  The Yoccoz inequality gives a $O(1/q)$ bound on the size of the $p/q$-limbs
of the Mandelbrot set. For over thirty years it has been conjectured that this
bound can be improved to $O(1/q^2)$. Using parabolic implosion, we provide the
first example of a family of limbs satisfying the improved bound, namely the
$1/q$-limbs. Additionally, we show that the $O(1/q^2)$ bound is optimal for
this family.
