I have trouble understanding the proof of theorem 5.4 from Cielibak's article "Pseudo-holomorphic curves and periodic orbits on cotangent bundles". At the bottom of page 267 he defines a function $v(s,t)=\frac{1}{2}\mu(1+\nu)(\delta^2-(s-s_1-\delta)^2)+\nu$ and says that by an easy calculation it follows that $(-\Delta-\mu)v\geq\mu$, where $s\in[s_1,s_2]$ and $\delta=\frac{s_2-s_1}{2}$. I am probably doing something wrong, because in my own caclulation I actually get the opposite inequality $(-\Delta-\mu)v\leq\mu$. Here is my reasoning:
As $v$ only depends on $s$ we get $-\Delta v=-\partial_s^2 v=\mu(1+\nu)$. Subtracting $\mu v$ from this yields $(-\Delta-\mu)v=\mu - \frac{1}{2}\mu^2(1+\nu)(\delta^2-(s-s_1-\delta)^2)$. Note that as $s$ ranges between $s_1$ and $s_2$, we get $s-s_1-\delta\in[-\delta,\delta]$. Thus $\delta^2-(s-s_1-\delta)^2\geq 0$, which means $(-\Delta-\mu)v$ equals $\mu$ minus something non-negative, and therefore is at most $\mu$.
I would appreciate for someone to tell me where I go wrong! Thanks for your help!
P.s. I also cannot find where it is used in the proof that $\mu\epsilon\leq \kappa$.
