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''indicator of an entire function''
 
''indicator of an entire function''
  
 
The quantity
 
The quantity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453701.png" /></td> </tr></table>
+
$$
 +
h ( \phi )  = \
 +
\overline{\lim\limits}\; _ {r \rightarrow \infty } \
 +
 
 +
\frac{ \mathop{\rm ln}  | f ( r e ^ {i \phi } ) | }{r  ^  \rho  }
 +
,
 +
$$
  
characterizing the growth of an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453702.png" /> of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453703.png" /> and finite type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453704.png" /> along the ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453705.png" /> for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453706.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453707.png" />). For instance, for the function
+
characterizing the growth of an entire function $  f( z) $
 +
of finite order $  \rho > 0 $
 +
and finite type $  \sigma $
 +
along the ray $  \mathop{\rm arg}  z = \phi $
 +
for large $  r $(
 +
$  z = r e ^ {i \phi } $).  
 +
For instance, for the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453708.png" /></td> </tr></table>
+
$$
 +
f ( z)  = e ^ {( a - i b ) z  ^  \rho  }
 +
$$
  
the order is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g0453709.png" /> and the growth indicatrix is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537010.png" />; for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537011.png" /> the order is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537013.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537014.png" /> is everywhere finite, continuous, has at each point left and right derivatives, has a derivative everywhere except possibly at a countable number of points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537015.png" /> always and there is at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537016.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537017.png" />, has the characteristic property of trigonometric convexity, i.e. if
+
the order is $  \rho $
 +
and the growth indicatrix is equal to $  h ( \phi ) = a  \cos  \rho \phi + b  \sin  \rho \phi $;  
 +
for the function $  \sin  z $
 +
the order is $  \rho = 1 $
 +
and $  h ( \phi ) = | \sin  \phi | $.  
 +
The function $  h ( \phi ) $
 +
is everywhere finite, continuous, has at each point left and right derivatives, has a derivative everywhere except possibly at a countable number of points, $  h ( \phi ) \leq  \sigma $
 +
always and there is at least one $  \phi $
 +
for which $  h ( \phi ) = \sigma $,  
 +
has the characteristic property of trigonometric convexity, i.e. if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537018.png" /></td> </tr></table>
+
$$
 +
h ( \phi _ {1} )  \leq  H ( \phi _ {1} ) ,\ \
 +
h ( \phi _ {2} )  \leq  H ( \phi _ {2} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537019.png" /></td> </tr></table>
+
$$
 +
H ( \phi )  = a  \cos  \rho \phi + b  \sin  \rho \phi ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537020.png" /></td> </tr></table>
+
$$
 +
\phi _ {1}  < \phi _ {2} ,\  \phi _ {2} - \phi _ {1}  < \min  \left (
 +
\frac \pi  \rho
 +
, 2 \pi \right ) ,
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537021.png" /></td> </tr></table>
+
$$
 +
h ( \phi )  \leq  H ( \phi ) ,\ \
 +
\phi _ {1} \leq  \phi \leq  \phi _ {2} .
 +
$$
  
 
The following inequality holds:
 
The following inequality holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537022.png" /></td> </tr></table>
+
$$
 +
| f ( r e ^ {i \phi } ) |
 +
\leq  e ^ {[ h ( \phi ) + \epsilon ] {r  ^  \rho  } } ,\ \
 +
r > r _ {0} ( \epsilon ) ,\ \
 +
\textrm{ for  all  }  \epsilon > 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537023.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537024.png" />.
+
where $  r _ {0} ( \epsilon ) $
 +
is independent of $  \phi $.
  
 
The growth indicatrix is also introduced for functions that are analytic in an angle and in this angle are of finite order or have a proximate order and are of finite type.
 
The growth indicatrix is also introduced for functions that are analytic in an angle and in this angle are of finite order or have a proximate order and are of finite type.
Line 31: Line 84:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.Ya. Levin,  "Distribution of zeros of entire functions" , Amer. Math. Soc.  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.Ya. Levin,  "Distribution of zeros of entire functions" , Amer. Math. Soc.  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Indicators of entire functions of several variables have been introduced also; e.g. Lelong's regularized radial indicator:
 
Indicators of entire functions of several variables have been introduced also; e.g. Lelong's regularized radial indicator:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537025.png" /></td> </tr></table>
+
$$
 +
L  ^ {*} ( z)  = \
 +
\overline{\lim\limits}\; _ {w \rightarrow z } \
 +
\overline{\lim\limits}\; _ {t \rightarrow \infty } \
 +
 
 +
\frac{ \mathop{\rm ln}  | f ( t, w) | }{t  ^  \rho  }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537026.png" /> is an entire function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537027.png" /> and of finite type on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537029.png" />. (If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537030.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537031.png" />.)
+
where $  f $
 +
is an entire function of order $  \rho > 0 $
 +
and of finite type on $  \mathbf C  ^ {n} $,  
 +
$  z \in \mathbf C  ^ {n} $.  
 +
(If $  n = 1 $:  
 +
$  h ( \phi ) = L  ^ {*} ( e ^ {i \phi } ) $.)
  
