Assume that $p>1$ and $e^f\in L^p[0,1]$ and assume that $f_n$ is a sequence of bounded function converging pointwise to $f$ and $\|f_n-f\|_{L^p[0,1]}\to 0$. Can we conclude that $\|e^{f_n}-e^f\|_{L^p[0,1]}\to 0$?
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$t \mapsto e^t$ is of class $C^1$ and it will be a Lipschitzian function on every compact interval. This will mean that provided $|f_n| \in [0, K]$ for all $n$ (and the limit too) then $|e^{f_n} - e^f| \leq \lambda |f_n - f|$ pointwisely. – William M. May 08 '23 at 14:49
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Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community May 08 '23 at 14:51