

Abstract: 
The wellknown second order moment HeisenbergWeyl inequality (or uncertainty relation) in Fourier Analysis states: Assume that is a complex valued function of a random real variable such that . Then the product of the second moment of the random real for and the second moment of the random real for is at least , where is the Fourier transform of , such that , and . This uncertainty relation is wellknown in classical quantum mechanics. In 2004, the author generalized the aforementioned result to higher order moments and in 2005, he investigated a HeisenbergWeyl type inequality without Fourier transforms. In this paper, a sharpened form of this generalized HeisenbergWeyl inequality is established in Fourier analysis. Afterwards, an open problem is proposed on some pertinent extremum principle.These results are useful in investigation of quantum mechanics.
