  Let $R$ be a $d$-dimensional standard graded ring over an Artin local ring.
Let $M$ be the unique maximal homogeneous ideal of $R.$ Let $h^i(R)_n$ denote
the length of $H^i_M(R)_n$, i.e. the nth graded component of the ith local
cohomology module of R with respect to M. Define the Eisenbud-Goto invariant of
$R$ to be the number
  $$EG(R)= \sum_{q=0}^{d-1} \binom{d-1}{q} h^q(R)_{1-q}.$$
  We prove that the $a$-invariant of $R$ satisfies
  $$ a(R) \leq e(R)-length(R_1)+(d-1)(length(R_0)-1)+ EG(R).$$
  Using this bound we get upper bounds for the reduction number of an
$m$-primary ideal of a Cohen-Macaulay local ring $(R,m)$ whose associated
graded ring $G(m)$ has almost maximal depth.
