Hölder bound for discounted CRRA consumption ($\mathbb{E}\int e^{-bt}U(c_t)dt\le U(x){\left(\mathbb{E}\int e^{-bt/R}{Z}_t^{1-1/R}dt\right)}^R$)

For $0<R<1$, nonnegative consumption with state-price budget $\mathbb{E}\int Z_t{c}_{t}dt\le x$ satisfies this bound under CRRA utility. Apply Holder inequality on probability-times-time measure to $(Zc{)}^{1-R}$ and $e^{-bt}{Z}^{R-1}$, with conjugate exponents $1/(1-R)$ and $1/R$. If the price integral $D$ is finite, equality forces full budget use and $c_t=(x/D)e^{-bt/R}{Z}_t^{-1/R}$. Financing this process is a separate feasibility issue, so the equality pattern alone does not assert attainability in an incomplete market.