On Heisenberg's inequality
It is proved that for every vectors \(\alpha, \beta, \gamma\) in an inner product space with \(\|\gamma\|=1\) we have \[ (\alpha, \beta)^2 \leq\|\alpha\|^2 \|\beta\|^2 -(\|\alpha\|(\beta, \gamma) -\|\beta\|(\alpha, \gamma))^2, \] which sharpens the Cauchy-Schwarz inequality. From this some refinements of the Heisenberg inequality are deduced, for instance \[ \int^{+\infty}_{-\infty} |f(t)|^2 dt\leq 4A^2 B^2 -C^2, \] where \[ A^2 =\int^{+\infty}_{-\infty} t^2 |f(t)|^2 dt, \quad B^2 =\int^{+\infty}_{-\infty} |f^{'}(t)|dt, \quad C=(Ax-By), \] \[ x=\int^{+\infty}_{-\infty} |tf(t)|[(1+t^2)\pi]^{-1/2} dt, \quad y=\int^{+\infty}_{-\infty}|f^{'}(t)|[(1+t^2)\pi]^{-1/2} dt. \] Equality is attained only if \(f(t)=C\exp (-ct^2)\), \(c>0.\)
scientific article; zbMATH DE number 1334736