{"title":"Applications of Briot-Bouquet differential subordination","authors":"N. Bohra, Sushil Kumar, V. Ravichandran","doi":"10.7153/jca-2021-18-02","DOIUrl":"https://doi.org/10.7153/jca-2021-18-02","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135835","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":"Number of zeros of a certain class of polynomials in a specific region","authors":"N. A. Rather, Ai az Bhat, Liyaqat Ali","doi":"10.7153/jca-2021-18-03","DOIUrl":"https://doi.org/10.7153/jca-2021-18-03","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135852","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}
Gupta in [6] introduced a general family of linear positive operators which produce large number of well known linear positive operators as particular cases. As the family of operators proposed by Gupta provides a unified approach this motivated us to extend the studies, and we establish some convergence estimates of these important operators. We estimate an asymptotic formula and the rate of convergence for these operators for the function having derivatives of bounded variation. Mathematics subject classification (2010): 41A25, 41A30.
{"title":"Direct estimates for Gupta type general operators","authors":"Ekta Pandey, R. K. Mishra","doi":"10.7153/JCA-2020-17-03","DOIUrl":"https://doi.org/10.7153/JCA-2020-17-03","url":null,"abstract":"Gupta in [6] introduced a general family of linear positive operators which produce large number of well known linear positive operators as particular cases. As the family of operators proposed by Gupta provides a unified approach this motivated us to extend the studies, and we establish some convergence estimates of these important operators. We estimate an asymptotic formula and the rate of convergence for these operators for the function having derivatives of bounded variation. Mathematics subject classification (2010): 41A25, 41A30.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135963","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":"Generalizations of Picard's theorem with moving hypersurfaces","authors":"Fei Li, Liu Yang","doi":"10.7153/jca-2021-18-11","DOIUrl":"https://doi.org/10.7153/jca-2021-18-11","url":null,"abstract":"","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136373","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 here a trilogarithmic expression that plays a similar role to Catalan’s constant G in many ways: The main purpose of this article is to demonstrate how G is a naturally occurring and useful expression that deserves to be recognized as a mathematical constant and as a natural trilogarithmic “extension” of Catalan’s constant G . Having identi fi ed this constant, we evaluate many new and non-trivial integrals, Euler-type sums, p F q series, and binomial-harmonic series using G , extending known results on the classical version of Catalan’s constant.
． 我们在这里考虑一个在许多方面与Catalan常数G起类似作用的三对数表达式:本文的主要目的是演示G如何是一个自然出现的有用表达式，它值得被认为是一个数学常数，并且是Catalan常数G的自然三对数“扩展”。在确定了这个常数之后，我们利用G计算了许多新的非平凡积分、欧拉型和、p F q级数和二项式调和级数，推广了经典版本Catalan常数的已知结果。
{"title":"A natural companion to Catalan's constant","authors":"John M. Campbell, P. Levrie, A. Nimbran","doi":"10.7153/jca-2021-18-09","DOIUrl":"https://doi.org/10.7153/jca-2021-18-09","url":null,"abstract":". We consider here a trilogarithmic expression that plays a similar role to Catalan’s constant G in many ways: The main purpose of this article is to demonstrate how G is a naturally occurring and useful expression that deserves to be recognized as a mathematical constant and as a natural trilogarithmic “extension” of Catalan’s constant G . Having identi fi ed this constant, we evaluate many new and non-trivial integrals, Euler-type sums, p F q series, and binomial-harmonic series using G , extending known results on the classical version of Catalan’s constant.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136178","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}
. Following the concept of statistical convergence, in this paper we introduce two more subtle notions, viz. statistical ω -limit set and statistical ω -cluster set than the general ω -limit set in a discrete dynamical system of a continuous function and study some properties related to these two points.
{"title":"On statistical ω-limit sets in a discrete dynamical system","authors":"B. Biswas","doi":"10.7153/jca-2021-18-08","DOIUrl":"https://doi.org/10.7153/jca-2021-18-08","url":null,"abstract":". Following the concept of statistical convergence, in this paper we introduce two more subtle notions, viz. statistical ω -limit set and statistical ω -cluster set than the general ω -limit set in a discrete dynamical system of a continuous function and study some properties related to these two points.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136163","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}
Thangjam Birkramjit Singh, Maisnam Triveni Devi, B. Chanam
. Let p ( z ) be a polynomial of degree n . The polar derivative of p ( z ) with respect to a real or complex number α is de fi ned by Govil and Mctume [Acta Math. 104, 115–126 (2004)] proved that if p is a of having all its | | 1, then for any complex number α with In this paper, we prove an improvement of the above inequality. Further, we prove an improve- ment of a result due to Govil [Proc. Natl. Acad. Sci., 50, 50–52 (1980)].
{"title":"Sharpening of Bernstein and Turán-type inequalities for polynomials","authors":"Thangjam Birkramjit Singh, Maisnam Triveni Devi, B. Chanam","doi":"10.7153/jca-2021-18-10","DOIUrl":"https://doi.org/10.7153/jca-2021-18-10","url":null,"abstract":". Let p ( z ) be a polynomial of degree n . The polar derivative of p ( z ) with respect to a real or complex number α is de fi ned by Govil and Mctume [Acta Math. 104, 115–126 (2004)] proved that if p is a of having all its | | 1, then for any complex number α with In this paper, we prove an improvement of the above inequality. Further, we prove an improve- ment of a result due to Govil [Proc. Natl. Acad. Sci., 50, 50–52 (1980)].","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"140 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71136264","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}
Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions involve exponential, Airy and Scorer functions and slowly varying analytic coefficient functions involving simple coefficients. The approximations are uniformly valid for large values of the parameter and unbounded real and complex values of the argument. Explicit and readily computable error bounds are either furnished or available for all approximations.
{"title":"Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations","authors":"T. M. Dunster","doi":"10.7153/JCA-2020-17-06","DOIUrl":"https://doi.org/10.7153/JCA-2020-17-06","url":null,"abstract":"Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions involve exponential, Airy and Scorer functions and slowly varying analytic coefficient functions involving simple coefficients. The approximations are uniformly valid for large values of the parameter and unbounded real and complex values of the argument. Explicit and readily computable error bounds are either furnished or available for all approximations.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71135573","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}
. An open problem concerning Riemann sums, posed by O. Furdui, is considered.
讨论了O.Furdui提出的一个关于Riemann和的开放问题。
{"title":"A problem concerning Riemann sums","authors":"I. Pinelis","doi":"10.7153/JCA-2020-16-07","DOIUrl":"https://doi.org/10.7153/JCA-2020-16-07","url":null,"abstract":". An open problem concerning Riemann sums, posed by O. Furdui, is considered.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49550402","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}
Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the bounds is illustrated numerically with an application to Bessel functions of large order.
{"title":"Sharp error bounds for turning point expansions","authors":"T. M. Dunster, A. Gil, J. Segura","doi":"10.7153/jca-2021-18-05","DOIUrl":"https://doi.org/10.7153/jca-2021-18-05","url":null,"abstract":"Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the bounds is illustrated numerically with an application to Bessel functions of large order.","PeriodicalId":73656,"journal":{"name":"Journal of classical analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49082549","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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