What is the inequality which is used to prove this inequality?

Question

Let $x,y,z,t$ be real numbers such that $x,y,z,t\geq 1$ and $xyzt=16$. How to prove

$$x-\frac{1}{x}+y-\frac{1}{y}+z-\frac{1}{z}+t-\frac{1}{t}\geq6$$

I want some hint. thank you very much

Answer 1

Consider $a=\log_2 x, b = \log_2 y, c = \log_2 z, d = \log_2 t$. In these variables, we have to show that for $a, b, c, d \ge 0$ and $a+b+c+d = 4 $, $$\sum (2^a-2^{-a}) \ge 6$$

This follows from applying Jensen's inequality to the convex function $2^a-2^{-a}$.

Answer 2

Hint: Assume two of the numbers are unequal, say $x$ and $y$. Show that you can replace $x$ and $y$ each with $\sqrt{xy}$ and your quantity will decrease. This effectively shows that the minimum occurs when all 4 variables are equal, i.e. equal to $2$.