= Deterministic claims maximize the adjustment coefficient at fixed mean and loading
{title2=$R_X\le R_{\mathbb EX}$}
Fix positive claim <expected value> $\mu$ and <relative safety loading> $\theta>0$ in a <classical risk model>. Assume the random claim law has an <adjustment coefficient> $R_X>0$. By <Jensen inequality>, $M_X(r)\ge e^{\mu r}$. Thus $e^{\mu R_X}-1-(1+\theta)\mu R_X\le0$. The expression on the left is strictly <convex>, has negative derivative at zero and tends to infinity, so its unique positive zero $R_\mu$ satisfies $R_X\le R_\mu$. If the claim size is not almost surely constant, strict <Jensen inequality> gives $R_X<R_\mu$. Consequently the <Lundberg inequality> upper bound $e^{-R_\mu u}$ for deterministic claims is smaller for $u>0$. Comparing these upper bounds alone does not establish an ordering of actual <ultimate ruin probabilities>.
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