The indicator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537032.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045370/g04537033.png" />-homogeneous [[Plurisubharmonic function|plurisubharmonic function]]. This corresponds with the convexity properties of the one-dimensional case. However, in general it is not a continuous function.
+
The indicator $  L  ^ {*} ( z) $
 +
is a $  \rho $-
 +
homogeneous [[Plurisubharmonic function|plurisubharmonic function]]. This corresponds with the convexity properties of the one-dimensional case. However, in general it is not a continuous function.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.I. Ronkin,  "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Lelong,  L. Gruman,  "Entire functions of several variables" , Springer  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.P. Boas,  "Entire functions" , Acad. Press  (1954)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.I. Ronkin,  "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Lelong,  L. Gruman,  "Entire functions of several variables" , Springer  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.P. Boas,  "Entire functions" , Acad. Press  (1954)</TD></TR></table>

Latest revision as of 19:42, 5 June 2020


indicator of an entire function

The quantity

$$ h ( \phi ) = \ \overline{\lim\limits}\; _ {r \rightarrow \infty } \ \frac{ \mathop{\rm ln} | f ( r e ^ {i \phi } ) | }{r ^ \rho } , $$

characterizing the growth of an entire function $ f( z) $ of finite order $ \rho > 0 $ and finite type $ \sigma $ along the ray $ \mathop{\rm arg} z = \phi $ for large $ r $( $ z = r e ^ {i \phi } $). For instance, for the function

$$ f ( z) = e ^ {( a - i b ) z ^ \rho } $$

the order is $ \rho $ and the growth indicatrix is equal to $ h ( \phi ) = a \cos \rho \phi + b \sin \rho \phi $; for the function $ \sin z $ the order is $ \rho = 1 $ and $ h ( \phi ) = | \sin \phi | $. The function $ h ( \phi ) $ is everywhere finite, continuous, has at each point left and right derivatives, has a derivative everywhere except possibly at a countable number of points, $ h ( \phi ) \leq \sigma $ always and there is at least one $ \phi $ for which $ h ( \phi ) = \sigma $, has the characteristic property of trigonometric convexity, i.e. if

$$ h ( \phi _ {1} ) \leq H ( \phi _ {1} ) ,\ \ h ( \phi _ {2} ) \leq H ( \phi _ {2} ) , $$

$$ H ( \phi ) = a \cos \rho \phi + b \sin \rho \phi , $$

$$ \phi _ {1} < \phi _ {2} ,\ \phi _ {2} - \phi _ {1} < \min \left ( \frac \pi \rho , 2 \pi \right ) , $$

then

$$ h ( \phi ) \leq H ( \phi ) ,\ \ \phi _ {1} \leq \phi \leq \phi _ {2} . $$

The following inequality holds:

$$ | f ( r e ^ {i \phi } ) | \leq e ^ {[ h ( \phi ) + \epsilon ] {r ^ \rho } } ,\ \ r > r _ {0} ( \epsilon ) ,\ \ \textrm{ for all } \epsilon > 0 , $$

where $ r _ {0} ( \epsilon ) $ is independent of $ \phi $.

The growth indicatrix is also introduced for functions that are analytic in an angle and in this angle are of finite order or have a proximate order and are of finite type.

References

[1] B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1980) (Translated from Russian)
[2] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)

Comments

Indicators of entire functions of several variables have been introduced also; e.g. Lelong's regularized radial indicator:

$$ L ^ {*} ( z) = \ \overline{\lim\limits}\; _ {w \rightarrow z } \ \overline{\lim\limits}\; _ {t \rightarrow \infty } \ \frac{ \mathop{\rm ln} | f ( t, w) | }{t ^ \rho } , $$

where $ f $ is an entire function of order $ \rho > 0 $ and of finite type on $ \mathbf C ^ {n} $, $ z \in \mathbf C ^ {n} $. (If $ n = 1 $: $ h ( \phi ) = L ^ {*} ( e ^ {i \phi } ) $.)

The indicator $ L ^ {*} ( z) $ is a $ \rho $- homogeneous plurisubharmonic function. This corresponds with the convexity properties of the one-dimensional case. However, in general it is not a continuous function.

References

[a1] L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , Transl. Math. Monogr. , 44 , Amer. Math. Soc. (1974) (Translated from Russian)
[a2] P. Lelong, L. Gruman, "Entire functions of several variables" , Springer (1986)
[a3] R.P. Boas, "Entire functions" , Acad. Press (1954)
How to Cite This Entry:
Growth indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Growth_indicatrix&oldid=13310
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article