Polynomial moment bounds for the adjustment coefficient ($R<r_2<r_1,r_1=2\rho \mu /m_2,m_3{r}_2^2+3m_2{r}_2-6\rho \mu =0$)

For positive claims with a finite positive moment-generating function neighbourhood, use $e^u>1+u+u^2/2$ and $e^u>1+u+u^2/2+u^3/6$ for $u>0$ in the defining adjustment coefficient equation. They imply the displayed strict upper bounds, where $r_2$ is the positive quadratic root. Its defining polynomial is negative at zero and positive at $r_1$, proving $r_2<r_1$. Higher exponential-series truncations similarly yield upper bounds from more raw moments.