Can we not argue on continuity grounds that the maximum modulus function of a non-constant, analytic function is a continuous, monotonic, and increasing function of the outward radius? And since and , f(z) is not a constant function and by Cauchy's inequality for the second derivative, where is the contour and . This then would imply . Therefore, the maximum of on must be larger than and so that the maximum modulus between and is a continuous function of r at least in the range of .
Also, I thought it interesting to verify this directly with the function which satisfies and in which case the maximum modulus as a function of r is .