For valued fields of equicharacteristic $0$ the valuation inequality asserts that the transcendence degree of the field (assumed finite here) is an upper bound for the sum of the rank of the value group and the transcendence degree of the residue field. This was subsequently generalized to the case of polynomially bounded o-minimal structures where transcendence degree is replaced by the usual o-minimal dimension defined via the pregeometry of Skolem closure. (The corresponding inequality was then applied to showing that the real exponential field has a model complete and o-minimal theory.)

In this talk I present a version of the valuation inequality for the complex field associated to an o-minimal structure where the notion of Skolem closure is defined via holomorphic definable functions. I then suggest applications of this inequality to the problem of showing that certain expansions of the complex field are quasi-minimal and report on one small success.