Let $f$ be integrable on $[0, t]$ , $t≥0$ then prove that $$\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,t)f(s)}ds$$

This clearly smells like Jensen's inequality where we instead have, $$\exp\left(\frac1t\int_0^t f(s)ds \right) \le \frac1t \int_0^t e^{f(s)}ds$$ I don't see a special function I can plug into the Jensen inequality to get the result.

Does anyone knows how the aforementioned inequality could be derived from Jensen inequality?


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