Need to find Closed form solution to interesting problem

Question

$A= (1+x)(1+x-y)^{1/y}$

$B= x(1+x)^{1/y}$

for $y\in(0,1)$ and $x > 0$;

I want to show that A > B, but could not prove it mathematically. I run a simulation and it shows that in fact A > B, but I need closed form solution. Will be glad if you help me.

Answer 1

Hint: Apply Bernoulli's inequality to $$ \frac AB = \frac{1+x}{x} \left( 1- \frac{y}{1+x}\right)^{1/y} $$