Complex Analysis: Theory and Applications · Problem 4.4.9

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · inequality-proof

Problem

Here we establish the validity of Inequality 4.17 in the proof of Theorem 4.17. ((a)) Show that

\left|\frac{s^n - t^n}{s - t}\right| = |s^{n-1} + s^{n-2}t + s^{n-3}t^2 + \cdots + st^{n-2} + t^{n-1}|

\leq |s^{n-1}| + |s^{n-2}t| + |s^{n-3}t^2| + \cdots + |st^{n-2}| + |t^{n-1}|,

where

s

and

t

are arbitrary complex numbers,

s \neq t

. ((b)) Explain why, in Inequality 4.17,

|z - \alpha| < r

and

|z_0 - \alpha| < r

. ((c)) Let

s = z - \alpha

and

t = z_0 - \alpha

in part (a) to establish Inequality 4.17.