$\frac{1}{\frac{1}{a}+\frac{1}{b}}+\frac{1}{\frac{1}{c}+\frac{1}{d}} \leq \frac{1}{\frac{1}{a+b}+\frac{1}{c+d}}$
I think it has something to do with Harmonic mean, but can't fighre it out.
I think it has something to do with Harmonic mean, but can't fighre it out.
This is wrong. For example, take $a=b=2$, $c=d=1$
It should be $\frac{1}{\frac{1}{a}+\frac{1}{b}}+\frac{1}{\frac{1}{c}+\frac{1}{d}} \leq \frac{1}{\frac{1}{a+d}+\frac{1}{b+c}}$ for positives $a$, $b$ $c$ and $d$, which is $(ac-bd)^2\geq0$.