zbMATH — the first resource for mathematics

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