The Oka principle for holomorphic fibre bundles of Holder-Zygmund classes on strongly pseudoconvex domains

Let $\bar{\mathrm{\Omega }}$ be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let $h:Z\to \bar{\mathrm{\Omega }}$ be a fibre bundle of Hölder-Zygmund class ${\mathrm{\Lambda }}^r$, $r>0$, which is holomorphic over $\mathrm{\Omega }$. Assuming that the fibre is an Oka manifold, we prove that every continuous section $f_0:\bar{\mathrm{\Omega }}\to Z$ is homotopic to a section $f_1:\bar{\mathrm{\Omega }}\to Z$ of class ${\mathrm{\Lambda }}^r(\bar{\mathrm{\Omega }})$ which is holomorphic on $\mathrm{\Omega }$. We also establish the parametric h-principle in this context. As an application, we obtain the Oka principle for the classification of vector bundles and principal bundles of Hölder-Zygmund classes on such domains.