Question

Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example

i). if z = a + ib, where a and b are integers, then \lvert 1 + z + z^2 + .... + z^n \rvert \geq \lvert z \rvert ^nif a>0

ii) If f(z) and \overline{f(z)} are analytic functions in a domain, then
f is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in D iff u(x, − y)
is harmonic in D .
iv) \lim_{n \to \infty }(n!)^{1/n} = \infty
v) The inequality \lvert e^a -e^b \rvert \leq \lvert a - b\rvert holds for a,b \in D = \left \{ w : Re\: w \leq 0 \right \}.
vi) If f(z) = \sum_{n=0}^{\infty}a_n(z-a)^n has the property that \sum_{n=0}^{\infty}f^{(n)}(a) converges, then f is necessarily an entire function.
vii) If a power series \sum_{n=0}^{\infty}a_nz^n converges for \lvert z \rvert< 1 and if
b_n \in C is such that \lvert b_n \rvert < n^2 \lvert a_n \rvert for all n \geq 0, then \sum_{n=0}^{\infty}b_nz^nconverges for \lvert z \rvert < 1.
viii) If f is entire and f (z) = f (-z) for all z, then there exists an entire function g such that f (z) = g(z^2) for all z \in C.
ix) A mobius transformation which maps the upper half plane \left \{ z : Im \: z > 0 \right \} onto itself and fixing 0, \infty and no other points, must be of the form Tz = \alpha z for some \alpha > 0 and \alpha \neq 1.
x) If f  is entire and Re f(z) is bounded as \lvert z \rvert \to \infty , then
f is constant.

07 Feb 2024
Answer :
Word Count : 511

i) False. Consider \( z = 1 + i \), \( n = 2 \), and \( a > 0 \). Then \( |1 + z + z^2| = |1 + (1+i) + (1+i)^2| = |3 + 3i| = \sqrt{18} \) while \( |z|^n = |1 + i|^2 = \sqrt{2}^2 = 2 \). Clearly, \( \sqrt{18} \nleq 2^2 \).

ii) True. If \( f(z) \) and \( \overline{f(z)} \) are analytic, then their sum and difference are analytic as well. \( f(z) + \overline{f(z)} = ___ ______ ___ ________ __________ ____ ______ ________ ______ ________ ________ __________.
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