Pub Date : 2023-12-28DOI: 10.3103/s1068362323060067
V. Tsagareishvili
Abstract
Convergence of classical Fourier series (trigonometric, Haar, Walsh, (dots) systems) of differentiable functions are trivial problems and they are well known. But general Fourier series, as it is known, even for the function (f(x)=1) does not converge. In such a case, if we want differentiable functions with respect to the general orthonormal system (ONS) ((varphi_{n})) to have convergent Fourier series, we must find the special conditions on the functions (varphi_{n}) of system ((varphi_{n})). This problem is studied in the present paper. It is established that the resulting conditions are best possible. Subsystems of general orthonormal systems are considered.
{"title":"Convergence of General Fourier Series of Differentiable Functions","authors":"V. Tsagareishvili","doi":"10.3103/s1068362323060067","DOIUrl":"https://doi.org/10.3103/s1068362323060067","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Convergence of classical Fourier series (trigonometric, Haar, Walsh, <span>(dots)</span> systems) of differentiable functions are trivial problems and they are well known. But general Fourier series, as it is known, even for the function <span>(f(x)=1)</span> does not converge. In such a case, if we want differentiable functions with respect to the general orthonormal system (ONS) <span>((varphi_{n}))</span> to have convergent Fourier series, we must find the special conditions on the functions <span>(varphi_{n})</span> of system <span>((varphi_{n}))</span>. This problem is studied in the present paper. It is established that the resulting conditions are best possible. Subsystems of general orthonormal systems are considered.</p>","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-28DOI: 10.3103/s1068362323060055
D. M. Martirosyan
Abstract
This work contributes to the research devoted to the recognition of a convex body by probabilistic characteristics of its lower-dimensional sections. In this paper, for any convex quadrilateral, five orientation-dependent characteristics are introduced and explicitly evaluated per direction. In terms of these characteristics, simple explicit representations of the orientation-dependent chord length distribution function and the covariogram are obtained not only for an arbitrary convex quadrilateral but also for any right prism based on it.
{"title":"Orientation-Dependent Chord Length Distribution in a Convex Quadrilateral","authors":"D. M. Martirosyan","doi":"10.3103/s1068362323060055","DOIUrl":"https://doi.org/10.3103/s1068362323060055","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>This work contributes to the research devoted to the recognition of a convex body by probabilistic characteristics of its lower-dimensional sections. In this paper, for any convex quadrilateral, five orientation-dependent characteristics are introduced and explicitly evaluated per direction. In terms of these characteristics, simple explicit representations of the orientation-dependent chord length distribution function and the covariogram are obtained not only for an arbitrary convex quadrilateral but also for any right prism based on it.</p>","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-28DOI: 10.3103/s1068362323060079
M.-H. Wang, J.-F. Chen
Abstract
In this paper, we investigate the uniqueness of meromorphic functions of finite order (f(z)) concerning their difference operators (Delta_{c}f(z)) and derivatives (f^{prime}(z)) and prove that if (Delta_{c}f(z)) and (f^{prime}(z)) share (a(z)), (b(z)), (infty) CM, where (a(z)) and (b(z)) are two distinct polynomials, then they assume one of following cases: ((1))(f^{prime}(z)equivDelta_{c}f(z)); ((2))(f(z)) reduces to a polynomial and (f^{prime}(z)-ADelta_{c}f(z)equiv(1-A)(c_{n}z^{n}+c_{n-1}z^{n-1}+cdots+c_{1}z+c_{0})), where (A(neq 1)) is a nonzero constant and (c_{n},c_{n-1},cdots,c_{1},c_{0}) are all constants. This generalizes the corresponding results due to Qi et al. and Deng et al.
