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(b) Let f be analytic on the unit disk D(0,1). If f (z) ± 0 for all z E D(0,1),and set m = min |f (z)|, prove that there is a point zo with |zo] = 1/2 such that |f (z,)| |z]<1/2 = m.

(b) Let f be analytic on the unit disk D(0,1). If f (z) ± 0 for all z E D(0,1),and set m = min |f (z)|, prove that there is a point zo with |zo] = 1/2 such that |f (z,)| |z]<1/2 = 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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need help with part b, using maximum modulus principle complex variables, thanks for clear explanation

(a) State the Maximum Modulus theorem/principle for analytic functions on region 2. Maximum Modulus Principle: Let f be an analytic function in a region 2 and a E N. If If (z)| < |f (a)| for all z E N, then f is constant on 2. In other words, any non-constant analytic function on a region n does not attain a maximum modulus on 2. (b) Let f be analytic on the unit disk D(0,1). If f (z) # 0 for all z e D(0,1),and set m = ¸min_ |f(z)|, prove that there is a point z, with |Zo| = 1/2 such that |ƒ (zo)| = m. |z|S1/2
Transcribed Image Text:(a) State the Maximum Modulus theorem/principle for analytic functions on region 2. Maximum Modulus Principle: Let f be an analytic function in a region 2 and a E N. If If (z)| < |f (a)| for all z E N, then f is constant on 2. In other words, any non-constant analytic function on a region n does not attain a maximum modulus on 2. (b) Let f be analytic on the unit disk D(0,1). If f (z) # 0 for all z e D(0,1),and set m = ¸min_ |f(z)|, prove that there is a point z, with |Zo| = 1/2 such that |ƒ (zo)| = m. |z|S1/2
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