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 n  if a>0.

ii) If f (z) and  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) .

v) The inequality  holds for 

vi) If  has the property that  converges, then f is necessarily an entire function.

vii) If a power series converges for and if bn  is such that for all  , then converges for .

viii) If f is entire and f (z) = f (−z) for all z , then there exists an entire function g such that

ix) A mobius transformation which maps the upper half plane onto itself and fixing 0, and no other points, must be of the form Tz =z for some  0 and .

x) If f is entire and Re f (z) is bounded as | z| →  , then f is constant. 

i) False. The statement appears to have some typographical errors, so I'll address a corrected version: If z = a + ib, where a and b are integers, then |1 + Z + Z^2 + ... + Z^n| ≥ |z|^n if a > 0. Counterexample: Take z = 1 (a = 1, b = 0). Then, 1 + Z + Z^2 + ... + Z^n = 1 + 1 + ... + 1 = n + 1, and |z|^n = 1. For n > 1, |1 + Z + Z^2 + ... + Z^n| < |z|^n.

ii) False. Counterexample: Let f(z) = z and g(z) = z*. Both f(z) and g(z) are analytic, but f(z) is not a constant.

iii) True. If u(x, y) is harmonic, then it satisfies Laplace's equation: u_{xx}(x, y) + u_{yy}(x, y) = 0. Let v(x, y) = u(x, -y). Then, v_{xx}(x, y) = _________ ____ _________ _________ _____.
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