  For an irrational number $\alpha\in\mathbb{R}$ we consider its irrationality
measure function $$ \psi_\alpha(x) = \min_{1\le q\le x,\, q\in\mathbb{Z}} \|
q\alpha \|. $$ It is known for all irrational numbers $\alpha$ and $\beta$
satisfying $\alpha\pm\beta\not\in\mathbb{Z}$, there exist arbitrary large
values of $t$ with \begin{equation*}
  | \psi_\alpha(t) - \psi_\beta(t) | \geqslant \left( \sqrt{\tau} - 1\right)
\cdot \min( \psi_\alpha(t), \psi_\beta(t) ), \end{equation*} where $\tau =
\frac{\sqrt{5} + 1}{2}$ and this result is optimal for certain numbers
equivalent to $\tau$. Here we prove that for all irrational numbers $\alpha$
and $\beta$, satisfying $\alpha\pm\beta\not\in\mathbb{Z}$, such that at least
one of them is not equivalent to $\tau$, there exist arbitrary large values of
$t$ with $$ | \psi_\alpha(t) - \psi_\beta(t) | \geqslant
(\sqrt{\sqrt2+1}-1)\cdot \min( \psi_\alpha(t), \psi_\beta(t) ). $$ Moreover, we
show that the constant on the right-hand side is optimal.
