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Filevich, P. V.

doi:10.1023/a:1010489609188

Cited by 6 publications

(6 citation statements)

References 2 publications

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“…as r → R − 0 (r / ∈ E), h-measE < +∞. In particular, as a corollary in the case R = +∞, we obtain one statement from article [2] (see also [10, p.58]).…”

Section: Statement Of Lemma 2 Is Proved Becausementioning

confidence: 65%

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Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets

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Kuryliak

2020

Carpathian Math. Publ.

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Let $\mathcal{E}_R$ be the class of analytic functions $f$ represented by power series of the form $f(z)=\sum\limits\limits_{n=0}^{+\infty}a_n z^n$ with the radius of convergence $R:=R(f)\in(0;+\infty].$ For $r\in [0, R)$ we denote the maximum modulus by $M_f(r)=\max\{|f(z)|\colon$ $ |z|=r\}$ and the maximal term of the series by $\mu_f(r)=\max\{|a_n| r^n\colon n\geq 0\}$. We also denote by $\mathcal{H}_R$, $R\leq +\infty$, the class of continuous positive functions, which increase on $[0;R)$ to $+\infty$, such that $h(r)\geq2$ for all $r\in (0,R)$ and $ \int^R_{r_{0}} h(r) d\ln r =+\infty $ for some $r_0\in(0,R)$. In particular, the following statements are proved. $1^0.$ If $h\in \mathcal{H}_R$ and $f\in \mathcal{E}_R,$ then for any $\delta>0$ there exist $E(\delta,f,h):=E\subset(0,R)$, $r_0 \in (0,R)$ such that $$ \forall\ r\in (r_0,R)\backslash E\colon\ M_f(r)\leq h(r) \mu_f(r) \big\{\ln h(r)\ln(h(r)\mu_f(r))\big\}^{1/2+\delta}$$ and $$\int\nolimits_E h(r) dr < +\infty. $$ $2^0.$ If we additionally assume that the function $f\in \mathcal{E}_R$ is unbounded, then $$ \ln M_f(r)\leq(1+o(1))\ln (h(r)\mu_f(r)) $$ holds as $r\to R$, $r\notin E$. Remark, that assertion $1^0$ at $h(r)\equiv \text{const}$ implies the classical Wiman-Valiron theorem for entire functions and at $h(r)\equiv 1/(1-r)$ theorem about the Kövari-type inequality for analytic functions in the unit disc. From statement $2^0$ in the case that $\ln h(r)=o(\ln\mu_f(r))$, $r\to R$, it follows that $ \ln M_f(r)=(1+o(1))\ln \mu_f(r) $ holds as $r\to R$, $r\notin E$.

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“…as r → R − 0 (r / ∈ E), h-measE < +∞. In particular, as a corollary in the case R = +∞, we obtain one statement from article [2] (see also [10, p.58]).…”

Section: Statement Of Lemma 2 Is Proved Becausementioning

confidence: 65%

“…If the functions f ∈ E and h ∈ H are such that ln + h(r) = o(ln 2 µ f (r)) (r → +∞), then for every ε > 0 inequality (2) holds for all r ∈ [1; +∞) \ E(ε, f , h), where a set E(ε, f , h) := E has finite h-logarithmic measure, i.e. h-meas E < +∞.…”

Section: Propositionmentioning

confidence: 99%

Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets

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Kuryliak

2020

Carpathian Math. Publ.

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“…Since in this case at least µ(x, F ) ≤ µ(x 1 , f 1 )µ(x 2 , ..., x p , F 1 ), for every x ∈ E = E 1 × R p−1 we have inequality (10). Given that meas…”

Section: Introduction For An Entire Function Of the Formmentioning

confidence: 96%

“…Ostrovskii in a conversation with the third co-author of this article formulated the following problem: what is the best possible description of the values of an exceptional set E? This problem was considered in a number of articles (for example, see [4,5,[7][8][9][10][11][12][13][14]) in view to many other relations obtained in the Wiman-Valiron theory. In this same conversation, Prof. Ǐ.V.…”

Section: Introduction For An Entire Function Of the Formmentioning

confidence: 99%

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Erdős-Macintyre type theorem’s for multiple Dirichlet series: exceptional sets and open problems

2023

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In the paper, we formulate some open problems related to the best description of the values of the exceptional sets in Wiman's inequality for entire functions and in the Erd\H{o}s-Macintyre type theorems for entire multiple Dirichlet series. At the same time, we clarify the statement of one \v{I}.V. Ostrovskii problem on Wiman's inequality. We also prove three propositions and one theorem. On the one hand, in a rather special case, these results give the best possible description of the values of the exceptional set in the Erd\H{o}s-Macintyre-type theorem. On the second hand, they indicate the possible structure of the best possible description in the general case.

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Wiman-Type Inequality in Multiple-Circular Domains: Lévy’s Phenomenon and Exceptional Sets

Kuryliak

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2022

Ukr Math J

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Cited by 6 publications

References 2 publications

Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets

Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets

Erdős-Macintyre type theorem’s for multiple Dirichlet series: exceptional sets and open problems

Wiman-Type Inequality in Multiple-Circular Domains: Lévy’s Phenomenon and Exceptional Sets

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