Growth indicatrix

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)

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