= Solution
Write $D=\mathbb E\int_0^\infty e^{-bt/R}Z_t^{1-1/R}\,dt$ and use the <product measure> $d\mathbb P\,dt$. For $0<R<1$, the <conjugate exponents> are $p=1/(1-R)$ and $q=1/R$. Factor the utility integrand as
$$
e^{-bt}c_t^{1-R}=(Z_tc_t)^{1-R}\bigl(e^{-bt}Z_t^{R-1}\bigr).
$$
The <Holder inequality> on this <product measure> gives
$$
\mathbb E\int_0^\infty e^{-bt}c_t^{1-R}\,dt\leq\left(\mathbb E\int_0^\infty Z_tc_t\,dt\right)^{1-R}D^R\leq X_0^{1-R}D^R.
$$
Divide by $1-R$ to obtain the <Hölder bound for discounted CRRA consumption>:
$$
\boxed{\mathbb E\int_0^\infty e^{-bt}U(c_t)\,dt\leq U(X_0)\left[\mathbb E\int_0^\infty e^{-bt/R}Z_t^{1-1/R}\,dt\right]^R.}
$$
For a finite right side this is ordinary Hölder; on the infinite measure space one can first truncate time and then pass by <monotone convergence theorem>. If $D=\infty$ and $X_0>0$, the bound is valid but uninformative. If $X_0=0$, the budget forces $c=0$ almost everywhere, so the utility integral is zero; handle this separately instead of writing an undefined $0\cdot\infty$.
When $D$ is finite and the bound is attained, equality requires full budget use and $c_t=(X_0/D)e^{-bt/R}Z_t^{-1/R}$. This is the equality pattern of Hölder, not an assertion that every market can finance that process.
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