<s>[INST] Prove that $\displaystyle\left|\int_L \frac{z^3}{z^2+1} dz\right| \le \frac{9\pi}{8}$ where $L=\big(z: |z|=3, Re(z) \ge 0\big)$ I tried solving this by using the fact that the length of L is half the circumference of a circle with radius 3, which would mean that
if
$$\left|\frac{z^3}{z^2+1}\right| \le \frac{3}{8}$$ this problem is solved.
however, It appears to me as though this cannot be true:
$$\left|\frac{z^3}{z^2+1}\right| = \frac{\left|z\right|^3}{|z^2+1|} \ge \frac{27}{10}$$
using the triangle inequallity.
Is there something wrong with the way I'm going about this or is the statement really false?
 [/INST] I believe the inequality you stated is not true. Firstly, if $|z|=3$, by the reverse triangle inequality we get that $|z^2+1|=|z^2-(-1)| \geq | |z^2|-1| = |3^2-1| = 8$. Taking reciprocals gives us that $\dfrac{1}{|z^2+1|} \leq \dfrac{1}{8}$.
If we multiply both sides of the inequality by $|z^3|$, we obtain for $|z|=3$ that
$$\left|\frac{z^3}{z^2+1}\right|\leq \dfrac{27}{8}$$
Now if we apply standard integral estimates, we get $\displaystyle\left|\int_L \dfrac{z^3}{z^2+1}dz\right| \leq \dfrac{27}{8}\cdot\operatorname{length}(L)$. As you said, the length of $L$ is half the circumference of a circle with radius $3$. That is, $\operatorname{length}(L) = \dfrac{6\pi}{2} = 3\pi$.
Thus, $$\displaystyle\left|\int_L \dfrac{z^3}{z^2+1}dz\right| \leq \dfrac{27}{8}\cdot3\pi= \dfrac{81\pi}{8} $$
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