# Growth indicatrix

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indicator of an entire function

The quantity characterizing the growth of an entire function of finite order and finite type along the ray for large ( ). For instance, for the function 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   then The following inequality holds: 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.

How to Cite This Entry:
Growth indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Growth_indicatrix&oldid=13310
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