 0854.26013 [8] Sándor, J, On certain inequalities for means. III, Archiv der Mathematik, 76, 34-40, (2001) · Zbl 0976.26015 [9] Shi, M-Y; Chu, Y-M; Jiang, Y-P, Optimal inequalities among various means of two arguments, No. 2009, 10, (2009) · Zbl 1187.26017 [10] Carlson, BC, The logarithmic Mean, The American Mathematical Monthly, 79, 615-618, (1972) · Zbl 0241.33001 [11] Sándor, J, On the identric and logarithmic means, Aequationes Mathematicae, 40, 261-270, (1990) · Zbl 0717.26014 [12] Sándor, J, A note on some inequalities for means, Archiv der Mathematik, 56, 471-473, (1991) · Zbl 0693.26005 [13] Lin, TP, The power mean and the logarithmic Mean, The American Mathematical Monthly, 81, 879-883, (1974) · Zbl 0292.26015 [14] Pittenger, AO, Inequalities between arithmetic and logarithmic means, Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, 678-715, 15-18, (1981) · Zbl 0469.26009 [15] Imoru, CO, The power mean and the logarithmic Mean, International Journal of Mathematics and Mathematical Sciences, 5, 337-343, (1982) · Zbl 0483.26012 [16] Chen, Ch-P, The monotonicity of the ratio between generalized logarithmic means, Journal of Mathematical Analysis and Applications, 345, 86-89, (2008) · Zbl 1160.26012 [17] Li, X; Chen, Ch-P; Qi, F, Monotonicity result for generalized logarithmic means, Tamkang Journal of Mathematics, 38, 177-181, (2007) · Zbl 1132.26326 [18] Qi, F; Chen, Sh-X; Chen, Ch-P, Monotonicity of ratio between the generalized logarithmic means, Mathematical Inequalities & Applications, 10, 559-564, (2007) · Zbl 1127.26021
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