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](https://mathematica.stackexchange.com/questions/47156/how-to-study-the-sign-of-an-expression-including-a-function-with-conditions-on#comment144011_47156):

>  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]