Result: Loewner chains and parametric representation of biholomorphic mappings in complex Banach spaces: Loewner chains and parametric representation of biholomorphic mappings in complex Banach spaces.

Let \(X\) be a complex Banach space and let \(\mathbb B\) be the unit ball of \(X\). The authors give improvements in the existence theorem and prove that solutions of the Loewner differential equation give rise to Loewner chains. Certain properties of Loewner chains in infinite dimensions are studied, and infinite dimensional versions of some well-known univalence criteria on \(\mathbb B\subset\mathbb C^n\) are obtained. A characterization of spirallikeness of type \(\alpha\) in terms of Loewner chains, and sufficient conditions for biholomorphic mappings on \(\mathbb B\) to have parametric representation are given. Let \(\alpha\in (-\pi/2,\pi/2)\) and \(f:\mathbb B\to X\) be a normalized locally biholomorphic mapping. Then \(f\) is spirallike of type \(\alpha\) iff \(f(z,t)= e^{(1-ia)t}f(e^{iat}z)\) is a Loewner chain, where \(a =\tan\alpha\). Hence \(f\) is starlike iff \(f(z,t) = e^tf(z)\) is a Loewner chain. A sufficient condition for close-to-starlikeness in the case of infinite dimensional Banach spaces, and an example which is a direct application of this result are also proposed.