We prove some new multidimensional Hardy-type inequalities involving general Hardy type operators with positive kernels for functions $\phi$ which may not necessarily be convex but satisfy the condition $A\psi(\x)\leq\phi(\x)\leq B\psi(\x)$, where $\psi $ is convex. Our approach is mainly the use of convexity argument and the results obtained are new even for the one-dimensional case and also unify and extend several inequalities of Hardy type known in the literature.