Let 0<x<y be real numbers. Let
H\equal{}\frac{2xy}{x\plus{}y} , G\equal{}\sqrt{xy} , A\equal{}\frac{x\plus{}y}{2} , Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}}
be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of x and y. As generally known H<G<A<Q. Arrange the intervals [H,G] , [G,A] and [A,Q] in ascending order by their length. quadraticsinequalitiesinequalities unsolved