Mahler, using the GM-AM ineq, has that $$ {\prod_{}^k \ (x_i +y_i)}^{\dfrac{1}{k}} \ge \prod_{}^k \ (x_i)^{\dfrac{1}{k}} + \prod_{}^k \ (y_i)^{\dfrac{1}{k}} \quad\quad x_i,y_i>0 $$
For trying to compare $\prod_{}^k (x_i - 1)$ with $\prod_{}^k x_i$, and $y_i = 1, x_i := x_i - 1$ etc.. this gives
$$ {\prod_{}^k \ (x_i)}^{\dfrac{1}{k}} \ge \prod_{}^k \ (x_i - 1)^{\dfrac{1}{k}} + 1 $$ $$ {\prod_{}^k \ (x_i)} \ge ({\prod_{}^k \ (x_i - 1)^{\dfrac{1}{k}} + 1})^k $$ but the RHS seems messy due to the fractional exponents and I tried turning it into a binomial but that doesn't get rid of the exponents so it's not really more useful it seems.I've been trying to find other inequalities or results that are applicable. Have looked through lists like ineq trying to find anything but can't find anything useful. Is there anyone who might have some insight?