Discussion
Consider the following statements :
(a) \(|e^{\sin z}|\) does not assume its maximum value in C.
(b) \(|\sin(e^{z})|\) does not assume its minimum value in C.
Then :
Answer: A
Only (a) is true.
(a) \(e^{\sin z}\) is a non-constant entire function, so by the maximum modulus principle its modulus has no maximum on C; indeed it is unbounded.
(b) is false. \(\sin(e^{z})\) has zeros: \(e^{z} = \pi\) has the solution \(z = \ln \pi\), and \(\sin \pi = 0\). So \(|\sin(e^{z})|\) attains its minimum value 0. The minimum modulus principle forbids an interior minimum only for functions without zeros.
So 'only (b)', 'both true' and 'both false' are wrong.
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