§3, p. 533 — $f(x) = K + f(S)$ for $x < s$

Let $0 \le \alpha < 1$ and let $s \le S$ be integers. In the inventory model with set-up cost $K \ge 0$, one-period cost $G_\alpha$ (convex, $G_\alpha(y) \to \infty$ as $|y| \to \infty$) and i.i.d. integer demands with finite mean, let $f(x \mid s, S)$ be the expected total discounted cost of the stationary $(s, S)$ policy started at $X_1 = x$. If the initial stock is below the reorder point, $x < s$, then

Starting below $s$, the policy immediately orders up to $S$; the identity says that the cost of such a start is the set-up cost plus the cost of starting at $S$. In particular $a_\alpha(x \mid s, S) = (1-\alpha)f(x \mid s, S)$ is constant on $x < s$, which is what makes the function $\mathcal L_\alpha(S, D)$ of the paper well defined.

Formalization Note $f$ is the expected discounted cost of the $(s,S)$ Markov chain (definition fCost), not a formula; the model's standing assumptions are fields of Model although this identity does not use convexity or coercivity.