The aim of this paper is to examine the families of monotonically stratified functions with respect to one parameter and the connections of such families of functions with certain results stemming from the Theory of Analytic Inequalities. The obtained results are applied to the Cusa-Huygens inequality and some related inequalities.
{"title":"A minimax approximant in the theory of analytic inequalities","authors":"Branko J. Malesevic, Bojana Mihailovic","doi":"10.2298/aadm210511032m","DOIUrl":"https://doi.org/10.2298/aadm210511032m","url":null,"abstract":"The aim of this paper is to examine the families of monotonically stratified functions with respect to one parameter and the connections of such families of functions with certain results stemming from the Theory of Analytic Inequalities. The obtained results are applied to the Cusa-Huygens inequality and some related inequalities.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353779","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}
Using the Bernstein theorem we prove the complete monotonicity of the three parameter Mittag?Leffler function E??,? (?w) for w ? 0 and suitably constrained parameters ?, ? and ?.
{"title":"On complete monotonicity of three parameter Mittag-Leffler function","authors":"K. Górska, A. Horzela, A. Lattanzi, K. T. Pogány","doi":"10.2298/aadm190226025g","DOIUrl":"https://doi.org/10.2298/aadm190226025g","url":null,"abstract":"Using the Bernstein theorem we prove the complete monotonicity of the three parameter Mittag?Leffler function E??,? (?w) for w ? 0 and suitably constrained parameters ?, ? and ?.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68352611","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}
As a sequel to our recent paper, its general approach was here extended to finite alternating trigonometric sums giving rise to polynomials which were systematically examined in full detail as well as in a unified manner using simple arguments. Two new general families of integer-valued polynomials (along with four other families derived from them, also integer-valued, including two already known) were deduced. Also, these polynomials enable closed-form summation of a great deal of general families of finite sums.
{"title":"Another two families of integer-valued polynomials associated with finite trigonometric sums","authors":"D. Cvijovic","doi":"10.2298/AADM200915004C","DOIUrl":"https://doi.org/10.2298/AADM200915004C","url":null,"abstract":"As a sequel to our recent paper, its general approach was here extended to finite alternating trigonometric sums giving rise to polynomials which were systematically examined in full detail as well as in a unified manner using simple arguments. Two new general families of integer-valued polynomials (along with four other families derived from them, also integer-valued, including two already known) were deduced. Also, these polynomials enable closed-form summation of a great deal of general families of finite sums.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353181","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}
Aleksandar Kartelj, Milana Grbić, Dragan Matic, V. Filipović
In this paper we study the Roman domination number of some classes of planar graphs - convex polytopes: An, Rn and Tn. We establish the exact values of Roman domination number for: An, R3k, R3k+1, T8k, T8k+2, T8k+3, T8k+5 and T8k+6. For R3k+2, T8k+1, T8k+4 and T8k-1 we propose new upper and lower bounds, proving that the gap between the bounds is 1 for all cases except for the case of T8k+4, where the gap is 2.
{"title":"The Roman domination number of some special classes of graphs - convex polytopes","authors":"Aleksandar Kartelj, Milana Grbić, Dragan Matic, V. Filipović","doi":"10.2298/AADM171211019K","DOIUrl":"https://doi.org/10.2298/AADM171211019K","url":null,"abstract":"In this paper we study the Roman domination number of some classes of planar graphs - convex polytopes: An, Rn and Tn. We establish the exact values of Roman domination number for: An, R3k, R3k+1, T8k, T8k+2, T8k+3, T8k+5 and T8k+6. For R3k+2, T8k+1, T8k+4 and T8k-1 we propose new upper and lower bounds, proving that the gap between the bounds is 1 for all cases except for the case of T8k+4, where the gap is 2.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68351515","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}
Let K be a number field. We investigate the indices I(K) and i(K) of K introduced respectively by Dedekind and Gunji-McQuillan. Let n be a positif integer, we then prove that for any prime p ? n, there exists K a number field of degree n over Q such that p divide i(K). This result is an analogue to Bauer''s one for i(K). We compute I(K) and i(K) for cubic fields and infinite families of simplest number fields of degree less than 7. We solve questions and disprove the conjecture stated in [1].
