Diameter estimate for planar $L_p$ dual Minkowski problem

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 $C^{\alpha }$.