Talk:Liouville's Theorem (complex analysis)
I think your upper bound for the integral should be M/R, not M/R^2. The estimate M/R is the usual Cauchy Estimate.
You can construct a counterexample to your bound by taking f(z) = z - z_sub_zero, taking C equal to the circle of radius R around z_sub_zero, and taking M to be the maximum of |f(z)| on C, which equals R.
Then both f'(z_sub_zero) and M/R equal 1, but M/R^2 equals 1/R. So your inequality is wrong when R > 1.
I hope this is helpful.