{"title":"On the indices in number fields and their computation for small degrees","authors":"A. Bayad, M. Seddik","doi":"10.2298/aadm191025032b","DOIUrl":"https://doi.org/10.2298/aadm191025032b","url":null,"abstract":"Let K be a number field. We investigate the indices I(K) and i(K) of K introduced respectively by Dedekind and Gunji-McQuillan. Let n be a positif integer, we then prove that for any prime p ? n, there exists K a number field of degree n over Q such that p divide i(K). This result is an analogue to Bauer''s one for i(K). We compute I(K) and i(K) for cubic fields and infinite families of simplest number fields of degree less than 7. We solve questions and disprove the conjecture stated in [1].","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68352662","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}
Our perpose in this work is the complete the study of Simsek numbers. We give answer to some open problems concerning polynomial representations and associated generating function. At the end of the study we investigate a new generalization of these numbers and obtain useful identities which connect Simsek numbers and Stirling numbers of second kind.
{"title":"Generating functions for generalization Simsek numbers and their applications","authors":"M. Goubi","doi":"10.2298/AADM200522005G","DOIUrl":"https://doi.org/10.2298/AADM200522005G","url":null,"abstract":"Our perpose in this work is the complete the study of Simsek numbers. We give answer to some open problems concerning polynomial representations and associated generating function. At the end of the study we investigate a new generalization of these numbers and obtain useful identities which connect Simsek numbers and Stirling numbers of second kind.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353130","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}
The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.
{"title":"Upper bounds on the energy of graphs in terms of matching number","authors":"S. Akbari, Abdullah J. Alazemi, Milica Andjelic","doi":"10.2298/aadm201227016a","DOIUrl":"https://doi.org/10.2298/aadm201227016a","url":null,"abstract":"The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353926","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}
In this paper we consider the generalized Jacobsthal Jn,m and the generalized Jacobsthal-Lucas numbers jn,m. Also, we introduce new sequences of numbers An,m, Bn,m, Cn,m and Dn,m. Namely, these new sequences are convolutions of the sequences Jn,m and jn,m. Further, we find the generating functions and some recurrence relations for these sequences of numbers.
{"title":"Convolutions of the generalized Jacobsthal and generalized Lucas numbers","authors":"G. Djordjevic, S. Djordjevic","doi":"10.2298/AADM191103008D","DOIUrl":"https://doi.org/10.2298/AADM191103008D","url":null,"abstract":"In this paper we consider the generalized Jacobsthal Jn,m and the generalized Jacobsthal-Lucas numbers jn,m. Also, we introduce new sequences of numbers An,m, Bn,m, Cn,m and Dn,m. Namely, these new sequences are convolutions of the sequences Jn,m and jn,m. Further, we find the generating functions and some recurrence relations for these sequences of numbers.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68352679","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}
Let p(z) = ?n?=0 a?z? be a polynomial of degree n, M(p,R) := max|z|=R?0 |p(z)|, and M(p,1) := ||p||. Then according to a well-known result of Ankeny and Rivlin, we have for R ? 1, M(p,R) ? (Rn+1/2) ||p||. This inequality has been sharpened among others by Govil, who proved that for R ? 1, M(p,R) ? (Rn+1/2) ||p||-n/2 (||p||2-4|an|2/||p||) {(R-1)||p||/||p||+2|an|- ln (1+ (R-1)||p||/||p||+2|an|)}. In this paper, we sharpen the above inequality of Govil, which in turn sharpens inequality of Ankeny and Rivlin. We present our result in terms of the LerchPhi function ?(z,s,a), implemented in Wolfram's MATHEMATICA as LerchPhi [z,s,a], which can be evaluated to arbitrary numerical precision, and is suitable for both symbolic and numerical manipulations. Also, we present an example and by using MATLAB show that for some polynomials the improvement in bound can be considerably significant.
