Proof that gmT2 is the reciprocal of gmR2

I just published an article in the journal Concepts in Magnetic Resonance Part A. The concept of this article came out of a debate I had with one of my examiners during my PhD defense <I was right ;-) >.

Abstract: When performing multiexponential analysis of relaxation times in the slow exchange regime, it is often convenient to simplify the resulting distribution to one or more relaxation time constants. In doing so, what averaging method should be used and should the rate constant or time constant be reported? This note outlines why the geometric mean is most appropriate and provides a proof that the geometric mean rate and geometric mean time constants are reciprocals, which is counter-intuitive unless one considers that averaging values in a distribution is different than physically averaging relaxation species.

Proof that gmT2 is the reciprocal of gmR2