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The Hardy–Littlewood theorem in the theory of functions of a complex variable: If  $  a _ {k} \geq  0 $, 
$  k = 0, 1 \dots $
and if the power series

$$ 
f ( z)  = \ 
\sum _ {k = 0 } ^  \infty  
a _ {k} z  ^ {k}
$$

with radius of convergence 1 satisfies on the real axis the asymptotic equality

$$ 
f ( x)  = \ 
\sum _ {k = 0 } ^  \infty  
a _ {k} x  ^ {k}  \sim \ 

\frac{1}{1 - x }
 ,\ \ 
x \uparrow 1,
$$

then the partial sums  $  s _ {n} $
satisfy the asymptotic equality

$$ 
s _ {n}  = \ 
\sum _ {k = 0 } ^ { n } 
a _ {n}  \sim  n,\ \ 
n \rightarrow \infty .
$$

This theorem was established by G.H. Hardy and J.E. Littlewood [[#References|[1]]] and is one of the [[Tauberian theorems|Tauberian theorems]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   "Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive"  ''Proc. London. Math. Soc. (2)'' , '''13'''  (1914)  pp. 174–191</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.C. Titchmarsh,   "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR></table>

''E.D. Solomentsev''

The Hardy–Littlewood theorem on a non-negative summable function. A theorem on integral properties of a certain function connected with the given one. It was established by G.H. Hardy and J.E. Littlewood [[#References|[1]]]. Let  $  f $
be a non-negative summable function on  $  [ a, b] $, 
and let

$$ 
\theta ( x)  = \ 
\theta _ {f} ( x)  = \ 
\sup _ {\begin{array}{c}
\xi \in [ a, b] \\
 \xi \neq x 
\end{array}
 } \ 

\frac{1}{x - \xi }

\int\limits _  \xi  ^ { x } 
f ( t)  dt.
$$

Then:

1) If  $  f \in L _ {p} ( a, b) $, 
$  1 < p < \infty $, 
then

$$ 
\int\limits _ { a } ^ { b } 
\theta  ^ {p} ( x) \ 
dx  \leq   2
\left (

\frac{p}{p - 1 }

\right )  ^ {p}
\int\limits _ { a } ^ { b } 
f ^ { p } ( x)  dx.
$$

2) If  $  f \in L _ {1} ( a, b) $, 
then for all  $  \alpha \in ( 0, 1) $,

$$ 
\int\limits _ { a } ^ { b } 
\theta  ^  \alpha  ( x)  dx  \leq  \ 

\frac{2 ( b - a) ^ {1 - \alpha } }{1 - \alpha }

\int\limits _ { a } ^ { b }  f ( x)  dx.
$$

3) If  $  f   \mathop{\rm ln}  ^ {+}  f \in L _ {1} ( a, b) $, 
then

$$ 
\int\limits _ { a } ^ { b } 
\theta ( x)  dx  \leq   4
\int\limits _ { a } ^ { b } 
f ( x)   \mathop{\rm ln}  ^ {+}  f ( x)  dx + A,
$$

where  $  A $
depends only on  $  b - a $. 
Here

$$ 
 \mathop{\rm ln}  ^ {+}  u  = \ 
\left \{ 
\begin{array}{ll}
 0  & \textrm{ if }  u < 1,  \\
 \mathop{\rm ln}  u  & \textrm{ if }  u \geq  1.  \\
\end{array}

 \right .$$

Let  $  f $
be a  $  2 \pi $-
periodic function that is summable on  $  [- \pi , \pi ] $, 
and let

$$ 
M ( x)  = \ 
M _ {f} ( x)  = \ 
\sup _ {0 < | t | \leq  \pi } \ 
{
\frac{1}{t}
 }
\int\limits _ { x } ^ { {x }  + t }
| f ( u) |  du.
$$

Then  $  M _ {f} ( x) \leq  \theta _ {| f | }  ( x) $, 
where  $  \theta _ {| f | }  ( x) $
is constructed for  $  [- 2 \pi , 2 \pi ] $. 
From the theorem for  $  \theta $
one obtains integral inequalities for  $  M $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   "A maximal theorem with function-theoretic applications"  ''Acta. Math.'' , '''54'''  (1930)  pp. 81–116</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>

''A.A. Konyushkov''

====Comments====
The function  $  M _ {f} $
is called the Hardy–Littlewood maximal function for  $  f $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.M. Stein,   G. Weiss,   "Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>