Bound on the a-invariant and reduction numbers of ideals
  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. 
Clare D'Cruz, Vijay Kodiyalam and Jugal. K. Verma