  Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically
closed field $k$. A base point free complete very special linear system $g^r_n$
on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor
$E$ on $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and
$r=(m^2+3m)/2-(md-n)$. In this paper, we prove the following: Theorem Let
$g^r_n$ be a base point free very special non-trivial complete linear system on
$C$. Write $r=(x+1)(x+2)/2-b$ with $x, b$ integers satisfying $x\ge 1, 0\le b
\le x$. Then $n\ge n(r):=(d-3)(x+3)-b$. Moreover, this inequality is best
possible.
