A note on improved variance bounds for certain bounded unimodal distributions

Gray and Odell have proved that no symmetric continuous unimodal density on the interval [a,b], with modes interior to (a,b), can have variance exceeding (b - a)2/12. Jacobson has derived more general sufficient conditions for the application of this bound and also has shown that no unimodal distribution on [a,b] can have variance larger than (b - a)2/9. Seaman, Odell and Young have presented even more general sufficient conditions for the smaller bound. In this note, we make use of a dispersion ordering to show that the previous conditions for the smaller bound are far too restrictive. Indeed, no continuous unimodal density [latin small letter f with hook] on [a, b], with [latin small letter f with hook](a) [less-than-or-equals, slant]1/(b - a) and [latin small letter f with hook](b) [less-than-or-equals, slant] 1/(b - a), can have variance larger than (b - a)2/12.