Please help with this problem. Thank you.
Let f be amalytic in D = {z in C | |z| <= 10} with f(1) = 0, f"(0) = -i. Prove that there exists a z0 in D such that |f(z0)|= sqrt2009.
Please help with this problem. Thank you.
Let f be amalytic in D = {z in C | |z| <= 10} with f(1) = 0, f"(0) = -i. Prove that there exists a z0 in D such that |f(z0)|= sqrt2009.
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 $\displaystyle f(1)=0$ and $\displaystyle f''(0)=-i$, f(z) is not a constant function and by Cauchy's inequality for the second derivative, $\displaystyle |f''(\gamma)|\leq \frac{2}{R^2}M$ where $\displaystyle \gamma$ is the contour $\displaystyle z=10e^{it}$ and $\displaystyle |f(\gamma)|\leq M$. This then would imply $\displaystyle 50\leq M$. Therefore, the maximum of $\displaystyle |f(z)|$ on $\displaystyle \gamma$ must be larger than $\displaystyle 50$ and $\displaystyle f(1)=0$ so that the maximum modulus between $\displaystyle r=1$ and $\displaystyle r=10$ is a continuous function of r at least in the range of $\displaystyle 1<\sqrt{2009}<50$.
Also, I thought it interesting to verify this directly with the function $\displaystyle f(z)=1/2 i z^3-i/2 z^2$ which satisfies $\displaystyle f(1)=0$ and $\displaystyle f''(0)=-i$ in which case the maximum modulus as a function of r is $\displaystyle r^2/2(r+1)$.