I would like to show the inequality (for $n\geq 4$ and $y>1$) below.
$$\frac{x+1}{x-3}\left(\frac{1}{x}\right)^{\frac{y}{y-1}}\left(-1+y\left(1-\left(\frac{1}{x}\right)^{\frac{1}{y-1}}\right)\right)<\frac{1}{2}$$
I already know that it is analytically true for $x=4$ and numerically I can be pretty sure that it is true for all $x\geq 4$ and $y>1$ . I tried to show that the partial derivative for $x$ is negative and the same for $y$, but everything without success. Do you have any ideas?