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Define $M(r,F)$ as the maximum value of the entire function $F$ on $|z|=r$

Also define the function $log_n = log(log(log(...)))$ n times with base e.

Now,

$ lim_{n\rightarrow \infty}sup\dfrac{log_nM(r,F)}{logr} < \infty$

what is the general form of the entire functions not satisfying the condition above?

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I doubt that there can anything reasonable be said about them in a sense of Hadamards theorem etc.. What kind of statement would you be looking for? –  Marc Palm Feb 26 '13 at 11:24

2 Answers 2

These classes of functions can be characterized in terms of the rate of growth of Taylor coefficients: MR0218567

Šeremeta, M. N. Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion. (Russian) Izv. Vysš. Učebn. Zaved. Matematika 1967 1967 no. 2 (57), 100–108.

MR0330452 Šeremeta, M. N. The coefficients of the power series expansion of entire functions. (Russian) Teor. Funkciĭ Funkcional. Anal. i Priložen. No. 16 (1972), 41–44, 216.

If you prefer something using Latin alphabet, there is an old book

O. Blumenthal, Principes de la theorie des fonctions entieres d'ordre infini, Paris, Gauthier-Vilars, 1910.

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This is a "no-answer" answer. If for some $n$ the quantity $M_n:=\sup_{r\to\infty}\frac{\log_n(r,F)}{\log r}$ is finite then $M_{n+k}=0$ for all $k>0$. Therefore you are asking for entire functions with $M_n=\infty$ for every integer $n$. In particular $F$ is of infinite order, and you don't even have the existence of interresting "cercles de remplissage". In other words you're out naked in the woods, without a single classical theorem at your disposal (see the edit below). I'm not sure you can even give a single "explicit" example of such a $F$, let alone address your question in such generality. See also the answer given here to be convinced that your question is (unfortunately) way too broad.

Edit: As has been pointed out by Alexandre Eremenko, there is a classical theorem, but its practical usefulness is arguably limited. Writing $F(z)=\sum_{n=0}^{\infty}a_n z^n$ define for $r>0$ the maximum $B(r):=\max_{n\in\mathbb N} |a_n|r^n$ and let $\mu(r)$ be the biggest rank for which this value is reached. Then

$B(r)\leq M(r,F)\leq B(r)(1+2\mu\(r+\frac{r}{\mu(r)}\))$

There is also the formula

$\ln B(r) - \ln B(r') = \int_{r'}^r \frac{\mu(u)}{u}\mathrm{d}u$

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I agree that these results are of limited practical use. And the question itself too. –  Alexandre Eremenko Feb 27 '13 at 2:04
    
I concour to both your observations… –  Loïc Teyssier Feb 28 '13 at 20:08

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