  For equations $ x'(t) = -x(t) + \zeta f(x(t-h)), x \in \R, f'(0)= -1, \zeta >
0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and
satisfies $xf(x) < 0$ for $x\not=0$, we prove convergence of all solutions to
zero when both $\zeta -1 >0$ and $h(\zeta-1)^{1/8}$ are less than some constant
(independent on $h,\zeta$). This result gives additional insight to the
conjecture about the equivalence between local and global asymptotical
stabilities in the Mackey-Glass type delay differential equations.
