Characteristic Function Of The Great Mold

In this article we have completely solved the first open problem in Zhang[Zhang, Z.M.,2007.A note on extreme magnitudes of characteristic functions.Statist.Probab. Lett.77,1641-1643],this is a problem about extreme magnitudes of characteristic functions.The special case of the problem thatα=0 is solved in Luo and Zhang[Luo,S.L., Zhang,Z.M.,2005.An extremal problem for Fourier transforms of probabilities.C. R.Math.Acad.Sci.Paris 341,293-296],and Zhang[2007].The general case thatα∈[0,1]is solved in this article,largely by making use of the methods and results in Luo and Zhang[2005],and Zhang[2007],the corresponding results we obtained here are natural generalizations of those in the special case.This problem was raised in the context of quantum mechanics,so naturally our results have certain applications in quantum mechanics.