Thread: maximum modulus (complex analysis)

For complex number z, how to show that max_|z|<=1 |azⁿ+b|=|a|+|b| where a, b are real number?

Originally Posted by shyuan

For complex number z, how to show that max_|z|<=1 |azⁿ+b|=|a|+|b| where a, b are real number?

Use the triangle inequality: . So



Therefore .

but this only prove that |a|+|b| is on eof the upper bound, not necessary to be the maximum value right?

Originally Posted by shyuan

but this only prove that |a|+|b| is on eof the upper bound, not necessary to be the maximum value right?

If it's the least upper bound (which it is) and a possible value (which it also is), then it is the maximum value.

thx for ur help bt cn i ask, hw to confirm that it is already the least upper bound?

Originally Posted by shyuan

this only prove that |a|+|b| is on eof the upper bound, not necessary to be the maximum value right?

Given a, b and n, try to find a z with |z| = 1 such that |ax^n + b| = |a| + |b|.

Thank you very much. I solved my problem with the helps from you guys. Thanks.