Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $eta(s)$, and hence Riemann's function $zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(ikappa)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $zeta^{pm}(s)$ and $eta^{pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.
{"title":"A series representation for Riemann's zeta function and some interesting identities that follow","authors":"M. Milgram","doi":"10.7153/jca-2021-17-09","DOIUrl":"https://doi.org/10.7153/jca-2021-17-09","url":null,"abstract":"Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $eta(s)$, and hence Riemann's function $zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(ikappa)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $zeta^{pm}(s)$ and $eta^{pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43093848","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}
Pub Date : 2020-08-21DOI: 10.1142/s1793557121501023
Manivannan Mathi, J. K. Prajapat
In this paper, we introduce a new class of sense preserving harmonic mappings [Formula: see text] in the open unit disk and prove that functions in this class are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the sections of functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.
{"title":"On a subclass of close-to-convex harmonic mappings","authors":"Manivannan Mathi, J. K. Prajapat","doi":"10.1142/s1793557121501023","DOIUrl":"https://doi.org/10.1142/s1793557121501023","url":null,"abstract":"In this paper, we introduce a new class of sense preserving harmonic mappings [Formula: see text] in the open unit disk and prove that functions in this class are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the sections of functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s1793557121501023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42427659","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 improve the upper bound of the third order Hankel determinant for the class of Ozaki close-to-convex functions. The sharp bound is conjectured.
本文改进了Ozaki类近凸函数的三阶Hankel行列式的上界。尖锐的界限被推测出来。
{"title":"Improved upper bound of third order Hankel determinant for Ozaki close-to-convex functions","authors":"M. Obradovi'c, N. Tuneski","doi":"10.7153/jca-2022-19-02","DOIUrl":"https://doi.org/10.7153/jca-2022-19-02","url":null,"abstract":"In this paper we improve the upper bound of the third order Hankel determinant for the class of Ozaki close-to-convex functions. The sharp bound is conjectured.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44712881","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}
. For a class of sets with multiple terms a space. We a solution f extends analytically an
． 对于一类具有多项的集合，一个空间。我们将解f解析地展开
{"title":"A moment problem in a weighted L^2 space on the real line","authors":"E. Zikkos","doi":"10.7153/jca-2020-16-03","DOIUrl":"https://doi.org/10.7153/jca-2020-16-03","url":null,"abstract":". For a class of sets with multiple terms a space. We a solution f extends analytically an","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":"71135055","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 honor of Frank-Olme Speck on the occasion of his 75th birthday Abstract In this paper, we introduce two new convolutions associated with the Mellin transform which exhibit factorization properties upon the use of certain weight functions. This is applied to the solvability analysis of classes of integral equations. In particular, we present sufficient conditions for the solvability of an integral equation and a system of integral equations of convolution type.
{"title":"New convolutions associated with the Mellin transform and their applications in integral equations","authors":"L. Castro, A. S. Silva, N. Tuan","doi":"10.7153/JCA-2020-16-08","DOIUrl":"https://doi.org/10.7153/JCA-2020-16-08","url":null,"abstract":"In honor of Frank-Olme Speck on the occasion of his 75th birthday Abstract In this paper, we introduce two new convolutions associated with the Mellin transform which exhibit factorization properties upon the use of certain weight functions. This is applied to the solvability analysis of classes of integral equations. In particular, we present sufficient conditions for the solvability of an integral equation and a system of integral equations of convolution type.","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":"71135234","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 generalized the Wijsman lacunary invariant statistical convergence of closed sets in metric space by introducing the Wijsman lacunary invariant statistical φ̃ convergence for the sets of triple sequences. We introduce the concepts of Wijsman invariant φ̃ -convergence, Wijsman invariant statistical φ̃ -convergence, Wijsman lacunary invariant φ̃ -convergence, Wijsman lacunary invariant statistical φ̃ -convergence for the sets of triple sequences. In addition, we investigate existence of some relations among these new notations for the sets of triple sequences.
{"title":"Wijsman lacunary invariant statistical convergence for triple sequences via Orlicz function","authors":"M. Huban, M. Gürdal","doi":"10.7153/JCA-2020-17-08","DOIUrl":"https://doi.org/10.7153/JCA-2020-17-08","url":null,"abstract":"In this paper, we generalized the Wijsman lacunary invariant statistical convergence of closed sets in metric space by introducing the Wijsman lacunary invariant statistical φ̃ convergence for the sets of triple sequences. We introduce the concepts of Wijsman invariant φ̃ -convergence, Wijsman invariant statistical φ̃ -convergence, Wijsman lacunary invariant φ̃ -convergence, Wijsman lacunary invariant statistical φ̃ -convergence for the sets of triple sequences. In addition, we investigate existence of some relations among these new notations for the sets of triple sequences.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":"119-128"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135672","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}
This paper is based on new generalization of q -analogue of the MKZ type operators using generating functions. We study approximation properties of the proposed operator using Korovkin type theorem. Further, we estimate the rate of convergence of these operators by using the modulus of continuity. In the last, we introduce and establish the uniform convergence of 2D -generalization of the q -MKZ operators using generating functions.
