  We provide a direct proof of the following theorem of Kalton, Hollenbeck, and
Verbitsky \cite{HKV}: let $H$ be the Hilbert transform and let $a,b$ be real
constants. Then for $1<p<\infty$ the norm of the operator $aI+bH$ from
$L^p(\mathbb R)$ to $L^p(\mathbb R)$ is equal to $$ \bigg(\max_{x\in \Bbb
R}\frac{|ax-b+(bx+a)\tan \frac{\pi}{2p}|^p+|ax-b-(bx+a)\tan
\frac{\pi}{2p}|^p}{|x+\tan \frac{\pi}{2p}|^p+|x-\tan \frac{\pi}{2p}|^p}
\bigg)^{\frac 1p}. $$ Our proof avoids passing through the analogous result for
the conjugate function on the circle, as in \cite{HKV}, and is given directly
on the line. We also provide new approximate extremals for $aI+bH$ in the case
$p>2$.
