For $\left|\frac{\alpha^n-\beta^n}{\alpha-\beta}\right| \leq \frac{|\alpha|^n-|\beta|^n}{|\alpha|-|\beta|}$, what will happen when $|\alpha|=|\beta|$? For the complex inequality
$$
\left|\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta}\right| \leq \frac{|\alpha|^{n}-|\beta|^{n}}{|\alpha|-|\beta|},\quad|\alpha|\neq|\beta|,\quad\alpha\in\mathbb{C},\quad\beta\in\mathbb{C}
$$
does it still hold for the case when $|\alpha|=|\beta|$, if not, how could we interpreted this case?
I try to rewrite $|\alpha|^n-|\beta|^n$ in form of
$$
|\alpha|^n-|\beta|^n=
(|\alpha|-|\beta|)(|\alpha|^{n-1}+|\beta||\alpha|^{n-2}+\cdots+|\alpha||\beta|^{n-2}+|\beta|^{n-1}),
$$
but is it true to obtain
$$
\frac{(|\alpha|-|\beta|)(|\alpha|^{n-1}+|\beta||\alpha|^{n-2}+\cdots+|\alpha||\beta|^{n-2}+|\beta|^{n-1})}{|\alpha|-|\beta|}=|\alpha|^{n-1}+|\beta||\alpha|^{n-2}+\cdots+|\alpha||\beta|^{n-2}+|\beta|^{n-1}.
$$
Thanks for helping.

You would just take the limit of the right-hand side as $|\alpha|$  approaches $|\beta|$, which gives you $n|\beta|^{n-1} = n|\alpha|^{n-1}$.
