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Minimum modulus of lacunary power series and h-measure of exceptional sets 虚幂级数的最小模与例外集的h测度
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-4-135
Salo Tetyana Mykhailivna, Skaskiv Oleh Bohdanovych
{"title":"Minimum modulus of lacunary power series and h-measure of exceptional sets","authors":"Salo Tetyana Mykhailivna, Skaskiv Oleh Bohdanovych","doi":"10.13108/2017-9-4-135","DOIUrl":"https://doi.org/10.13108/2017-9-4-135","url":null,"abstract":"We consider some generalizations of Fenton theorem for the entire functions represented by lacunary power series. Let f(z) = ∑︀+∞ k=0 fkz nk , where (nk) is a strictly increasing sequence of non-negative integers. We denote by Mf (r) = max{|f(z)| : |z| = r}, mf (r) = min{|f(z)| : |z| = r}, μf (r) = max{|fk|rk : k > 0} the maximum modulus, the minimum modulus and the maximum term of f, respectively. Let h(r) be a positive continuous function increasing to infinity on [1,+∞) with a nondecreasing derivative. For a measurable set E ⊂ [1,+∞) we introduce h − meas (E) = ∫︀ E dh(r) r . In this paper we establish conditions guaranteeing that the relations Mf (r) = (1 + o(1))mf (r), Mf (r) = (1 + o(1))μf (r) are true as r → +∞ outside some exceptional set E such that h − meas (E) < +∞. For some subclasses we obtain necessary and sufficient conditions. We also provide similar results for entire Dirichlet series.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"139 1","pages":"135-144"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77823908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Two-sided estimates for the relative growth of functions and their derivatives 函数及其导数相对增长的双边估计
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-3-18
G. G. Braichev
{"title":"Two-sided estimates for the relative growth of functions and their derivatives","authors":"G. G. Braichev","doi":"10.13108/2017-9-3-18","DOIUrl":"https://doi.org/10.13108/2017-9-3-18","url":null,"abstract":"We provide an extended presentation of a talk given at the International mathematical conference on theory of functions dedicated to centenary of corresponding member of AS USSR A.F. Leont’ev. We propose a new method for obtaining uniform two-sided estimates for the fraction of the derivatives of two real functions on the base of the information of two-sided estimates for the functions themselves. At that, one of the functions possesses certain properties and serves as a reference for measuring a growth and introduces some scale. The other function, whose growth is compared with that of the reference function, is convex, increases unboundedly or decays to zero on a certain interval. The method is also applicable to some class of functions concave on an interval. We consider examples of applications of the obtained results to the behavior of entire functions.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"56 1","pages":"18-25"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77859193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation 极限方程特征外缘附近椭圆型边值问题解参数的渐近性
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-3-137
Yurii Zakirovich Shaygardanov
{"title":"Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation","authors":"Yurii Zakirovich Shaygardanov","doi":"10.13108/2017-9-3-137","DOIUrl":"https://doi.org/10.13108/2017-9-3-137","url":null,"abstract":". In a bounded domain 𝑄 ⊂ R 3 with a smooth boundary Γ we consider the boundary value problem Here 𝐴 is a second order elliptic operator, 𝜀 is a small parameter. The limiting equation, as 𝜀 = 0, is the first order equation. Its characteristics are the straight lines parallel to the axis 𝑂𝑥 3 . For the domain 𝑄 we assume that the characteristic either intersects Γ at two points or touches Γ from outside. The set of touching point forms a closed smooth curve. In the paper we construct the asymptotics as 𝜀 → 0 for the solutions to the studied problem in the vicinity of this curve. For constructing the asymptotics we employ the method of matching asymptotic expansions.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"42 1","pages":"137-147"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78214780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Integration of equation of Toda periodic chain kind Toda周期链类方程的积分
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-2-17
B. Babajanov, A. B. Khasanov
{"title":"Integration of equation of Toda periodic chain kind","authors":"B. Babajanov, A. B. Khasanov","doi":"10.13108/2017-9-2-17","DOIUrl":"https://doi.org/10.13108/2017-9-2-17","url":null,"abstract":"In this work we apply the method of the inverse spectral problem to integrating an equation of Toda periodic chain kind. For the one-band case we write out explicit formulae for the solutions to an analogue of Dubrovin system of equations and thus, for our problem. These solutions are expressed in term of Jacobi elliptic functions.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"24 1","pages":"17-24"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74555563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Lower bound for the Hardy constant for an arbitrary domain in $mathbb{R}^n$ $mathbb{R}^n$中任意定义域Hardy常数的下界
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-2-102
I. K. Shafigullin
{"title":"Lower bound for the Hardy constant for an arbitrary domain in $mathbb{R}^n$","authors":"I. K. Shafigullin","doi":"10.13108/2017-9-2-102","DOIUrl":"https://doi.org/10.13108/2017-9-2-102","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"44 1","pages":"102-108"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77502512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Third Hankel determinant for the inverse of reciprocal of bounded turning functions has a positive real part of order alpha 第三，有界转动函数的倒数的逆的汉克尔行列式有阶的正实部
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-2-109
