In this section, we assume that x and y are two different positive numbers. Let , , , and be the arithmetic, geometric, logarithmic, and identric means, respectively. Without loss of generality, we set . By the transformation , we can compute and obtain
where .
Now, the four results in Section 1 are equivalent to the following ones for four classical means.
Theorem 5.1 Let , and x and y be positive real numbers with . Then
(5.1)
holds if and only if and .
Theorem 5.1 can deduce the following one, which is from Zhu [8].
Corollary 5.2 ([[8], Theorem 1])
Let , and x and y be positive real numbers with . Then
(5.2)
holds if and only if and .
When letting in Theorem 5.1, one can obtain the result (see [12–14], [[15], Theorem 1]).
Corollary 5.3 Let x and y be positive real numbers with . Then
(5.3)
holds if and only if and .
When letting in the right-hand inequality of (5.3), one can obtain the well-known inequality by Carlson [16]
(5.4)
Theorem 5.4 Let . Then
(1) if , the double inequality
(5.5)
holds if and only if and ;
(2) if , the double inequality
(5.6)
holds if and only if and .
The part (2) of Theorem 5.4 is a result of Trif [17].
When letting and in the right-hand inequality of (5.6), one can obtain the following result, which is from Sándor and Trif [18].
(5.7)
When letting in the double inequality (5.5), one can obtain the following result (see [12], [[15], Theorem 2]).
Corollary 5.5 Let x and y be positive real numbers with . Then
(5.8)
holds if and only if and .
When letting in the left-hand inequality in (5.8), one can obtain the following result, which is from Sándor [19].
(5.9)
Theorem 5.6 Let , x and y be positive real numbers with . Then
(5.10)
holds if and only if and .
Theorem 5.6 can deduce the following result (see Zhu [15]).
Corollary 5.7 ([[15], Theorem 3])
Let x and y be positive real numbers with . Then
(5.11)
holds if and only if and .
When letting in the left-hand inequality of (5.11), one can obtain the following result, which is from Sándor [4, 19].
(5.12)
Finally, we give the bounds for in terms of and , and obtain the following new result.
Theorem 5.8 Let x and y be positive real numbers with , and . Then
(5.13)
holds if and only if and .
Theorem 5.8 can deduce a result of Zhu [15]:
Corollary 5.9 ([[15], Theorem 4])
Let x and y be positive real numbers with . Then
(5.14)
holds if and only if and .
Obviously, the right-hand side of (5.14) is an extension of the following inequality:
(5.15)
which was given by Alzer [5].