Find maximum value of $(ab+1)(bc+1)(cd+1)(da+1)$ if $abcd=1$ and $\frac{1}{2}≤a,b,c,d≤2$
$$ab+cd≥2(abcd)^{1/2}=2$$ $$da+bc≥2(abcd)^{1/2}=2$$ $$(ab+1)(bc+1)(cd+1)(da+1)≤(\frac{ab+1+bc+1+cd+1+da+1}{4})^4$$ $$\implies ≤(\frac{2+2+4}{4})^4=16$$
However, $(2,2,1/2,1/2)$ gives us $25$, which is clearly greater than $16$. Where is the mistake in my solution? (I already have a different solution that gives the right answer of 25 but if someone has a method similar to my approach which does produce the correct answer, then please share)
The direction of your second inequlity sign in the followilng should be reversed
$$(ab+1)(bc+1)(cd+1)(da+1)≤(\frac{ab+1+bc+1+cd+1+da+1}{4})^4$$ $$\implies ≤(\frac{2+2+4}{4})^4=16$$
Thus your implication is not valid.