On the norm of the operator $aI+bH$ on $L^p(\mathbb R)$

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 $p>2$.