{"title":"Derivatives of Meromorphic Functions Sharing Polynomials with Their Difference Operators","authors":"M.-H. Wang, J.-F. Chen","doi":"10.3103/s1068362323060079","DOIUrl":"https://doi.org/10.3103/s1068362323060079","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper, we investigate the uniqueness of meromorphic functions of finite order <span>(f(z))</span> concerning their difference operators <span>(Delta_{c}f(z))</span> and derivatives <span>(f^{prime}(z))</span> and prove that if <span>(Delta_{c}f(z))</span> and <span>(f^{prime}(z))</span> share <span>(a(z))</span>, <span>(b(z))</span>, <span>(infty)</span> CM, where <span>(a(z))</span> and <span>(b(z))</span> are two distinct polynomials, then they assume one of following cases: <span>((1))</span>\u0000<span>(f^{prime}(z)equivDelta_{c}f(z))</span>; <span>((2))</span>\u0000<span>(f(z))</span> reduces to a polynomial and <span>(f^{prime}(z)-ADelta_{c}f(z)equiv(1-A)(c_{n}z^{n}+c_{n-1}z^{n-1}+cdots+c_{1}z+c_{0}))</span>, where <span>(A(neq 1))</span> is a nonzero constant and <span>(c_{n},c_{n-1},cdots,c_{1},c_{0})</span> are all constants. This generalizes the corresponding results due to Qi et al. and Deng et al.</p>","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum to: Some Results on Nonlinear Difference-Differential Equations","authors":"L. L. Wu, M. L. Liu#, P. C. Hu","doi":"10.3103/s1068362323300015","DOIUrl":"https://doi.org/10.3103/s1068362323300015","url":null,"abstract":"<p>An Erratum to this paper has been published: https://doi.org/10.3103/S1068362323300015</p>","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138509344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.3103/s1068362323050023
M. A. Khachaturyan, V. N. Margaryan
{"title":"Comparison of Polynomials and Weighted-Hyperbolic Operators","authors":"M. A. Khachaturyan, V. N. Margaryan","doi":"10.3103/s1068362323050023","DOIUrl":"https://doi.org/10.3103/s1068362323050023","url":null,"abstract":"","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135963494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.3103/s1068362323050096
Xi Fu, Q. Shi#
{"title":"A Hardy–Littlewood Type Theorem for Harmonic Bergman–Orlicz Spaces and Applications","authors":"Xi Fu, Q. Shi#","doi":"10.3103/s1068362323050096","DOIUrl":"https://doi.org/10.3103/s1068362323050096","url":null,"abstract":"","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135963668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.3103/s1068362323050059
A. Mir, A. Hussain
{"title":"Operator Preserving Bernstein-Type Inequalities between Polynomials","authors":"A. Mir, A. Hussain","doi":"10.3103/s1068362323050059","DOIUrl":"https://doi.org/10.3103/s1068362323050059","url":null,"abstract":"","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135963505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.3103/s1068362323050047
S. A. Malik, B. A. Zargar
Abstract Let $$W(zeta)=(a_{0}+a_{1}zeta+...+a_{n}zeta^{n})$$ be a polynomial of degree $$n$$ having all its zeros in $$mathbb{T}_{k}cupmathbb{E}^{-}_{k}$$ , $$kgeq 1$$ , then for every real or complex number $$alpha$$ with $$|alpha|geq 1+k+k^{n}$$ , Govil and McTume [7] showed that the following inequality holds $$maxlimits_{zetainmathbb{T}_{1}}|D_{alpha}W(zeta)|geq nleft(frac{|alpha|-k}{1+k^{n}}right)||W||+nleft(frac{|alpha|-(1+k+k^{n})}{1+k^{n}}right)minlimits_{zetainmathbb{T}_{k}}|W(zeta)|.$$ In this paper, we have obtained a generalization of this inequality involving sequence of operators known as polar derivatives. In addition, the problem for the limiting case is also considered.
{"title":"On a Generalization of an Operator Preserving Turán-Type Inequality for Complex Polynomials","authors":"S. A. Malik, B. A. Zargar","doi":"10.3103/s1068362323050047","DOIUrl":"https://doi.org/10.3103/s1068362323050047","url":null,"abstract":"Abstract Let $$W(zeta)=(a_{0}+a_{1}zeta+...+a_{n}zeta^{n})$$ be a polynomial of degree $$n$$ having all its zeros in $$mathbb{T}_{k}cupmathbb{E}^{-}_{k}$$ , $$kgeq 1$$ , then for every real or complex number $$alpha$$ with $$|alpha|geq 1+k+k^{n}$$ , Govil and McTume [7] showed that the following inequality holds $$maxlimits_{zetainmathbb{T}_{1}}|D_{alpha}W(zeta)|geq nleft(frac{|alpha|-k}{1+k^{n}}right)||W||+nleft(frac{|alpha|-(1+k+k^{n})}{1+k^{n}}right)minlimits_{zetainmathbb{T}_{k}}|W(zeta)|.$$ In this paper, we have obtained a generalization of this inequality involving sequence of operators known as polar derivatives. In addition, the problem for the limiting case is also considered.","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135963522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.3103/s1068362323050084
G. M. Sofi, W. M. Shah
Abstract According to the Gauss–Lucas theorem, the critical points of a complex polynomial $$p(z):=sum_{j=0}^{n}a_{j}z^{j}$$ where $$a_{j}inmathbb{C}$$ always lie in the convex hull of its zeros. In this paper, we prove certain relations between the distribution of zeros of a polynomial and its critical points. Using these relations, we prove the well-known Sendov’s conjecture for certain special cases.
{"title":"Distribution of Zeros and Critical Points of a Polynomial, and Sendov’s Conjecture","authors":"G. M. Sofi, W. M. Shah","doi":"10.3103/s1068362323050084","DOIUrl":"https://doi.org/10.3103/s1068362323050084","url":null,"abstract":"Abstract According to the Gauss–Lucas theorem, the critical points of a complex polynomial $$p(z):=sum_{j=0}^{n}a_{j}z^{j}$$ where $$a_{j}inmathbb{C}$$ always lie in the convex hull of its zeros. In this paper, we prove certain relations between the distribution of zeros of a polynomial and its critical points. Using these relations, we prove the well-known Sendov’s conjecture for certain special cases.","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135963691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.3103/s1068362323050035
Xh. Z. Krasniqi
{"title":"Effectiveness of the Even-Type Delayed Mean in Approximation of Conjugate Functions","authors":"Xh. Z. Krasniqi","doi":"10.3103/s1068362323050035","DOIUrl":"https://doi.org/10.3103/s1068362323050035","url":null,"abstract":"","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135964017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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