= Solution
By <Gibbs inequality>,
$$
g(a)=\sum_jp_j\log\frac{p_j}{u_j}-D(p\Vert a)\leq\sum_jp_j\log\frac{p_j}{u_j},
$$
with equality exactly when \b[$\boxed{a_j=p_j}$]. Put $q_j=u_j/U$, a <probability distribution>. The optimal <log-optimal multiplicative betting> rate is
$$
\boxed{g_* =D(p\Vert q)-\log U.}
$$
For $U<1$ this is strictly positive by <Gibbs inequality>; taking $0<\varepsilon<g_*$ in the preceding <strong law of large numbers> bound proves growth with probability tending to one.
For $U>1$ the casino can set $u_j=Up_j$. Every optimal return multiplier is then exactly $1/U$, so the fortune decreases deterministically as $fU^{-m}$. More generally $g_*=-\log U<0$ for that choice. \b[An arbitrary choice with $U>1$ need not cause decline.] For example, take $p_1=p_2=1/2$, $u_1=0.01$, $u_2=1.99$. Then $U=2$ but $g_*=(1/2)\log[0.25/(0.01\cdot1.99)]>0$. For any fixed positive $p$, making one $u_j$ sufficiently small while preserving $U$ also makes the optimal rate positive.
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