  Let $(R, m)$ be a $d$-dimensional Cohen-Macaulay local ring.
  In this note we prove, in a very elementary way, an upper bound of the first
normalized Hilbert coefficient of a $m$-primary ideal $I\subset R$ that
improves all known upper bounds unless for a finite number of cases. We also
provide new upper bounds of the Hilbert functions of $I$ extending the known
bounds for the maximal ideal.
