Connection between $\mu(r)$, $\nu(r)$, $M(r)$ and $N(r)$

For an entire function $f(z)$ let

The convergence of the series

is assured, excluded or undecided according as

For the Weierstrass $\sigma (z)$ function, in the case when the period parallelogram is a square with sides of length $1$,

as $r$ tends to $\infty$.

There is an entire function for which

for $n = 1, 2, 3, \ldots$ (see Problem 50 of Part I). Construct an entire function for which

for $n = 1, 2, 3, \ldots$.

For the entire functions

determine the limits of the quotients

as $r$ (or $n$ in the first quotient) tends to infinity, and collect the results in a table.

For any entire function

(it is obvious that the inequality $\leq$ holds). The common value of these two limits is called the order of the entire function.

The order of an entire function may also be defined as the exponent of convergence (this is the notation of Part I, Chapter 3, Section 2) of the sequence $\rho_{1}, \rho_{2}, \ldots , \rho_{n}, \ldots$.

The order of an entire function $f(z) = \sum_{n=0}^{\infty } a_{n} z^{n}$ with positive coefficients $a_{0}, a_{1}, a_{2}, \ldots , a_{n}, \ldots$ may also be defined as the following limit:

For entire functions of finite order we have the following result (stronger than problem 51):

For transcendental entire functions the integral

is convergent for $l > 1$.

For any entire function, given any arbitrary $\varepsilon$, $\varepsilon > 0$, there exist arbitrarily large values of $r$ such that

For an entire function of finite order $\lambda$ we have the following result (stronger than problem 56): given any arbitrary $\varepsilon$, $\varepsilon > 0$, there exist arbitrarily large values of $r$ such that

The order of an entire function of genus zero is equal to the exponent of convergence of its zeros (see Volume I, pp. 25-26, Titchmarsh p. 250, Hille Vol. II, p. 188).

For an entire function of finite order $\lambda$, $\lambda > 0$, whose central index has the jump-points $\rho_{1}, \rho_{2}, \rho_{3}, \ldots , \rho_{n}, \ldots$ which are regularly spaced in the sense of Chapter 4, Section 1 of Part II, we have

For entire functions of finite order $\lambda$, $\lambda > 0$, without any special conditions of regularity we have

Let $\lambda$ be the exponent of convergence of the sequence $r_{1}, r_{2}, r_{3}, \ldots , r_{n}, \ldots$, which is regularly spaced in the sense of Chapter 4, Section 1 of Part II, and let $0 < \lambda < 1$. For the entire function of genus zero

we have

if $\vartheta$ is a fixed number, $-\pi < \vartheta < \pi$.

For the entire function discussed in problem 61 we have, even without assuming the regular spacing of the sequence $r_{1}, r_{2}, r_{3}, \ldots , r_{n}, \ldots$,

For an entire function whose zeros have a finite exponent of convergence $\lambda$

Let $f(z)$ be an entire function and let $\mathfrak {G}(r)$ and $\mathfrak {g}(r)$ be defined as the geometric means of $\left|f(z)\right|$ on the circle $\left|z\right| = r$, and in the circular disc $\left|z\right| \leq r$, respectively (this is the notation of Problem 121 of Part III). If the moduli of the zeros of $f(z)$ are regularly spaced (this is the notation of Chapter 4, Section 1 of Part II), then

where $\lambda$, $0 < \lambda < \infty$, denotes the exponent of convergence of the zeros. For polynomials the above limit also exists and equals $e^{-1/2}$.

(Continued from problem 64.) Without assuming the regular spacing of the zeros we have

Let $f(z)$ be an entire function of finite order $\lambda$ and $\mathfrak {M}(r)$ and $\mathrm{m}(r)$ the arithmetic means of $\left|f(z)\right|^{2}$ on the circle $\left|z\right| = r$ and in the circular disc $\left|z\right| \leq r$ respectively. Then