10
Sharp Inequalities Involving Neuman Means of the Second Kind with Applications
Authors: Lin-Chang Shen, Yue-Ying Yang, Wei-Mao Qian
Number of views: 142
In this paper, we give the explicit formulas for Neuman means of the second kind
NGQ(a, b) and NQG(a, b), and find the best possible parameters αi, βi ∈ (0, 1)(i = 1, 2, 3, · · · , 6)
such that the double inequalities
α1Q(a, b) + (1 − α1)G(a, b) < NQG(a, b) < β1Q(a, b) + (1 − β1)G(a, b),
α2
G(a, b)
+
1 − α2
Q(a, b)
<
1
NQG(a, b)
<
β2
G(a, b)
+
1 − β2
Q(a, b)
,
α3Q(a, b) + (1 − α3)G(a, b) < NGQ(a, b) < β3Q(a, b) + (1 − β3)G(a, b),
α4
G(a, b)
+
1 − α4
Q(a, b)
<
1
NGQ(a, b)
<
β4
G(a, b)
+
1 − β4
Q(a, b)
,
α5Q(a, b) + (1 − α5)V (a, b) < NQG(a, b) < β5Q(a, b) + (1 − β5)V (a, b),
α6Q(a, b) + (1 − α6)U(a, b) < NGQ(a, b) < β6Q(a, b) + (1 − β6)U(a, b),
holds for all a, b > 0 with a 6= b, where G(a, b) and Q(a, b) are the classical geometric and quadratic
means, V (a, b), U(a, b), NQG(a, b) and NGQ(a, b) are Yang and Neuman mean of the second kind.