It is well-known (see Grafakos' Classical Fourier Analysis, Exercise 5.1.12) that if $f$ is a real valued $L^p(\mathbb R)$ function and $1<p<2$ , then we have the following inequality: $$ \|Hf\|_{L^p(\mathbb R)}\leq A_p\|f\|_{L^p(\mathbb R)}, $$ where $A_p=\tan \frac {\pi}{2p}$. Moreover, this bound is sharp.
A nice and elegant proof and also be found in https://faculty.missouri.edu/~grafakosl/preprints/pichorides.pdf.
However, I was wondering if this bound is also true for complex valued functions $f$. The proof above used the fact that if $f$ is real valued, then the real part and the imaginary part of the integral $$ \frac i \pi\int_{-\infty}^\infty \frac {f(t)}{z-t}dt $$ converges to $f(x)$ and $Hf(x)$ respectively, as $y\to 0^+$, where $z=x+iy$. For complex valued $f$, this is no longer true.