Namespaces
Variants
Actions

Growth indicatrix

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


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=47146
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