behaves linearly for $|\phi|>M_{\text{p}}$. This minimum may be in dS or AdS depending on the choice for $D$.

-   iii\) Those two cases saturate the bound but are not duality invariant. If however one insists in a duality invariance $\phi \leftrightarrow -\phi$ one has V()\_[extr]{} =  (\_2(-ie\^) - \_2(ie\^)) +   (-)   The potential is then symmetric with a minimum at $\phi=0$. For $A=2, B=1$, $C=D=0$ this is depicted in fig.\[extremal\] in black. This class of potentials is interesting in its own right and may have interesting physical applications e.g. in inflation. In particular, given its asymptotic linear behaviour, it should give rise to a variation of linear inflation.

![Examples of extremal potentials.[]{data-label="extremal"}](extremal.pdf)

Constraints on some simple potentials
-------------------------------------

It is interesting to see what are the constraints for a scalar potential of the form V() = m\^2\^2 + \^4  . It is easy to see one finds the constraint (\^2 - m\^2)  (m\^2+ \^2)\^2  . Note that the term in the right is strongly suppressed by the $M_p^{-2}$ factor, so that in practice (for $\phi^2\ll M_p^2$) the constraint amounts to the left hand side being positive. This is automatic for $m^2<0$ and $\lambda$ positive. This is similar to the situation 