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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:
26E60 Means
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