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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:
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