In a dynamic model of electoral competition ($t=1,2,\ldots$), I want to prove incumbency advantage, i.e., that the incumbent is always re-elected (unless an exogenous event occurs). It all boils down to the following result, which I am confident is true (all simulations work), but I have not been able to prove yet. Let $c,\eta,\mu,i$ be parameters such that $0<c$, $1<\eta \leq 2$, and $0<i<\mu$. Define $r\in (i,\mu)$ and $l\in (-\mu,i)$ such that $$2(\mu-r)=c\eta (r-i)^{\eta-1}$$ and $$2(l+\mu)=c\eta (i-l)^{\eta-1}.$$ It is easy to see that $r$ and $l$ are uniquely defined within the given ranges. Then, I would like to prove that $$r^2+c(r-i)^{\eta} \leq l^2+c(i-l)^{\eta}.$$ In this problem, $i$ is the previous policy position, $c$ and $\eta$ determine the cost of moving policies, $\mu$ is the degree of polarization of political parties, and $r$ ($l$) is the policy that a right-wing (left-wing) candidate candidate would choose if elected. If $\eta=2$, the result is straightforward. Any help or hint would be great. Thank you for your time!