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Inequalities for certain means in two arguments. (English) Zbl 0938.26011
Let \(G= \sqrt{ab}\); \(L= (b-a)/(\ln b-\ln a)\); \(I= {1\over e}(b^b/a^a)^{1/(b-a)}\); \(A={a+b\over 2}\), \(S= a^{a/(a+ b)} b^{b/(a+b)}\). The following inequalities are valid: \[ A^2/I< (4A^2- G^2)/3I< S< A^4/I^3< A^2/G, \] \[ AL+ SI< 2A^2< S^2+ G^2, \] \[ (4A^2- 2G^2)/e< SI< A^2 L^2/G^2, \] \[ (S/A)^2< (I/G)^3, \] \[ (A^2- G^2)/A^2< \ln S/G< (A^2- G^2)/G, \] \[ (S-G)/(S-A)> \sqrt 2. \]

MSC:
26D15 Inequalities for sums, series and integrals
26E60 Means
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