Logarithm inequality including square root:minimum of $\sqrt{\log_2^2a+1}+\sqrt{\log_2^2b+1}+\sqrt{\log_2^2c+1}$

Question

I just encountered this problem and tried a few ways but seems like it didn't work. Can some one help me?

Edited

I forgot to including what i have tried.

I tried to take the logarithm of both side of the hypothesis and then replace them in P, but it can not reduce the variables.

I also tried to calculate $P^2$, but did not seem to work out well though.

Let $a,b,c>0$ such that $abc=8$, find \begin{align*} \min P= \sqrt{\log_2^2a+1}+\sqrt{\log_2^2b+1}+\sqrt{\log_2^2c+1}. \end{align*} Any helps is appreciated, thanks in advance!

Answer 1

let $\log_2a=x,\log_2b=y,\log_2c=z$

we have $x+y+z=3$

we have to minimise $$P=\sum_{cyc}\sqrt{x^2+1}$$ by C-S $$\sqrt{x^2+1}\ge \frac{|x+1|}{\sqrt{2}}$$ Thus by C-S along with triangle ineqality$$\sum_{cyc}\sqrt{x^2+1}\ge \sum \frac{1}{\sqrt{2}}|x+1|\ge \frac{1}{\sqrt{2}}|x+1+y+1+z+1|=?$$