In this paper, by using standard techniques we shall obtain a result that gives regions containing all the zeros of a polynomial with real coefficients. Our result not only generalizes several well-known results concerning the location of zeros of polynomials but also yields an answer to a question raised by Professor N. K. Govil. We also obtain a similar result for analytic functions. In addition to this, we show by examples that our result gives better information about the bounds of zeros of polynomials than some known results. Mathematics subject classification (2010): 30A10, 30C15.
在本文中，通过使用标准技术，我们将得到一个结果，该结果给出了包含实系数多项式的所有零的区域。我们的结果不仅推广了几个著名的关于多项式零点位置的结果，而且对N. K. Govil教授提出的一个问题给出了答案。对于解析函数，我们也得到了类似的结果。除此之外，我们还通过实例表明，我们的结果比一些已知的结果提供了关于多项式零点边界的更好信息。数学学科分类(2010):30A10, 30C15。
{"title":"Generalization of Eneström-Kakeya theorem and its extension to analytic functions","authors":"N. A. Rather, Ishfaq Dar, A. Iqbal","doi":"10.7153/jca-2020-16-05","DOIUrl":"https://doi.org/10.7153/jca-2020-16-05","url":null,"abstract":"In this paper, by using standard techniques we shall obtain a result that gives regions containing all the zeros of a polynomial with real coefficients. Our result not only generalizes several well-known results concerning the location of zeros of polynomials but also yields an answer to a question raised by Professor N. K. Govil. We also obtain a similar result for analytic functions. In addition to this, we show by examples that our result gives better information about the bounds of zeros of polynomials than some known results. Mathematics subject classification (2010): 30A10, 30C15.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135170","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 present two conjectures about the characterization of functions by conditions on their divided differences. To analyze the conjectures and prove some results, we recall some facts about the Hermite interpolation problem including the computation of divided differences for positive and negative powers of x .
{"title":"On the characterization of polynomials and rational functions using divided differences","authors":"F. Dubeau","doi":"10.7153/JCA-2020-16-09","DOIUrl":"https://doi.org/10.7153/JCA-2020-16-09","url":null,"abstract":". In this paper we present two conjectures about the characterization of functions by conditions on their divided differences. To analyze the conjectures and prove some results, we recall some facts about the Hermite interpolation problem including the computation of divided differences for positive and negative powers of x .","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135349","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}
. The core of the present paper is represented by the calculation of two essential har- monic series with a weight 5 structure, involving harmonic numbers of the type H 2 n . The two main series are evaluated by also exploiting results and strategies presented in the book, (Almost) Impossible Integrals, Sums, and Series , 2019.
{"title":"On the calculation of two essential harmonic series with a weight 5 structure, involving harmonic numbers of the type H_2n","authors":"C. Vălean","doi":"10.7153/jca-2020-16-01","DOIUrl":"https://doi.org/10.7153/jca-2020-16-01","url":null,"abstract":". The core of the present paper is represented by the calculation of two essential har- monic series with a weight 5 structure, involving harmonic numbers of the type H 2 n . The two main series are evaluated by also exploiting results and strategies presented in the book, (Almost) Impossible Integrals, Sums, and Series , 2019.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135436","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 investigate the simultaneous approximation properties of the de la Vallée-Poussin means in weighted Orlicz spaces in terms of the modulus of smoothness. In terms of the modulus of smoothness the direct theorem of simultaneous approximation is proved. Also, in weighted Orlicz spaces the modulus of smoothness are estimated from below and above in terms of n -th partial Fourier sums and de la Vallée-Poussin means.
我们研究了加权Orlicz空间中de la vall - poussin均值在光滑模方面的同时逼近性质。从光滑模的角度，证明了同时逼近的直接定理。同样，在加权的Orlicz空间中，平滑的模量是根据n次偏傅里叶和和de la vall - poussin均值从下往上估计的。
{"title":"Simultaneous approximation properties of de la Vallée-Poussin means in weighted Orlicz spaces","authors":"S. Jafarov","doi":"10.7153/JCA-2020-17-12","DOIUrl":"https://doi.org/10.7153/JCA-2020-17-12","url":null,"abstract":"We investigate the simultaneous approximation properties of the de la Vallée-Poussin means in weighted Orlicz spaces in terms of the modulus of smoothness. In terms of the modulus of smoothness the direct theorem of simultaneous approximation is proved. Also, in weighted Orlicz spaces the modulus of smoothness are estimated from below and above in terms of n -th partial Fourier sums and de la Vallée-Poussin means.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":"189-198"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135663","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":"A Poisson logarithmic integral for integer order powers n = 0, 1, 2, and 3","authors":"Ting Zhu","doi":"10.7153/JCA-2020-17-01","DOIUrl":"https://doi.org/10.7153/JCA-2020-17-01","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135281","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 study about the sum and product of relative ( p , q , t ) L -th type and relative ( p , q , t ) L -th lower type of an entire function with respect to another entire function in the light of a special type of non-decreasing, unbounded function Ψ .
