Does $E[X]>E[Y]$ imply $\operatorname E\left[\int_0^1 t^{X} \, dt \right]<E\left[\int_0^1 t^{Y} \, dt \right]$?

Question

Suppose $X$ and $Y$ are arbitrary random variables and $E[X]>E[Y]$ where $X>0$ and $Y>0$.

Then would the following inequality hold? $\operatorname E\left[\int_0^1 t^{X} \, dt \right]<E\left[\int_0^1 t^{Y} \, dt \right]$.

I tried to solve it using Jensen's inequality but Jensen's inequality only provides lower bound for LHS or RHS.