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Suppose that f(z) is analytic and satisfies |ƒ(z)| ≤ 1 for |z| < 1. Show that if f(z) has a zero of order m at zo, then |zo|™ ≥ |ƒ(0)|. Hint. Let (z) = (z − zo)/(1 — Zoz), which is a fractional linear transformation mapping the unit disk onto itself, and show that |ƒ(z)| ≤ |v(z)|m.

Suppose that f(z) is analytic and satisfies |ƒ(z)| ≤ 1 for |z| < 1. Show that if f(z) has a zero of order m at zo, then |zo|™ ≥ |ƒ(0)|. Hint. Let (z) = (z − zo)/(1 — Zoz), which is a fractional linear transformation mapping the unit disk onto itself, and show that |ƒ(z)| ≤ |v(z)|m.

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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Suppose that f(z) is analytic and satisfies |ƒ(z)| ≤ 1 for |z| < 1. Show that if f(z) has a zero of order m at zo, then |zo|m ≥ |ƒ(0)|. Hint. Let (z) = (z − zo)/(1 — Zoz), which is a fractional linear transformation mapping the unit disk onto itself, and show that |ƒ(z)| ≤ |v(z)|m.
Transcribed Image Text:Suppose that f(z) is analytic and satisfies |ƒ(z)| ≤ 1 for |z| < 1. Show that if f(z) has a zero of order m at zo, then |zo|m ≥ |ƒ(0)|. Hint. Let (z) = (z − zo)/(1 — Zoz), which is a fractional linear transformation mapping the unit disk onto itself, and show that |ƒ(z)| ≤ |v(z)|m.
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