<p>In the first part of this paper, we give a biholomorphicity criterion for holomorphic mappings on a convex domain in a complex Banach space. In the second part of this paper, we study the biholomorphicity of Schwarz mappings. We also study <i>A</i>-normalized subordination chains on the unit ball of a reflexive complex Banach space and obtain the biholomorphicity, surjectivity and generation by its transition mapping under some assumptions. We further obtain that if an <i>A</i>-normalized subordination chain on the unit ball <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {B}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">B</mi> </math></EquationSource> </InlineEquation> of a separable reflexive complex Banach space is locally absolutely continuous on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\([0,\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> locally uniformly with respect to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(z\in \mathbb {B}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">B</mi> </mrow> </math></EquationSource> </InlineEquation>, then it is a solution to the Loewner PDE. These results give some positive answers to the questions and conjectures proposed by Graham, Hamada, Kohr and Kohr in 2013. In the third part of this paper, we consider the connections between standard solutions and a biholomorphic standard solution or the canonical solution of the Loewner PDE. These results give improvements of the results by Graham, Hamada, Kohr and Kohr in 2013 and also a generalization of the results by Becker in 1973 and Duren, Graham, Hamada and Kohr in 2010 to infinite dimensions.</p>