دورية أكاديمية
Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters
| العنوان: | Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters |
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| المؤلفون: | Wei-Mao Qian, Yu-Ming Chu |
| المصدر: | Journal of Inequalities and Applications, Vol 2017, Iss 1, Pp 1-10 (2017) |
| بيانات النشر: | SpringerOpen, 2017. |
| سنة النشر: | 2017 |
| المجموعة: | LCC:Mathematics |
| مصطلحات موضوعية: | quasi-arithmetic mean, complete elliptic integral, Gaussian hypergeometric function, arithmetic mean, geometric mean, Mathematics, QA1-939 |
| الوصف: | Abstract In the article, we present the best possible parameters λ = λ ( p ) $\lambda=\lambda (p)$ and μ = μ ( p ) $\mu=\mu(p)$ on the interval [ 0 , 1 / 2 ] $[0, 1/2]$ such that the double inequality G p [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] A 1 − p ( a , b ) < E ( a , b ) < G p [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] A 1 − p ( a , b ) $$\begin{aligned}& G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a \bigr]A^{1-p}(a,b) \\& \quad< E(a,b) < G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}$$ holds for any p ∈ [ 1 , ∞ ) $p\in[1, \infty)$ and all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , where A ( a , b ) = ( a + b ) / 2 $A(a, b)=(a+b)/2$ , G ( a , b ) = a b $G(a,b)=\sqrt{ab}$ and E ( a , b ) = [ 2 ∫ 0 π / 2 a cos 2 θ + b sin 2 θ d θ / π ] 2 $E(a,b)=[2\int_{0}^{\pi /2}\sqrt{a\cos^{2}\theta+b\sin^{2}\theta}\,d\theta/\pi]^{2}$ are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively. |
| نوع الوثيقة: | article |
| وصف الملف: | electronic resource |
| اللغة: | English |
| تدمد: | 1029-242X |
| Relation: | http://link.springer.com/article/10.1186/s13660-017-1550-5; https://doaj.org/toc/1029-242X |
| DOI: | 10.1186/s13660-017-1550-5 |
| URL الوصول: | https://doaj.org/article/241a6ecf929e42afb5aed3eaeadd36f9 |
| رقم الأكسشن: | edsdoj.241a6ecf929e42afb5aed3eaeadd36f9 |
| قاعدة البيانات: | Directory of Open Access Journals |
Find this article in full text from Springer Verlag. 
Full Text Finder | تدمد: | 1029242X |
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| DOI: | 10.1186/s13660-017-1550-5 |