Claims inflation of a fixed reinsurance layer ($\mathbb{E}[L_{M,A}(hX)\mid hX>M]=hC(\mu ,\sigma ;M/h,A/h)$)

For positive inflation factor $h$, $L_{M,A}(hX)=h{L}_{M/h,A/h}(X)$. The conditioning event also changes to $X>M/h$. If claims have a lognormal distribution, the same result follows by replacing their log-mean $\mu$ by $\mu +\log h$ and keeping the contractual attachment and limit fixed. Merely multiplying the original conditional mean by $h$ would hold those effective thresholds incorrectly fixed.