{"title":"A note on q-MKZ operators using generating functions","authors":"Honey Sharma, Ramapati Maurya","doi":"10.7153/JCA-2020-17-11","DOIUrl":"https://doi.org/10.7153/JCA-2020-17-11","url":null,"abstract":"This paper is based on new generalization of q -analogue of the MKZ type operators using generating functions. We study approximation properties of the proposed operator using Korovkin type theorem. Further, we estimate the rate of convergence of these operators by using the modulus of continuity. In the last, we introduce and establish the uniform convergence of 2D -generalization of the q -MKZ operators using generating functions.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":"177-187"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135987","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 primary focus of this article is to introduce the ideas of asymptotically deferred statistical equivalence and strongly asymptotically deferred statistical equivalence by considering two nonnegative real-valued Lebesgue measurable functions on (1,∞) . Additionally, we examine some main properties of these two concepts. Mathematics subject classification (2010): 40H05, 40C05.
{"title":"On asymptotically deferred statistical equivalent measurable functions","authors":"R. Savaş","doi":"10.7153/JCA-2020-16-14","DOIUrl":"https://doi.org/10.7153/JCA-2020-16-14","url":null,"abstract":"The primary focus of this article is to introduce the ideas of asymptotically deferred statistical equivalence and strongly asymptotically deferred statistical equivalence by considering two nonnegative real-valued Lebesgue measurable functions on (1,∞) . Additionally, we examine some main properties of these two concepts. Mathematics subject classification (2010): 40H05, 40C05.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":"141-147"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135269","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 establish some upper bound estimates for the polar derivative of a polynomial not vanishing in a disk | z | < k , k (cid:2) 1 with a zero of multiplicity s , 0 (cid:3) s (cid:3) n − 1 at the origin. The obtained results enable us to derive polar derivative analogues of some well known Bernstein-type inequalities as special cases.
． 本文建立了一个不消失于圆盘| z | < k, k (cid:2) 1的多项式的极导数的上界估计，该多项式在原点处的多重性为s, 0 (cid:3) s (cid:3) n−1。所得到的结果使我们能够推导出一些著名的伯恩斯坦型不等式的极导数类似物作为特殊情况。
{"title":"On Bernstein-type inequalities for polynomials involving the polar derivative","authors":"A. Hussain, A. Mir, Abrar Ahmad","doi":"10.7153/jca-2020-16-02","DOIUrl":"https://doi.org/10.7153/jca-2020-16-02","url":null,"abstract":". In this paper, we establish some upper bound estimates for the polar derivative of a polynomial not vanishing in a disk | z | < k , k (cid:2) 1 with a zero of multiplicity s , 0 (cid:3) s (cid:3) n − 1 at the origin. The obtained results enable us to derive polar derivative analogues of some well known Bernstein-type inequalities as special cases.","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":"71135450","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 purpose of this paper is to introduce more general results on the existence of solutions for nonlinear dynamic equations on time scales with impulses and nonlocal initial conditions. We establish the existence of solutions by applying a fixed point result due to O’Regan, while the uniqueness of solutions is obtained through the contraction mapping principle. Our results extend previous work in the literature and an example is discussed to illustrate the obtained results. Mathematics subject classification (2010): 34N05, 34A12, 34A37, 39A12.
{"title":"Nonlinear dynamic equations on time scales with impulses and nonlocal conditions","authors":"Sanket Tikare, C. Tisdell","doi":"10.7153/JCA-2020-16-13","DOIUrl":"https://doi.org/10.7153/JCA-2020-16-13","url":null,"abstract":"The purpose of this paper is to introduce more general results on the existence of solutions for nonlinear dynamic equations on time scales with impulses and nonlocal initial conditions. We establish the existence of solutions by applying a fixed point result due to O’Regan, while the uniqueness of solutions is obtained through the contraction mapping principle. Our results extend previous work in the literature and an example is discussed to illustrate the obtained results. Mathematics subject classification (2010): 34N05, 34A12, 34A37, 39A12.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":"125-140"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135220","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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