I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrassingly weak -
I would like to show that, for a real number $p \geq 1$ and complex numbers $\alpha, \beta$, I have $\begin{equation} |\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p) \end{equation}$
I thought it would be best to rewrite this as $\begin{equation} \left|\frac{\alpha + \beta}{2}\right|^p \leq \frac{|\alpha|^p + |\beta|^p}{2} \end{equation}$
but then I am unsure what to do next - is this a sensible start anyways ? Any help would be great !
(P.S. this is not a homework question - I am currently trying to brush up my knowledge of $L^p$ spaces, and this inequality came up as a statement. I thought it might be worthwhile to make sure I can fill in the gaps to improve my skills in computing inequalities.)