I am trying to find if the function $4u^4x^{2.5}y^{-5}$ is convex over $u>0, y>0$ and $x>0$.
The thing which comes to my mind immediately is to check the positive semi-definiteness of the Hessian over the domain and I tried finding the hessian using MATLAB. It's very complicated and i don't think it is that complicated. Should I use the basic definition of convexity?($f(\lambda x+ (1-\lambda x) <= \lambda f(x) + (1-\lambda) f(x))$
Function defined by $f(x,y,u) \ = \ 4 x^{5/2}y^{-5}u^4$ is NOT convex on this domain, i.e.,
$$f(\lambda X_1+ (1-\lambda) X_2) \ \ \leq \ \ \lambda f(X_1) + (1-\lambda) f(X_2) \ \ \ \ \ (*)$$
is false for certain points and certain values of $\lambda \in (0,1)$.
It suffices for example to take the midpoint ($\lambda=1/2$) of
$$X_1=(x_1,y_1,u_1)=(0.5,1,1) \ \ \ \ and \ \ \ \ X_2=(x_2,y_2,u_2)=(1,1,0.5) \ \ \ (*)$$
The value of the left side of (*) is $0.61\cdots$ whereas the value on the right is $0.47\cdots$.
These cases are not rare, but aren't dominant either: random simulations in the limited range $(0,1)\times(0,1)\times(0,1)$ show that (*) is not true for about 1 out of 6 cases .