Analytic functions complex

Let f = u + iv be an analytic function on an open connected set G in C ( C = complex plane) where u and v are its real and imaginary parts. assume u(z) >= u(a) for some a in G and all z in G. Prove that f is constant.

Complex/entire function

Let f be an entire function such that |f(z)| =<10|z+1| for all |z|>100 Show that f is a linear function, f(z)= pz + q

Cross ratio complex

Evaluate cross ratio (infinity,0,i,1) give answer in the form a + ib where a,b in R.

Complex analysis, singularities.

If f : G -> C ( C here is complex plane) is analytic except for poles show that the poles of f cannot have limit point in G.

Complex analysis/singularities

One can classify isolated singularities by examining the equations: lim (z -> a) |z - a|^s |f(z)| = 0 lim(z -> a) |z - a|^s |f(z)| = infinity Now, prove that a function f has an essential singularity at z = a iff neither of the above holds for any real number s.