Conditional Second-order Derivative and Degree of Homogeneity

I have a function with first-order derivatives and second-order derivatives .

Additionally I know that the degree of homogeneity of the function is larger than but smaller than .

In an economics paper I found the statement that given these assumptions the derivative of with regard to is negative if the first-order derivative with regard to is held constant:

Why is this so?

I understand that I can decompose into

Now, I would have to show that but I don't know how to do that using and the information about the degree of homogeneity.

Re: Conditional Second-order Derivative and Degree of Homogeneity

Just a small correction. It should read:

I know that I can decompose into ...