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The basic form of the theory of regular variation, a subject initiated in 1930 by the Yugoslav mathematician J. Karamata.

Viewed from a modern perspective, Karamata theory is the study of asymptotic relations of the form

$$ \tag{a1 }
{
\frac{f ( \lambda x ) }{f ( x ) }
 } \rightarrow g ( \lambda ) \in ( 0, \infty )  ( x \rightarrow \infty ) ,  \forall \lambda > 0,
$$

together with their consequences and ramifications. The case  $  g \equiv 1 $
is particularly important; measurable functions  $  f $
satisfying (a1) with  $  g \equiv 1 $
are called slowly varying; such slowly varying functions are often written  $  L $
or  $  {\mathcal l} $(
for  "lente" ).

Many useful and interesting properties are implied by such relations. For instance:

i) The uniform convergence theorem: for  $  f $
slowly varying, (a1) holds uniformly on compact  $  \lambda $-
sets in  $  ( 0, \infty ) $. 
There is a topological analogue, with measurability replaced by the [[Baire property|Baire property]].

ii) The representation theorem:  $  f $
is slowly varying if and only if, for  $  x $
large enough,  $  f $
is of the form

$$ 
f ( x ) = c ( x ) { \mathop{\rm exp} } \left ( \int\limits _ { a } ^ { x }  {\epsilon ( u ) }  { {
\frac{du }{u}
 } } \right ) ,
$$

where  $  c ( \cdot ) $, 
$  \epsilon ( \cdot ) $
are measurable,  $  c ( x ) \rightarrow c \in ( 0, \infty ) $, 
$  \epsilon ( x ) \rightarrow 0 $
as  $  x \rightarrow \infty $.

iii) The characterization theorem: for measurable  $  f $, 
$  g ( \lambda ) $
in (a1) must be of the form  $  g ( \lambda ) \equiv \lambda  ^  \rho  $
for some  $  \rho \in \mathbf R $, 
called the index of regular variation:  $  f \in R _  \rho  $. 
Then  $  f ( x ) = x  ^  \rho  {\mathcal l} ( x ) $
with  $  {\mathcal l} $
slowly varying ( $  {\mathcal l} \in R _ {0} $).

iv) Karamata's theorem: if  $  f \in R _  \rho  $
and  $  \sigma > - ( \rho + 1 ) $, 
then

$$ \tag{a2 }
{
\frac{x ^ {\rho + 1 } f ( x ) }{\int\limits _ { a } ^ { x }  {t  ^  \sigma  f ( t ) }  {dt } }
 } \rightarrow \sigma + \rho + 1   ( x \rightarrow \infty ) .
$$

(That is, the  $  {\mathcal l} $
in  $  f ( x ) = x  ^  \rho  {\mathcal l} ( x ) $"
behaves asymptotically like a constant"  under integration.) Conversely, (a2) implies  $  f \in R _  \rho  $.

Perhaps the most important application of Karamata theory to analysis is Karamata's Tauberian theorem (or the Hardy–Littlewood–Karamata theorem): if  $  f \in R _  \rho  $(
$  \rho \geq  0 $) 
is increasing, with Laplace–Stieltjes transform  $  {\widehat{f}  } ( s ) = \int _ {0}  ^  \infty  {e ^ {- sx } }  {df ( x ) } $, 
then  $  f ( x ) \sim c { {x  ^  \rho  {\mathcal l} ( x ) } / {\Gamma ( 1 + \rho ) } } $(
$  x \rightarrow \infty $) 
with  $  c \geq  0 $, 
$  {\mathcal l} \in R _ {0} $
if and only if  $  {\widehat{f}  } ( s ) \sim c s ^ {- \rho } {\mathcal l} ( {1 / s } ) $
$  ( s \downarrow 0 ) $.

For details, background and references on these and other results, see e.g. [[#References|[a1]]], Chap. 1.

The union over all  $  \rho \in \mathbf R $
of the classes  $  R _  \rho  $
gives the class  $  R $
of regularly varying functions. This is contained in the larger class  $  ER $
of extended regularly varying functions, itself included in the class  $  OR $
of  $  O $-
regularly varying functions:  $  R \subset  ER \subset  OR $. 
Just as a function  $  f \in R $
has an index  $  \rho $
of regular variation, and then  $  f \in R _  \rho  $, 
so a function  $  f \in ER $
has a pair  $  c ( f ) , d ( f ) $
of upper and lower Karamata indices (and these are equal, to  $  \rho $
say, if and only if  $  f \in R _  \rho  $), 
and a function  $  f \in OR $
has a pair  $  \alpha ( f ) , \beta ( f ) $
of upper and lower Matuszewska indices. These larger classes  $  ER $, 
$  OR $
have analogues of the results above; for instance, uniform convergence and representation theorems. For details, see e.g. [[#References|[a1]]], Chap. 2.

Karamata theory may be regarded as the  "first-order"  theory of regular variation. There is a corresponding  "second-order"  theory: [[De Haan theory|de Haan theory]] [[#References|[a1]]], Chap. 3.

Karamata theory has found extensive use in several areas of analysis, such as Abelian, Tauberian and Mercerian theorems ([[#References|[a1]]], Chap. 4, 5; cf. also [[Tauberian theorems|Tauberian theorems]]; [[Mercer theorem|Mercer theorem]]; [[Abel theorem|Abel theorem]]) and the Levin–Pfluger theory of completely regular growth of entire functions ([[#References|[a1]]], Chap. 6; cf. also [[Entire function|Entire function]]), and is also useful in asymptotic questions in [[Analytic number theory|analytic number theory]] [[#References|[a1]]], Chap. 7. It has been widely used also in [[Probability theory|probability theory]], following the work of W. Feller [[#References|[a2]]]; [[#References|[a1]]], Chap. 8.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.H. Bingham, C.M. Goldie, J.L. Teugels, "Regular variation", ''Encycl. Math. Appl.'', '''27''', Cambridge Univ. Press (1989) (Edition: Second)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Springer (1976) (Edition: Second)</TD></TR></table>