An upper bound on common stabilizations of Heegaard splittings

We show that for any two Heegaard splittings of genus $p$ and $q$ for the same closed 3-manifold, there is a common stabilization of genus at most 3/2 p + 2q - 1. One may compare this to recent examples of Heegaard splittings whose smallest common stabilizations have genus at least $p+q$ or $p + 1/2 q$ depending on the notion of equivalence.