I'm working with some fluid mechanics problems, where I had to prove that a pipe with an elliptic cross-section has lower flow rate than an otherwise similar pipe of circular cross-section that has the same cross-sectional area (see problem 4 in the link for the interested: https://farside.ph.utexas.edu/teaching/336L/Fluid/node144.html)
The flow rate $Q$ for an ellipse is given by the expression:
$$Q_{\text{ellipse}} = \frac{\pi G}{4\mu}\frac{a^3b^3}{a^2+b^2}$$
So for a circle of radius $r$, the flow rate is given by just plugging in $r\equiv a=b$
$$Q_{\text{circle}} = \frac{\pi Gr^4}{8\mu}$$
In order to prove what is sought, I just have to prove that
$$ \frac{a^3b^3}{a^2+b^2} \leq \frac{r^4}{2}$$ under the constraint that $\pi ab = \pi r^2$. Rewriting the numerator on the LHS gives us
$$ \frac{r^6}{a^2+b^2}$$
where we are left with showing that $a^2+b^2 \geq 2r^2$. I'd be glad if anyone could provide any hints on how to proceed from here.