Given $x,y,n\in\mathbb{N}^+$, and $x<y$, which quantity among
$$ u=\frac{x^n}{y^n}\,\,\,\text{ and }\,\,\,v=\frac{\binom{x+n-1}{n}}{\binom{y+n-1}{n}} $$
is greater?
This is probably a trivial question (sorry, in case), but I have troubles to discuss it rigorously. Thanks for your help!
As a sketch:
The left hand side is $$\frac x y \qquad\times\qquad \frac x y \qquad\times \cdots \times \quad\frac x y \quad\times \quad\frac x y$$
The right hand side is $$\frac{x+n-1}{y+n-1} \times \frac{x+n-2}{y+n-2} \times \cdots \times \frac{x+1}{y+1} \times \quad\frac x y\,\,$$
When $0 \le x \lt y$, the terms on the left hand side are smaller than the corresponding terms on the right hand side, apart for the last terms which are equal
So the left hand side is smaller than the right hand side when $n\gt 1$