On the Minimum Modulus of Analytic Functions of Moderate Growth in the Unit Disc

We study the behavior of the minimum modulus of analytic functions in the unit disc in terms of $$\rho _\infty $$ρ∞-order, which is the limit of the orders of $$L_p$$Lp-norms of $$\log |f(re^{i\theta })|$$log|f(reiθ)| over the circle as $$p\rightarrow \infty $$p→∞. This concept coincides with the usual order of the maximum modulus function if the order is greater than one. New results are obtained for analytic functions of order smaller than 1.


Introduction and Main Results
Let D R = {z ∈ C : |z| < R}, 0 < R ≤ ∞, and D = D 1 . For an analytic function f on D R , we define the minimum modulus μ(r, f ) = min{| f (z)| : |z| = r }, 0 < r < R, Interplay between μ(r, f ) and M(r, f ) has been studied in a large number of papers. In the case of entire functions, i.e., R = ∞, a survey of results up to 1989 can be found in Hayman's book ( [16,Chap. 6]).
The orders of the growth of an analytic function f in D ∞ , and in D, respectively, are defined as For entire functions of order ρ[ f ] ≤ 1, there are a lot of sharp results on the behavior such as cos πρ-theorem ( [1,16]).
Theorem ( [1]) Suppose that 0 ≤ ρ < α < 1. If f is an entire function of order ρ and f (z) ≡ const then One of the most interesting open problems for entire functions of order greater than 1 is to find the asymptotic behavior of the minimum modulus with respect to the maximum modulus, especially for values of ρ[ f ] close to 1 ( [14,15]). The most precise results concerning the minimum modulus of entire and subharmonic functions of order zero can be found in [2][3][4][11][12][13].
For analytic functions in the unit disc D the situation, in a certain sense, is the opposite. Known results are much weaker in accuracy than the statements of the cos πρ-theorem type. Moreover, these results mainly concern analytic functions with We start with an old result of M. Heins.
then there exist a constant K > 0 and a sequence (r n ), r n 1 such that For the function f (z) = exp( 1 z−1 ), we have Thus, inequality (1) is sharp in the class of bounded analytic functions in the unit disc. A description of exceptional sets for the relation (1 − |z|) log |B(z)| → 0, |z| 1, where B is a Blaschke product, has been very recently obtained in [17].
In the general case, we have the following theorem of C.N. Linden.
Theorem B [20] Let f (z) be an analytic function, for some sequence of number (r n ), r n 1.
The following theorem plays a key role for the estimates of minimum modulus.
Theorem C [20] Let f be analytic in D, and suppose that 1 2 ≤ α < 1. Then, there Such an approach for analytic functions f of order ρ M [ f ] < 1 allows us to get the following results.
Let (a n ) be a sequence of zeros of analytic function f in D. For this sequence, we define If there are r 0 ∈ (0, 1) and a constant B such that A characteristic feature of Theorems D and E is that their conclusions do not depend on the corresponding value of order ρ M [ f ] ∈ [0, 1]. It appears that, in this case, the value ρ M [ f ] does not allow the behavior of the minimum modulus in terms of conditions on zeros of f to be described more precisely. The aim of this paper is to correct this defect. Note that, some classes of bounded analytic functions satisfying the inequality were found in [5].
For an analytic function f (z), z ∈ D, f ≡ 0 and p ≥ 1, we define We write We define the order ρ ∞ [ f ] of the function f as The limit exists since ρ p is a non-decreasing function in p ( [24]). This quantity appeared for the first time in a work of Linden [23], who proved that , but he did not study the classes of functions defined by the order Applications of this concept to factorization of analytic functions in D, and logarithmic derivative estimates can be found in [6,9]. Let a sequence (a n ) in D satisfy the condition Consider the canonical product, s ∈ N, where is the Weierstrass primary factor, and A n (z) = 1−|a n | 2 1−ā n z . The function P(z) is analytic in the unit disc with the zero sequence (a n ) provided that (2) holds. We note that if s = 0, we have P 0 (z) = C B(z), where C = n |a n |, is the Blaschke product corresponding to the sequence (a n ) provided that n (1 − |a n |) < ∞. We define Let E ⊂ [0, 1) be a measurable set. The upper density of E is defined by where λ 1 (E ∩[r, 1)) denotes the Lebesgue measure of E ∩[r, 1). Theorem 1 describes the minimum modulus of canonical products of genus s ∈ N.
In the general case, we need the following factorization theorem.

where P(z) is a canonical product of form (3) displaying the zeros of f , p is a nonnegative integer, g is non-zero and both P and g are analytic, and ρ
Let u(z) be a harmonic function in D. We then define Denote M ∞ (r, u) = max{|u(z)| : |z| = r }. The following statement is of some independent interest. It gives another way to compute ρ ∞ -order of an analytic function without zeros.

Proposition 1 Let u(z) be a harmonic function in D. Then, we have
The main result of this paper is the following.
Some generalizations of Theorems 1 and 2 are considered in Sect. 3.

Proof of the Main Results
The following lemma is important in our investigation.

Lemma 1 [21] For a given value θ let S m,k denote the region
where k and m are integers such that k > 0 and −2 k−1 ≤ m ≤ 2 k−1 − 1. Let k 0 be a positive integer and β > 0. Suppose that there are a finite number of points a n in {|z| < 1 − 2 −k 0 } and that for some value θ such that 0 ≤ θ < 2π there are at most C2 kβ points a n in each region (6) for k ≥ k 0 . Then, if s is an integer greater than β the function P defined by (3) is analytic in D and where K depends on s, β, C.
Proof of Theorem 1 Without loss of generality, we may assume that r 0 = 0. Otherwise, where C r 0 = max 1, n 0 We denote r N = 1 − 3 4 N , and use the following lemma.
Lemma 2 [15] Suppose that r > 0, h > 0 and that for |z| = r we have n z (h) ≤ n 0 . Then there exist a set E ⊂ [r, r + h 2 ] having measure at least 1 4 h such that, for R in E and |z| = R, where A is an absolute constant.
From Lemmas 3 and 1, and (2) we get where K > 1 is an arbitrary constant. Theorem 1 is proved.
Let P(z, w) = Re w+z w−z be the Poisson kernel. The Poisson formula together with Hölder's inequality yields For the estimate of the Poisson kernel, we use the following lemma  O(ψ(x)), x → ∞. Then, one should replace the factor 1 (1−r ) β log 1 1−r byψ 1 1−r log 1 1−r in the conclusion of Theorem 1, whereψ(x) = x 1 ψ(t) t dt (see [7], for details).
If we have additional information on the factors in the factorization formula (4) we can state more, using the same method.