  In this paper, given a prescribed measure on $\mathbb{S}^1$ whose density is
bounded and positive, we establish a uniform diameter estimate for solutions to
the planar $L_p$ dual Minkowski problem when $0<p<1$ and $q\ge 2$. We also
prove the uniqueness and positivity of solutions to the $L_p$ Minkowski problem
when the density of the measure is sufficiently close to a constant in
$C^\alpha$.
