  Threshold phenomena are investigated using a general approach, following
Talagrand [Ann. Probab. 22 (1994) 1576--1587] and Friedgut and Kalai [Proc.
Amer. Math. Soc. 12 (1999) 1017--1054]. The general upper bound for the
threshold width of symmetric monotone properties is improved. This follows from
a new lower bound on the maximal influence of a variable on a Boolean function.
The method of proof is based on a well-known logarithmic Sobolev inequality on
$\{0,1\}^n$. This new bound is shown to be asymptotically optimal.
