= Solution
Take <expected values> in the <sharp power-call inequality>. If $M(1+\varepsilon)$ is finite, then for every $K\ge0$,
$$
K^\varepsilon C(K)\le\frac{\varepsilon^\varepsilon}{(1+\varepsilon)^{1+\varepsilon}}M(1+\varepsilon).
$$
The bound is independent of the strike, so
$$
\boxed{\sup_{K\ge0}K^\varepsilon C(K)\le\frac{\varepsilon^\varepsilon}{(1+\varepsilon)^{1+\varepsilon}}M(1+\varepsilon)<\infty.}
$$
Combined with the previous part, this relates finite moments to polynomial decay of expected <European call option> payoffs, while retaining the distinction at the moment threshold.
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