GM-AM-QM inequalities for complex numbers?! To my surprise I could not find the HM-GM-AM-QM inequalities for complex numbers! ( inequality between Arithmetic-Geometric and Quadratic mean for complex numbers)
Can anyone help me finding them? I am specially interested in inequalities between quadratic mean and arithmetic means of $n$ complex numbers. 
Thanks.
 A: A comment that got too long to try and clarify the problem - 
the paper linked optimizes an inequality of the type $(\Pi_{k=1}^n |z_k|)^{1/n} \le C(n,\phi)|\sum_{k=1}^n {\frac{z_k}{n}}|$ in the case $|\arg z_k| \le \phi < \pi/2$ (which is a necessary condition for such inequalities since otherwise, the sum can just cancel)
However it is not hard to prove that when the arguments are like that, $|\sum {\frac{z_k}{n}}| \ge A(\phi)\sum {\frac{|z_k|}{n}}$, so the usual GM-AM inequality can be applied and the question reduces to the optimization of the final constant; in your case, it's not clear what you want qualitatively as again with appropriate argument bounds one can get reverse triangle inequalities with some constant so one can apply the usual AG-QM inequality etc
For example if $|\arg z_k| \le \psi < \pi/4$ there is a constant $A(\psi)$ (for example $\cos (2\psi)$ works) st $|\sum \frac{z_k^2}{n}| \ge A(\psi)(\sum \frac{|z_k^2|}{n}) \ge A(\psi)(\sum \frac{|z_k|}{n})^2 \ge A(\psi)|\sum \frac{z_k}{n}|^2$ and again the argument condtion is necessary since otherwise one can easily make $\sum \frac{z_k^2}{n}=0$ and then one can look into how to optimize $A(\psi)$ for the final inequality etc, but not sure exactly what the problem here requires
Edit later - to prove the above inequality one notices that for $|\arg z_k| \le \psi < \pi/4$ we get $|\arg z_k^2| \le 2\psi < \pi/2$, so $\Re z_k^2=|z_k^2|\cos \arg z_k^2 \ge |z_k|^2 \cos 2\psi$ so $|\sum \frac{z_k^2}{n}|\ge \sum \frac{\Re z_k^2}{n} \ge \cos 2\psi(\sum \frac{|z_k^2|}{n})$ so we can take $A(\psi)=\cos 2\psi$ as noted and then 
$|\sum \frac{z_k}{n}|^2 \le \frac{1}{\cos 2\psi}|\sum \frac{z_k^2}{n}|$
Note that as per the paper quoted in the Op, there is a chance that the constant can be optimized further
