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For the $L^\infty$ norms, nothing at all apart from Alexander Eremenko's answer, because of Borel's theorem: any sequence of real (resp. complex) numbers can be realized as the sequence of Taylor coefficients of some real (resp. complex) valued smooth function $f$ at any given point. Assuming, for instance, that all Taylor coefficients at a given point are positive, one can also arrange the $L^1$ norms of the derivatives of the function $f$ provided by Borel's theorem to blow up arbitrarily fast. Since you assumed the domain $\Omega$ to be bounded, this implies the same property for the $L^p$ norms for all $1\leq p\leq\infty$.