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This is not true in general. Indeed, let $x$ be a zero-mean random variable such that $Ee^{tx}<\infty$ for $t\in[0,\tau)$ but $Ee^{\tau x}=\infty$. Then for all $t\in(0,\tau)$ the left-hand side of the conjectured inequality will be finite but the right-hand side of the inequality will be $\infty$.