Growth indicatrix
indicator of an entire function
The quantity
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characterizing the growth of an entire function of finite order
and finite type
along the ray
for large
(
). For instance, for the function
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the order is and the growth indicatrix is equal to
; for the function
the order is
and
. The function
is everywhere finite, continuous, has at each point left and right derivatives, has a derivative everywhere except possibly at a countable number of points,
always and there is at least one
for which
, has the characteristic property of trigonometric convexity, i.e. if
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then
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The following inequality holds:
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where is independent of
.
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:
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where is an entire function of order
and of finite type on
,
. (If
:
.)
The indicator is a
-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) |
Growth indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Growth_indicatrix&oldid=13310