Q:

Proof of algebraic inequality with restrictions on the variables

How to prove the following inequality?
$$
(1-y)(x^y) + y(x^{y-1}) -1 > 0 \quad x>1,\quad  0 < y < 1
$$

A:

Using weighted AM-GM, 
$$(1-y)x^y + yx^{y-1} \geqslant x^{y(1-y)}\cdot x^{(y-1)y}=1$$
As equality is not possible (requires $x^y = x^{y-1}$), the inequality is strict.

