Solution of the inequality $2ec{\sqrt{ad}}\lt dc^2+ae^2$

Question

I just want to find a way to prove that the inequality $2ec{\sqrt{ad}}\lt dc^2+ae^2$ is true because I need it for a prove. Thanks for your help!

Answer 1

Instead of starting with a magic inequality, here is a derivation:

$\begin{array}\\ 2ec{\sqrt{ad}}\lt dc^2+ae^2\\ \iff\\ 4e^2c^2ad\lt d^2c^4+2dc^2ae^2+a^2e^4 \quad\text{(squaring both sides)}\\ \iff\\ 0\lt d^2c^4-2dc^2ae^2+a^2e^4 \quad\text{(subtracting }4e^2c^2ad\text{ from both sides)}\\ \iff\\ 0\lt (dc^2-ae^2)^2 \quad\text{(noticing the factorization)}\\ \end{array} $