The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function

Хабибуллин, Б. Н.; Baiguskarov, T. Yu.

doi:10.1134/s0001434616030317

Cited by 7 publications

(3 citation statements)

References 9 publications

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“…then, for each function (16) with (19) and with a number P ∈ R + if D := C and for any number a > 0, there exists a function f ∈ Hol(D) such that f = 0, f (Z) = 0 and…”

Section: Main Results On Sufficient Conditions For Zero Subsets In Do...mentioning

confidence: 99%

Necessary and sufficient conditions for zero subsets of holomorphic functions

Хабибуллин¹,

Khabibullin²

2020

Preprint

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Let D be a domain in the complex plane, M be an extended real function on D. If f is a non-zero holomorphic function on D with an upper constraint |f | ≤ exp M on this domain D, then it is natural to expect that there must be some upper constraints on the distribution of zeros of this holomorphic function exclusively in terms of the function M and the geometry of the domain D. We have investigated this question in detail in our previous works in the case when M is a subharmonic function and the domain D is arbitrary or with a non-polar boundary. The answer was given in terms of limiting the distribution of zeros of f from above via the Riesz measure of the subharmonic function M . In this article, the function M is the difference of subharmonic functions, or a δ-subharmonic function, and the upper constraints are given in terms of the Riesz charge of this δ-subharmonic function M . These results are also new to a certain extent for the subharmonic function M . The case when the domain D is the complex plane is considered separately. For the complex plane, it is possible to reach the criterion level.

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“…then, for each function (16) with (19) and with a number P ∈ R + if D := C and for any number a > 0, there exists a function f ∈ Hol(D) such that f = 0, f (Z) = 0 and…”

Section: Main Results On Sufficient Conditions For Zero Subsets In Do...mentioning

confidence: 99%

Necessary and sufficient conditions for zero subsets of holomorphic functions

Хабибуллин¹,

Khabibullin²

2020

Preprint

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“…The first one is to restrict to inequalities for points z lying outside some (small, if possible) exceptional set E ⊂ C. The second approach is to obtain lower bounds for integral averages of the form (2.10) or (2.7) over discs or circles of variable small radius r : C → R + \ 0. The latter option is often preferable because, first, it is additive, unlike the most commonly used lower bounds for the supremum of the function u on discs D z r(z) , and second, this approach is also capable of delivering lower bounds outside the exceptional sets E, as reflected in Theorem 2 of [50]. In this regard, we have the following result.…”

Section: Shifts and Balayage Of Amentioning

confidence: 92%

“….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.…”

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confidence: 99%

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

Khabibullin

2024

Izv. Math.

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Let $\mathrm Z$ and $\mathrm W$ be distributions of points on the complex plane $\mathbb C$. The following problem dates back to F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane, and others. For which $\mathrm Z$ and $\mathrm W$, for an entire function $g\neq 0$ of exponential type which vanishes on $\mathrm W$, there exists an entire function $f\neq 0$ of exponential type that vanishes on $\mathrm Z$ and is such that $|f|\leqslant |g|$ on the imaginary axis? The classical Malliavin-Rubel theorem of the early 1960s completely solves this problem for "positive" $\mathrm Z$ and $\mathrm W$ (which lie only on the positive semiaxis). Several generalizations of this criterion were established by the author of the present paper in the late 1980s for "complex" $\mathrm Z \subset \mathbb C$ and $\mathrm W\subset \mathbb C$ separated by angles from the imaginary axis, with some advances in the 2020s. In this paper, we solve more involved problems in a more general subharmonic framework for distributions of masses on $\mathbb C$. All the previously mentioned results can be obtained from the main results of this paper in a much stronger form (even for the initial formulation for distributions of points $\mathrm Z$ and $\mathrm W$ and entire functions $f$ and $g$ of exponential type). Some results of the present paper are closely related to the famous Beurling-Malliavin theorems on the radius of completeness and a multiplier.

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Balayage of measures and subharmonic functions to a system of rays. I. The classical case

Хабибуллин¹,

Shmeleva²

2019

St. Petersburg Math. J.

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The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function

Cited by 7 publications

References 9 publications

Necessary and sufficient conditions for zero subsets of holomorphic functions

Necessary and sufficient conditions for zero subsets of holomorphic functions

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

Balayage of measures and subharmonic functions to a system of rays. I. The classical case

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