Regularity of quasi-symbolic and bracket powers of Borel type ideals

In this paper, we show that the regularity of the q-th quasi-symbolic power $I^{((q))}$ and the regularity of the $q$-th bracket power $I^{[q]}$ of a monomial ideal of Borel type $I$, satisfy the relations $reg(I^{((q))})\leq q \cdot reg(I)$, respectively $reg(I^{[q]})\geq q\cdot reg(I)$. Also, we give an upper bound for $reg(I^{[q]})$.