$2.3\%$. It follows from that the bound in is only saturated for free theories (where $\hat{c}(\mu^2) \propto
\delta(\mu^2-m^2)$) for general interacting theories will the relative overshoot be smaller than $|\min_\rho \Upsilon (\rho)|$. We show in figure \[fig:boundfig\] the type of behaviour that we expect and which is confirmed for the minimal models below.

The actual choice of $k=4$ then is a compromise between maximising the range of analyticity (small $k$) and minimising the undershoot in $I({\rho'}_m)$ (large $k$).

Application to different 2D models
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In this section we calculate the central charge for the free bosonic and fermionic theory perturbed by a mass term. These theories are gaussian and we can calculate $F(s)$ exactly, the central charge can then be written as $\lim_{\rho \rightarrow \infty}I(\rho)$ as the contribution from the cut then vanishes. The infra-red limit of these theories is trivial ($c_{IR}=0$) as all the degrees of freedom are massive, and therefore decouple when approaching the infra-red fixed point where the scale goes to infinity. The free theories are none the less important to consider as we can here obtain the exact function $I(\rho)$.

In 4.3 we consider the unitary minimal models perturbed by the relevant operator $\phi_{(1,3)}$, this theory 