。本文针对一类特殊的非递减无界函数Ψ，研究了一个完整函数相对(p, q, t) L -型和相对(p, q, t) L -下型对另一个完整函数的和与积。
{"title":"Some results on sum and product of relative growth factors of composite entire functions","authors":"S. Datta, Banani Dutta, Nityagopal Biswas","doi":"10.7153/jca-2019-14-12","DOIUrl":"https://doi.org/10.7153/jca-2019-14-12","url":null,"abstract":". In this paper we study about the sum and product of relative ( p , q , t ) L -th type and relative ( p , q , t ) L -th lower type of an entire function with respect to another entire function in the light of a special type of non-decreasing, unbounded function Ψ .","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71134658","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 2000, the notion of a subsequence of a double sequence was introduced [3]. Using this de fi nition, a multidimensional analogue to a result from H. Steinhaus, that states that for any regular matrix A there exists a sequence of zeros and ones that is not A -summable, was proved. Additionally, an analogue of a result of R. C. Buck that states that a sequence x is convergent if and only if there exists a regular matrix A that sums every subsequence of x was presented. However, this de fi nition imposes a restrictive condition on the entries of the double sequence that can be considered for the subsequence. In this article, we introduce a less restrictive new de fi nition of a subsequence. We denote them by β -subsequences of a double sequence and show that analogues to these two fundamental theorems of summability still hold for these new subsequences.
{"title":"Fundamental theorems of summability theory for a new type of subsequences of double sequences","authors":"R. Dumitru, Jose A. Franco","doi":"10.7153/jca-2019-15-03","DOIUrl":"https://doi.org/10.7153/jca-2019-15-03","url":null,"abstract":". In 2000, the notion of a subsequence of a double sequence was introduced [3]. Using this de fi nition, a multidimensional analogue to a result from H. Steinhaus, that states that for any regular matrix A there exists a sequence of zeros and ones that is not A -summable, was proved. Additionally, an analogue of a result of R. C. Buck that states that a sequence x is convergent if and only if there exists a regular matrix A that sums every subsequence of x was presented. However, this de fi nition imposes a restrictive condition on the entries of the double sequence that can be considered for the subsequence. In this article, we introduce a less restrictive new de fi nition of a subsequence. We denote them by β -subsequences of a double sequence and show that analogues to these two fundamental theorems of summability still hold for these new subsequences.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71134763","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, a boundary version of Carath´eodory’s inequality on the right half plane is investigated. Here, the function Z ( s ) , is given as Z ( s ) = 1 + c 1 ( s − 1 )+ c 2 ( s − 1 ) 2 + ... be an analytic in the right half plane with ℜ Z ( s ) (cid:2) A ( A > 1 ) for ℜ s (cid:3) 0. We derive inequalities for the modulus of Z ( s ) function, | Z (cid:2) ( 0 ) | , by assuming the Z ( s ) function is also analytic at the boundary point s = 0 on the imaginary axis and fi nally, the sharpness of these inequalities is proved.
． 本文研究了Carath ' eodory不等式在右半平面上的边界形式。这里，函数Z (s)表示为Z (s) = 1 + c1 (s−1)+ c2 (s−1)2 +…在右半平面上是一个解析函数，对于1 (cid:3) 0，对1 (Z) (s) (cid:2) A (A >1)。通过假设Z (s)函数在虚轴上的边界点s = 0处也是解析函数，导出了Z (s)函数的模量| Z (cid:2)(0) |的不等式，最后证明了这些不等式的尖锐性。
{"title":"A sharp Carathéodory's inequality on the right half plane","authors":"B. Örnek","doi":"10.7153/jca-2019-14-04","DOIUrl":"https://doi.org/10.7153/jca-2019-14-04","url":null,"abstract":". In this paper, a boundary version of Carath´eodory’s inequality on the right half plane is investigated. Here, the function Z ( s ) , is given as Z ( s ) = 1 + c 1 ( s − 1 )+ c 2 ( s − 1 ) 2 + ... be an analytic in the right half plane with ℜ Z ( s ) (cid:2) A ( A > 1 ) for ℜ s (cid:3) 0. We derive inequalities for the modulus of Z ( s ) function, | Z (cid:2) ( 0 ) | , by assuming the Z ( s ) function is also analytic at the boundary point s = 0 on the imaginary axis and fi nally, the sharpness of these inequalities is proved.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71134524","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 develop a generalization of q -Bernstein-Kantorovich type operators. We first study some fundamental properties of these operators and then investigate approximation properties of a sequence of these operators using Korovkin theorem. Finally, we estimate rate of approximation by modulus of continuity. Mathematics subject classification (2010): 41A10, 41A25, 41A36.
{"title":"On approximation properties of generalized q-Bernstein-Kantorovich operators","authors":"L. Mishra, Dhawal J. Bhatt","doi":"10.7153/jca-2019-15-09","DOIUrl":"https://doi.org/10.7153/jca-2019-15-09","url":null,"abstract":"In this paper, we develop a generalization of q -Bernstein-Kantorovich type operators. We first study some fundamental properties of these operators and then investigate approximation properties of a sequence of these operators using Korovkin theorem. Finally, we estimate rate of approximation by modulus of continuity. Mathematics subject classification (2010): 41A10, 41A25, 41A36.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71134591","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":"Power means of the Hurwitz zeta function over large intervals","authors":"Yusuke Nishizawa","doi":"10.7153/jca-2019-15-05","DOIUrl":"https://doi.org/10.7153/jca-2019-15-05","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71134908","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}
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