Growth indicatrix

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.

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:

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)