This is an announcement for the paper "On holomorphic domination, I" by Imre Patyi. Abstract: Let $X$ be a separable Banach space and $u{:}\,X\to\Bbb{R}$ locally upper bounded. We show that there are a Banach space $Z$ and a holomorphic function $h{:}\,X\to Z$ with $u(x)<\|h(x)\|$ for $x\in X$. As a consequence we find that the sheaf cohomology group $H^q(X,\Cal{O})$ vanishes if $X$ has the bounded approximation property (i.e., $X$ is a direct summand of a Banach space with a Schauder basis), $\Cal{O}$ is the sheaf of germs of holomorphic functions on $X$, and $q\ge1$. As another consequence we prove that if $f$ is a $C^1$-smooth $\overline\partial$-closed $(0,1)$-form on the space $X=L_1[0,1]$ of summable functions, then there is a $C^1$-smooth function $u$ on $X$ with $\overline\partial u=f$ on $X$. Archive classification: math.CV math.FA Mathematics Subject Classification: 32U05; 32L10; 46G20 The source file(s), holodom-I-3.tex: 35922 bytes, is(are) stored in gzipped form as 0910.0476.gz with size 12kb. The corresponding postcript file has gzipped size 82kb. Submitted from: i355p113@speedpost.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.0476 or http://arXiv.org/abs/0910.0476 or by email in unzipped form by transmitting an empty message with subject line uget 0910.0476 or in gzipped form by using subject line get 0910.0476 to: math@arXiv.org.