I'm trying to study the sign of an expression, having information on some of my parameters. I don't know how to code these conditions. For exemple, I want to know when this expression-

k [2(a'[x])^2 - a[x]a"[x]] / (a[x]^2) 

-is positive, knowing that k>0, x>0 and that my function "a" is positive and increasing for all x>0. 

So basically, since a"[x] is the only term of the equation that I don't know anything about, I want Mathematica to gives me:

Out= a"[x] < a'[x]^2/a[x]

Thank you so much for your help,

PS: Alexei Boulbitch thank you for your answer, I hope it's clearer this time. I clearly have to get better at formulating programming questions!