Degenerate Tur\'an problems for hereditary properties

Let $H$ be a graph and $t\ge s\ge 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $\lambda (G)=O\left(n^{1-1\mathrm{/}s}\right)$ where $\lambda (G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.