Let's suppose a function $f$ is analytic on $A_r := \{z\in\mathbb{C}\mid 1/r<|z|<r\}$ and $|f(z)|$ is bounded on $A_r$ by $M$. On a disk $B_r$, Cauchy's estimate would claim that, on a smaller disk $B_s$, $s<r$ the derivative satisfies $|f'(z)|\leq M/(r-s)$.
I'm trying to prove the same for the derivative of $f$ on a smaller annulus.
By the maximum modulus principle it suffices to check $|f'(z)|$ on the boundaries of the smaller annulus, so for $|z| = 1/s$ or $|z| = s$. The second case is easy, because that's just the normal application of Cauchy's estimate for disks.
However in the second case, the difference between the original and the new inner boundary isn't $r-s$, but instead $1/s-1/r$, which yields the estimate
$|f'(z)|\leq (Mrs)/(r-s)$ on the inner boundary.
Is there any known theorem to improve this or obtain the form with $(r-s)$ in the denominator and a constant in the numerator?
Thanks!