inequalities – Verification of inequality

I want to check an inequality in mathematica but I do not know where I was wrong. Please help me, thanks in advance.

$ psi (t): to[0infty)$[0infty)$[0infty)$[0infty)$ and $ phi (t): to[0infty)$[0infty)$[0infty)$[0infty)$

or;

$ psi (t) = 3/2 * t ^ 2 $ and $ phi (t) = t $

$ d (x, y) = | x-y | $ or $ x, y epsilon mathbb {R} $

The above are the conditions that I want to put on the given inequality

begin {equation} 3 * t ^ 2 * d (x, y) -3 / 2 * t ^ 2 * d (x, y)] leq (3/2 * t ^ 2 * d (x, y)) ^ alpha * (t * d (x, y)) ^ {1- alpha} end {equation}
or $ alpha epsilon (0,1) $