Let $2\le p <\infty$, and let $X$ be a complex Banach space. It is shown that $X$ is $p$-uniformly PL-convex if and only if there exists $\lambda >0$ such that $ \|f\|_{H^p(X)}\,{\ge}\, (\|f(0)\|^p+\lambda\int_{\mathbb D} (1-|z|^2)^{p-1}\|f^{\prime}(z)\|^p\,dA(z))^{1/p}$, for all $f\in H^p(X)$. Applications to embeddings between vector-valued BMOA spaces defined via Poisson integral or Carleson measures are provided.