Sharp inequalities for polygamma functions

The main aim of this paper is to prove that the double inequality \frac{(k-1)!}{\Bigl{x+\Bigl[\frac{(k-1)!}{|\psi^{(k)}(1)|}\Bigr]^{1/k}\Bigr}^k} +\frac{k!}{x^{k+1}}<\bigl|\psi^{(k)}(x)\bigr|<\frac{(k-1)!}{\bigl(x+\frac12\bigr)^k}+\frac{k!}{x^{k+1}} holds for $x>0$ and $k\in \mathbb{N}$ and that the constants $[\frac{(k-1)!}{\mathrm{\mid }{\psi }^{(k)}(1)\mathrm{\mid }}{]}^{1\mathrm{/}k}$ and $\frac{1}{2}$ are the best possible. In passing, some related inequalities and (logarithmically) complete monotonicity results concerning the gamma, psi and polygamma functions are surveyed.