SHARP BOUNDS FOR TOADER MEAN IN TERMS OF CONTRAHARMONIC MEAN WITH APPLICATIONS
SHARP BOUNDS FOR TOADER MEAN IN TERMS OF CONTRAHARMONIC MEAN WITH APPLICATIONS
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DOI：
10.7153/jmi-07-15
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发表时间：
2013-06-01
影响因子：
2.9
通讯作者：
Ma, Xiao-Yan
中科院分区：
文献类型：
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作者：
Chu, Yu-Ming;Wang, Miao-Kun;Ma, Xiao-Yan
我们找到 (1/2, 1) 中的最大值 lambda 和最小值 mu，使得双重不等式 C(lambda a + (1 - lambda)b,lambda b + (1 - lambda)a) < T(a,b) < C(mu a + (1 - mu)b, mu b + (1 - mu)a) 对于所有 a, b > 0 且 a 不等于 b 成立，并给出椭圆周长的新界限。这里，T(a,b) = 2/pi积分(pi/2)(0)root a(2)cos(2)theta + b(2)sin(2)theta d theta，C(a,b) = (a(2) + b(2))/(a + b)分别表示Toader和两个正数a和b的反调和平均值。
We find the greatest value lambda and the least value mu in (1/2, 1) such that the double inequality C(lambda a + (1 - lambda)b,lambda b + (1 - lambda)a) < T(a,b) < C(mu a + (1 - mu)b, mu b + (1 - mu)a) holds for all a, b > 0 with a not equal b, and give new bounds for the perimeter of an ellipse. Here, T(a,b) = 2/pi integral(pi/2)(0)root a(2)cos(2)theta + b(2)sin(2)theta d theta, and C(a,b) = (a(2) + b(2))/(a + b) denote the Toader, and contraharmonic means of two positive numbers a and b, respectively.