B. Venkateswarlu, N. Rani
{"title":"Third Hankel determinant for the inverse of reciprocal of bounded turning functions has a positive real part of order alpha","authors":"B. Venkateswarlu, N. Rani","doi":"10.13108/2017-9-2-109","DOIUrl":"https://doi.org/10.13108/2017-9-2-109","url":null,"abstract":"In this paper we obtain the best possible upper bound to the third Hankel determinants for the functions belonging to the class of reciprocal of bounded turning functions using Toeplitz determinants. Mathematics subject classification: 30C45, 30C50.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"509 1","pages":"109-118"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86847447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
On multi-dimensional partial differential equations with power nonlinearities in first derivatives 一阶导数为幂非线性的多维偏微分方程
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-1-98
I. V. Rakhmelevich
{"title":"On multi-dimensional partial differential equations with power nonlinearities in first derivatives","authors":"I. V. Rakhmelevich","doi":"10.13108/2017-9-1-98","DOIUrl":"https://doi.org/10.13108/2017-9-1-98","url":null,"abstract":"We consider a class of multi-dimensional partial differential equations involving a linear differential operator of arbitrary order and a power nonlinearity in the first derivatives. Under some additional assumptions for this operator, we study the solutions of multi-dimensional travelling waves that depend on some linear combinations of the original variables. The original equation is transformed to a reduced one, which can be solved by the separation of variables. Solutions of the reduced equation are found for the cases of additive, multiplicative and combined separation of variables.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"112 1","pages":"98-108"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87638467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Analytic functions with smooth absolute value of boundary data 边界数据绝对值光滑的解析函数
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-3-148
F. Shamoyan
{"title":"Analytic functions with smooth absolute value of boundary data","authors":"F. Shamoyan","doi":"10.13108/2017-9-3-148","DOIUrl":"https://doi.org/10.13108/2017-9-3-148","url":null,"abstract":"Abstract. Let f be an analytic function in the unit circle D continuous up to its boundary Γ, f(z) 6= 0, z ∈ D. Assume that on Γ, the function f has a modulus of continuity ω(|f |, δ). In the paper we establish the estimate ω(f, δ) 6 Aω(|f |, √ δ), where A is a some non-negative number, and we prove that this estimate is sharp. Moreover, in the paper we establish a multi-dimensional analogue of the mentioned result. In the proof of the main theorem, an essential role is played by a theorem of Hardy-Littlewood type on Hölder classes of the functions analytic in the unit circle.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"15 1","pages":"148-157"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82998344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On commutant of differentiation and translation operators in weighted spaces of entire functions 整个函数加权空间中微分与平移算子的交换子
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-3-37
O. Ivanova, S. N. Melikhov, Y. N. Melikhov
{"title":"On commutant of differentiation and translation operators in weighted spaces of entire functions","authors":"O. Ivanova, S. N. Melikhov, Y. N. Melikhov","doi":"10.13108/2017-9-3-37","DOIUrl":"https://doi.org/10.13108/2017-9-3-37","url":null,"abstract":". We describe continuous linear operators acting in a countable inductive limit 𝐸 of weighted Fr´echet spaces of entire functions of several complex variables and commuting in these spaces with systems of partial differentiation and translation operators. Under the made assumptions, the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by an arbitrary continuous linear functional on 𝐸 . At that, we do not assume that the set of the polynomials is dense in 𝐸 . In the space 𝐸 ′ topological dual to 𝐸 , we introduce the natural multiplication. Under this multiplication, the algebra 𝐸 ′ is isomorphic to the aforementioned commutant with the usual multiplication, which is the composition of the operators. This isomorphism is also topological if 𝐸 ′ is equipped by the weak topology, while the commutant is equipped by the weak operator topology. This implies that the set of the polynomials of the differentiation operators is dense in the commutant with topology of pointwise convergence. We also study the possibility of representing an operator in the commutant as an infinite order differential operator with constant coefficients. We prove the immediate continuity of linear operators commuting with all differentiation operators in a weighted (LF)-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a real multi-dimensional space.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"10 6 1","pages":"37-47"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83441725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Existence tests for limiting cycles of second order differential equations 二阶微分方程极限环的存在性检验
IF 0.5
Ufa Mathematical Journal Pub Date : 2017-01-01 DOI: 10.13108/2017-9-4-3
Mullosharaf Kurbonovich Arabov, E. Muhamadiev, I. D. Nurov, Khurshed Ilkhomiddinovich Sobirov
{"title":"Existence tests for limiting cycles of second order differential equations","authors":"Mullosharaf Kurbonovich Arabov, E. Muhamadiev, I. D. Nurov, Khurshed Ilkhomiddinovich Sobirov","doi":"10.13108/2017-9-4-3","DOIUrl":"https://doi.org/10.13108/2017-9-4-3","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"14 1","pages":"3-11"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74351911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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