forms of potentials verifying the extremal case in which the inequality is saturated. One finds solutions of the general form V’ =   \^[-1]{}(e\^) + C  . Further integration yields for the extremal potential V()\_[extr]{} =  ( \_2(-ie\^) - \_2(ie\^)) +  C + D  . where $\text{Li}_2$ is the dilogarithm function and $i=\sqrt{-1}$. In spite of its complex appearance the potential is real (for real constants). One can see that the first term grows linear with $\phi$ as $\phi\rightarrow +\infty$ and is damped exponentially as $\phi \rightarrow - \infty$. This asymptotic linear behavior is in agreement with our comment above that linear potentials always satisfy the differential constraint.

This class of potentials depend on 4 real constants $A,B,C,D$. For $A=0$ one just gets straight lines. For other choices one may get also runaway potentials as well as minima which may be dS, AdS or Minkowski depending on the choice for D. Some simple interesting cases are as follows:

-   i\) $A=B=1;  C=D=0$, fig.\[extremal\] in blue. The potential is linear at large $\phi$ and exponentially decreasing for large negative $\phi$. So this is an example of a runaway potential.

-   ii\) $A=B=1; C=-1$,$D=0$, fig.\[extremal\] in red. The potential shows a minimum and 