令p(z) = ?n?= 0 z ?是n次多项式，M(p,R) = max|z|=R?0 |p(z)|， and M(p,1):= ||p||。然后根据Ankeny和Rivlin的一个著名的结果，我们有R ?1 M(p,R) ?p (Rn + 1/2) | | | |。这种不平等被Govil强化了，他证明了R ?1 M(p,R) ?p (Rn + 1/2) | | | | - n / 2 (| | p | | 2 - 4 | | 2 / | | p | |) {(r1) | | p | |和| | p | | + 2 | | - ln (1 + (r1) | | p | |和| | p | | + 2 | |)}。在本文中，我们锐化了Govil的上述不等式，这反过来锐化了Ankeny和Rivlin的不等式。我们用LerchPhi函数?(z,s,a)来表示我们的结果，该函数在Wolfram的MATHEMATICA中实现为LerchPhi [z,s,a]，它可以计算为任意数值精度，并且适用于符号和数值操作。此外，我们还给出了一个例子，并通过MATLAB证明了对某些多项式的界的改进是相当显著的。
{"title":"A note on sharpening of a theorem of Ankeny and Rivlin","authors":"Aseem Dalal, K. Govil","doi":"10.2298/AADM200206012D","DOIUrl":"https://doi.org/10.2298/AADM200206012D","url":null,"abstract":"Let p(z) = ?n?=0 a?z? be a polynomial of degree n, M(p,R) := max|z|=R?0 |p(z)|, and M(p,1) := ||p||. Then according to a well-known result of Ankeny and Rivlin, we have for R ? 1, M(p,R) ? (Rn+1/2) ||p||. This inequality has been sharpened among others by Govil, who proved that for R ? 1, M(p,R) ? (Rn+1/2) ||p||-n/2 (||p||2-4|an|2/||p||) {(R-1)||p||/||p||+2|an|- ln (1+ (R-1)||p||/||p||+2|an|)}. In this paper, we sharpen the above inequality of Govil, which in turn sharpens inequality of Ankeny and Rivlin. We present our result in terms of the LerchPhi function ?(z,s,a), implemented in Wolfram's MATHEMATICA as LerchPhi [z,s,a], which can be evaluated to arbitrary numerical precision, and is suitable for both symbolic and numerical manipulations. Also, we present an example and by using MATLAB show that for some polynomials the improvement in bound can be considerably significant.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353320","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}
In this paper, we mainly concerned with an alternate form of the generating functions for a certain class of combinatorial numbers and polynomials. We give matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type formula, recurrence relation and derivative formula for the aforementioned combinatorial numbers. Besides, we present some plots of the generating functions for these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, but the two-variable Hermite polynomials and the number of cyclic derangements. We also present some applications of these relationships. By applying Laplace transform and Mellin transform respectively to the aforementioned functions, we give not only an infinite series representation, but also an interpolation function of these combinatorial numbers. We also provide a contour integral representation of these combinatorial numbers. In addition, we construct exponential generating functions for a new family of numbers arising from the linear combination of the numbers of cyclic derangements in the wreath product of the finite cyclic group and the symmetric group of permutations of a set. Finally, we analyse the aforementioned functions in probabilistic and asymptotic manners, and we give some of their relationships with not only the Laplace distribution, but also the standard normal distribution. Then, we provide an asymptotic power series representation of the aforementioned exponential generating functions.
{"title":"Matrix representations for a certain class of combinatorial numbers associated with Bernstein basis functions and cyclic derangements and their probabilistic and asymptotic analyses","authors":"Irem Kucukoglu, Y. Simsek","doi":"10.2298/AADM201017009K","DOIUrl":"https://doi.org/10.2298/AADM201017009K","url":null,"abstract":"In this paper, we mainly concerned with an alternate form of the generating functions for a certain class of combinatorial numbers and polynomials. We give matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type formula, recurrence relation and derivative formula for the aforementioned combinatorial numbers. Besides, we present some plots of the generating functions for these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, but the two-variable Hermite polynomials and the number of cyclic derangements. We also present some applications of these relationships. By applying Laplace transform and Mellin transform respectively to the aforementioned functions, we give not only an infinite series representation, but also an interpolation function of these combinatorial numbers. We also provide a contour integral representation of these combinatorial numbers. In addition, we construct exponential generating functions for a new family of numbers arising from the linear combination of the numbers of cyclic derangements in the wreath product of the finite cyclic group and the symmetric group of permutations of a set. Finally, we analyse the aforementioned functions in probabilistic and asymptotic manners, and we give some of their relationships with not only the Laplace distribution, but also the standard normal distribution. Then, we provide an asymptotic power series representation of the aforementioned exponential generating functions.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353708","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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