How to study the sign of an expression including a function, with conditions on the function?

Question

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 example, I want to know when this expression:

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

is positive given that k > 0, x > 0 and that my function a is positive and increasing for all x > 0 (k is a constant, a is a function)

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

(* a''[x] < a'[x]^2/a[x] *)

Answer 1

Questions need to be answered, so I guess I will shoot:

Reduce[k (2 (a'[x])^2 - a[x] a''[x])/(a[x]^2) > 0 && k > 0]

$a'(x)\in \mathbb{R}\land \left(\left(a(x)<0\land a''(x)>\frac{2 a'(x)^2}{a(x)}\land k>0\right)\lor \left(a(x)>0\land a''(x)<\frac{2 a'(x)^2}{a(x)}\land k>0\right)\right)$

which is the result that you expected.

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Regarding your comment:

Is there any way I wouldn't have to repeat k>0 in every function but settle it beforehand?

You can do:

assum = k > 0;
Reduce[k (2 (a'[x])^2 - a[x] a''[x])/(a[x]^2) > 0 && assum]