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(a) Let z, w be two complex numbers such that zw 1. Prove that w z Wz and also that w z 1 - wz < 1 1 if |2 < 1 and |w| < 1, if |z| = 1 or |w| = 1. [Hint: Why can one assume that z is real? It then suffices to prove that (rw) (rw) ≤ (1-rw)(1-rw) with equality for appropriate r and wl.]

(a) Let z, w be two complex numbers such that zw 1. Prove that w z Wz and also that w z 1 - wz < 1 1 if |2 < 1 and |w| < 1, if |z| = 1 or |w| = 1. [Hint: Why can one assume that z is real? It then suffices to prove that (rw) (rw) ≤ (1-rw)(1-rw) with equality for appropriate r and wl.]

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Let z, w be two complex numbers such that zw 1. Prove that W P 1 - wz and also that Z W 1 - wz 2 < 1 if |z| < 1 and |w| < 1, = 1 if |z| = 1 or |w| = 1. [Hint: Why can one assume that z is real? It then suffices to prove that (r − w) (r − w) ≤ (1 − rw)(1 – rw) - - - with equality for appropriate r and w
Transcribed Image Text:(a) Let z, w be two complex numbers such that zw 1. Prove that W P 1 - wz and also that Z W 1 - wz 2 < 1 if |z| < 1 and |w| < 1, = 1 if |z| = 1 or |w| = 1. [Hint: Why can one assume that z is real? It then suffices to prove that (r − w) (r − w) ≤ (1 − rw)(1 – rw) - - - with equality for appropriate r and w
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