Path-integral action in the generalized uncertainty principle framework

Various gedanken experiments of quantum gravity phenomenology in search a complete theory near the Planck scale indicate modification Heisenberg uncertainty principle to generalized (GUP). This leads nontrivial contributions on Hamiltonian nonrelativistic particle moving an arbitrary potential. In this paper we study path integral representation potential using most general form GUP containing both linear and quadratic momentum. First work out action hence find upper bound velocity free particle. interestingly imposes restrictions relation between parameters $\alpha$ $\beta$. Analysis shows that $ \beta > 4 \alpha^2$. We then deduce mathematical expressions classical fluctuations for harmonic oscillator systems.