  A polynomial is called self-reciprocal (or palindromic) if the sequence of
its coefficients is palindromic. In this paper we obtain improved error bounds
for the number of irreducible polynomials and self-reciprocal irreducible monic
polynomials with prescribed coefficients over a finite field. The improved
bounds imply that self-reciprocal irreducible monic polynomials with degree
$2d$ always exist provided that the number of prescribed leading coefficients
is slightly less than $ d/2$.
