The well-known second order moment Heisenberg-Weyl 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 well-known in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to higher order moments and in 2005, he investigated a Heisenberg-Weyl type inequality without Fourier transforms. In this paper, a sharpened form of this generalized Heisenberg-Weyl 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.