The Modified Heinz’s Inequality

In my paper [1], we aimed to determine the best possible range of r such that the modified Heinz’s inequality holds for any bounded linear operators A and B on a Hilbert space H such as and for any given a and b such as a>0 and b>0. But the counter-examples prepared in [1] and also in [2] were not sufficient and, in this paper, we shall constitute the sufficient counter-examples which will satisfy all the lacking parts.