# Thread: complex: question with application of Cauchy's Inequality

1. ## complex: question with application of Cauchy's Inequality

2. 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 $f(1)=0$ and $f''(0)=-i$, f(z) is not a constant function and by Cauchy's inequality for the second derivative, $|f''(\gamma)|\leq \frac{2}{R^2}M$ where $\gamma$ is the contour $z=10e^{it}$ and $|f(\gamma)|\leq M$. This then would imply $50\leq M$. Therefore, the maximum of $|f(z)|$ on $\gamma$ must be larger than $50$ and $f(1)=0$ so that the maximum modulus between $r=1$ and $r=10$ is a continuous function of r at least in the range of $1<\sqrt{2009}<50$.
Also, I thought it interesting to verify this directly with the function $f(z)=1/2 i z^3-i/2 z^2$ which satisfies $f(1)=0$ and $f''(0)=-i$ in which case the maximum modulus as a function of r is $r^2/2(r+1)$.