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

IF 1.2 4区 数学 Q2 MATHEMATICS, APPLIED
Tie-hong Zhao, Yu-ming Chu
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引用次数: 0

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.

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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.40
自引率
10.00%
发文量
453
审稿时长
>12 weeks
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