An analytic function \(f\) in a hyperbolic plane domain \(\Omega\) is locally uniformly \(p\)-valent if \(f\) assumes every value at most \(p\)-times when restricted to any subset of \(\Omega\) with hyperbolic diameter at most 1. For \(\Omega= D=\) unit disk, the authors show that \(f\) is locally uniformly \(p\)-valent iff \(f\) preserves BMO. Using this they show that if \(f: D\to \mathbb{C}\) is locally uniformly \(p\)-valent and if \(f(G)\) is an univalent quasidisk image of a domain \(G\subset D\), then \(G\) itself is a quasidisk. For the quasiconformal version of this subinvariance result see [\textit{J. L. Fernández}, the first author and the reviewer, J. Anal. Math. 52, 117-132 (1989; Zbl 0677.30012)]. The last result is used to show that certain domains associated to Julia set, arising from rational approximation, and to Koenigs functions are quasidisks.