Research Article
Sharp Coefficient Related Results for Nephroid Shaped Domain
Dolly Jain*
Issue:
Volume 13, Issue 4, August 2024
Pages:
51-58
Received:
24 June 2024
Accepted:
25 July 2024
Published:
7 August 2024
DOI:
10.11648/j.pamj.20241304.11
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Abstract: In this study, the focus is on two main objectives related to starlike functions associated with a nephroid-shaped domain. Firstly, the aim is to determine sharp bounds for the coefficients of these functions up to the fifth order. These bounds are crucial as they provide a detailed understanding of the behavior of the coefficients, which is important for further analysis and various applications of these functions. The sharp determination of these coefficients can aid in refining mathematical models and theoretical frameworks involving starlike functions. Secondly, the sharp bound for the third order Hankel determinant for functions in this class is also derived. The Hankel determinant is a significant tool in complex analysis, as it provides insights into the growth, distortion, and other important properties of functions. By deriving these sharp bounds, this study improves upon the existing results in the literature, thereby contributing to a more sharp characterization of starlike functions associated with nephroid-shaped domains. This advancement has the potential to lead to enhanced applications, such as in geometric function theory and fluid dynamics, and offers a deeper understanding of these mathematical functions. By addressing these objectives, the study not only fills gaps in the current research but also opens new avenues for future exploration in the field of complex analysis.
Abstract: In this study, the focus is on two main objectives related to starlike functions associated with a nephroid-shaped domain. Firstly, the aim is to determine sharp bounds for the coefficients of these functions up to the fifth order. These bounds are crucial as they provide a detailed understanding of the behavior of the coefficients, which is import...
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Research Article
The Genesis of a Theorem in the Galois Theory of p-Extensions of ℚ with Restricted Tame Ramification
John Labute*
Issue:
Volume 13, Issue 4, August 2024
Pages:
59-65
Received:
26 July 2024
Accepted:
20 August 2024
Published:
26 August 2024
DOI:
10.11648/j.pamj.20241304.12
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Abstract: This article traces the genesis of a theorem that gives for the first time examples of the Galois group GS of the maximal p-extension of ℚ, unramified outside a finite set of primes not containing an odd p, that are of cohomlogical dimension 2 if the primes in S satisfy a certain linking condition. Because the ramification is tame the pro-p-group GS has all of its derived factors finite which is a strong finitenesss condition on GS. The paper starts with a question of Serre on one relator pro-p-groups and then a detour to discrete groups where the notion of strong freeness for a sequence of homogeneous Lie elements is given and a criterion for strong freeness is established. These notions are then carried over to pro-p-groups where the linking condtion on the primes of S is translated into a cohomological criterion for a pro-p-group to have cohomological dimension 2. An analysis is given of the work of Koch where he gives a weaker criterion for a pro-p-group to have have cohomological dimension 2. A connecttion is made with this work of Koch and that of the author which would have been sufficient to prove the fact that GS was of cohomological dimension 2 for certain sets S had it been applied to investigate whether the linking condition was true for certain sets S. It is not known if the cohomological dimension of GS is 2 if S does not satisfy this linking condition.
Abstract: This article traces the genesis of a theorem that gives for the first time examples of the Galois group GS of the maximal p-extension of ℚ, unramified outside a finite set of primes not containing an odd p, that are of cohomlogical dimension 2 if the primes in S satisfy a certain linking condition. Because the ramification is tame the pro-p-group G...
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