# 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.

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