In the present paper, the authors have studied the nonexistence of positive solutions of the following integrodifferential inequalities \[ (-1)^{n+ 1}y^{(n)}(t)+ \int^t_0 f(t- s,y(s))ds\leq 0 \] on \([0,\infty)\) \((n = 1,2,4)\) assuming that the inequalities \[ -\lambda^n+ \int^\infty_0 e^{\lambda^s} K(s)ds> 0,\quad\text{for all }\lambda>0\quad (n= 1,2,4) \] hold, where \(f\) and \(K\) satisfy certain suitable conditions.