Let $f$ be a complex valued function of a complex variable. Does $ \overline{\int f(z) dz} = \int \overline{f(z)}dz \text{ ?} $
If $f$ is a function of a real variable, the answer is yes as $ \int f(t) dt = \int \text{Re}(f(t))dt + i\text{Im}(f(t))dt. $
If $f$ is a complex valued function of a complex variable and belong to $L^2$, the answer is also yes as $L^2$ is a Hilbert space and, by conjugate symmetry of the inner product, $ \overline{\langle f,g\rangle}=\langle g,f\rangle $ where $g(z)=1$ is the identity function.
Apart from these two cases, is it otherwise true?
Is it true in $L^1$?