Salo Tetyana Mykhailivna, Skaskiv Oleh Bohdanovych
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.
{"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}
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.
{"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}
. 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.
{"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}
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.
{"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}
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
. 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.
{"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}
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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：