On Fractal Peano Curves

It is shown that, for a fractal Peano curve $p(t)$ that maps a unit segment onto a unit square, there always exists a pair of points $t,t'$ of the segment that satisfy the inequality $|p(t)-p(t')|^2\ge 5|t-t'|$. As is clear from the classical Peano–Hilbert curve, the number $5$ in this inequality cannot be replaced by a number greater than $6$ (the result of K. Bauman).