The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function. II. The complex plane

Distributions of zeros and masses of entire and subharmonic functions with restrictions on their growth along the strip

“….3)If, for this subharmonic function u ̸ ≡ −∞, its Riesz mass distribution ∆ u is of finite order ord[∆ u ] < +∞, then, for each d ∈ (0, 2], there exists an entire function f ̸ ≡ 0 satisfying (8.2),(8.3) and such that Proof. Using[51], § 1.3, Corollary 2, with a comment after this corollary, which is based on[51], § 2.4, the proof of Corollary 2, see also assertion 4 of Lemma 5.1 in[52] and Theorem 1 in[50], we find that, for each subharmonic function u ̸ ≡ −∞ on C and any number P ∈ R + with particular function P ̸ ≡ 0 such that ln |f P (z)| ⩽ u •p (z) for all z ∈ C. Hence, for the function r, with lower bounds (2.24) from Remark 2, we have r(z) ⩾ p(z) and ln |f P (z)| ⩽ u •r (z) for all z ∈ C \ RD with sufficiently large R ∈ R + . At the same time, by inequality (2.24) from Remark 2 we have r(z) ⩾ c for all z ∈ RD for some number c ∈ R + \ 0.…”