The Hölder condition for a function 
on an arc 
in the complex plane requires
that
 |
for all 
, with fixed positive real numbers 
and 
.
In some literature, functions 
satisfying the Hölder condition are said to be 
-Hölder continuous
on 
.
The quantities 
and 
are sometimes called the Hölder exponent and Hölder
constant of 
, respectively.
The Hölder condition comes up frequently in several branches of mathematics, notable among which is the study of Brownian motion in probability.
