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)

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)