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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
As an application, Yang’s inequality is refined.
📌 3 Key Takeaways
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Objective & Hypothesis:In this article, a new generalization of Jordan’s inequality $sumlimits_{k=1}^n mu_k (theta^t -x^t)^k leq frac{sin x} {x} - frac{sin theta} {theta} leq sumlimits_{k=1}^n omega_k (theta^t -x^t)^k$ for $t geq 2, n in Bbb{N}$ and $theta in (0, pi ]$ is established, where the coefficients $mu_k$ and $omega_k$ are defined by recursion formulas, and are the best possible.
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Key Findings & Data:In this article, a new generalization of Jordan’s inequality $sumlimits_{k=1}^n mu_k (theta^t -x^t)^k leq frac{sin x} {x} - frac{sin theta} {theta} leq sumlimits_{k=1}^n omega_k (theta^t -x^t)^k$ for $t geq 2, n in Bbb{N}$ and $theta in (0, pi ]$ is established, where the coefficients $mu_k$ and $omega_k$ are defined by recursion formulas, and are the best possible.
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Conclusion & Impact:As an application, Yang’s inequality is refined.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
In this article, a new generalization of Jordan’s inequality $sumlimits_{k=1}^n mu_k (theta^t -x^t)^k leq frac{sin x} {x} - frac{sin theta} {theta} leq sumlimits_{k=1}^n omega_k (theta^t -x^t)^k$ for $t geq 2, n in Bbb{N}$ and $theta in (0, pi ]$ is established, where the coefficients $mu_k$ and $omega_k$ are defined by recursion formulas, and are the best possible. As an application, Yang’s inequality is refined.
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Cite This Article
HUO, Z. H., NIU, D. W., CAO, J., QI, F. (2022). A generalization of Jordan's inequality and an application. Hacettepe Journal of Mathematics and Statistics, Cilt: 40 - Sayı: 1, 53–61. https://ajindex.com/articles/530546