Please think it easy because it is not an assignment.
I'm trying to show the following problem.
Show that the inequality $$ 12(x^{5/2}+y^{5/2})+15(\sqrt{x}y^{2}+x^{2}\sqrt{y})>25(\sqrt{x}+\sqrt{y})xy+18(x-y)^{5/2} $$ holds on $\{(x,y)\in[0,1]^{2}\mid\frac{2}{3}x<y\le x\}$.
For the function $$ f(x,y)=12(x^{5/2}+y^{5/2})+15(\sqrt{x}y^{2}+x^{2}\sqrt{y})-25(\sqrt{x}+\sqrt{y})xy-18(x-y)^{5/2}, $$ I could prove that it is positive on $y=x$ and $y=\frac{2}{3}x$, and also know numerically that this function is positive on the above set. I however can not give a rigorous proof. It seems to be no use only by evaluating it below simply.
I'm glad if you give me a hint or solution.
Thank you in advance.
Progress :
By setting $z=\frac{y}{x}$, I know that the above inequality is reduced to $$ 12(z^{5/2}+1)+15(z^{2}+\sqrt{z})>25(\sqrt{z}+1)z+18(1-z)^{5/2} $$ on $\{z\in(0,1]\mid\frac{2}{3}<z\le 1\}$.