Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

Tie-hong Zhao, Yu-ming Chu
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Abstract

In the article, we prove that the inequalities

$${H_p}({\cal K}(r),{\cal E}(r)) > {\pi \over 2},\,\,\,\,\,\,{L_q}({\cal K}(r),{\cal E}(r)) > {\pi \over 2}$$

hold for all r ∈ (0, 1) if and only if p ≥ −3/4 and q ≥ −3/4, where Hp(a, b) and Lq(a, b) are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b, and \({\cal K}(r)\) and \({\cal E}(r)\) are respectively the complete elliptic integrals of the first and second kinds.

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完全椭圆积分的锐幂型Heronian和Lehmer意味着不等式
本文证明了不等式$${H_p}({\cal K}(r),{\cal E}(r)) > {\pi \over 2},\,\,\,\,\,\,{L_q}({\cal K}(r),{\cal E}(r)) > {\pi \over 2}$$对所有r∈(0,1)成立,当且仅当p≥−3/4和q≥−3/4,其中Hp(a, b)和Lq(a, b)分别是a和b的p次幂型Heronian均值和q次Lehmer均值,\({\cal K}(r)\)和\({\cal E}(r)\)分别是第一类和第二类完全椭圆积分。
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来源期刊
自引率
10.00%
发文量
33
期刊介绍: Applied Mathematics promotes the integration of mathematics with other scientific disciplines, expanding its fields of study and promoting the development of relevant interdisciplinary subjects. The journal mainly publishes original research papers that apply mathematical concepts, theories and methods to other subjects such as physics, chemistry, biology, information science, energy, environmental science, economics, and finance. In addition, it also reports the latest developments and trends in which mathematics interacts with other disciplines. Readers include professors and students, professionals in applied mathematics, and engineers at research institutes and in industry. Applied Mathematics - A Journal of Chinese Universities has been an English-language quarterly since 1993. The English edition, abbreviated as Series B, has different contents than this Chinese edition, Series A.
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