Explanation of homogenous function

Question

Is there someone, who can explain why the function $g(s)=f(e^s,e^s)$ is not homogeneous when it can be written as $\frac{9}{4}e^{s/2}s$.

I got the function $f(x,y)=\sqrt x +2\sqrt y +\frac{3y}{\sqrt x+\sqrt y}$.

Answer 1

Let's calculate it: $g(t s) = f(e^{t s},e^{t s}) = e^{\frac{t s}{2}} + 2 e^{\frac{t s}{2}} + \frac{3e^{t s}}{2e^{\frac{t s}{2}}} = \frac{9}{2}e^{\frac{t s}{2}} \not= t^n (\frac{9}{2}e^\frac{s}{2})$ for any $n$. Hence $g$ does not satisfy the definition: https://en.wikipedia.org/wiki/Homogeneous_function.