Of course, we may assume without loss of generality that and (and the same for , , , ).
The proof begins with the crucial lemma.
Lemma 11
Let the real numbers
and
be such that
(19)
Then the inequality
(20)
is satisfied if and only if the relation
holds, or equivalently, if and only if
holds.
Proof Let us denote by . Then, from (19), it follows
and we find
(23)
In view of (19) and , , one can show that
(24)
Indeed, let us verify the relations (24). We have
which hold true since and . Similarly, we have
which holds true since and . Also, we have
which hold true since and . One can show in the same way that , , , so that (24) has been verified.
We prove now that the inequality (21) holds if and only if (22) holds. Indeed, using (23)2,4 and (24) we get
since the function is decreasing for .
Let us prove next that the inequalities (20) and (21) are equivalent. To accomplish this, we introduce the function by
(25)
Taking into account (23) and (24)1, the inequality (20) can be written equivalently as
which is equivalent to
since the function f defined by (25) is monotone increasing on , as we show next. To this aim, we denote by
Then the function (25) can be written as
(27)
We have to show that is decreasing with respect to . We compute the first derivative
(28)
The function (28) has the same sign as the function
(29)
i.e., the function given by
(30)
In order to show that for all , we remark that for fixed and we compute
since implies and .
Consequently, the function is decreasing with respect to r and for any we have that
(31)
From (29) and (31), it follows that is decreasing with respect to . This means that is increasing as a function of , i.e., the relation (26) is indeed equivalent to and the proof is complete. □
Consequence 12
Let the real numbers
and
be such that
Then one of the following inequalities holds:
(32)
or
(33)
The inequalities (32) and (33) are satisfied simultaneously if and only if , and .
Proof According to Lemma 11, the inequality (32) is equivalent to
while the inequality (33) is equivalent to
Since one of the relations (34) and (35) must hold, we have proved that one of the inequalities (32) and (33) is satisfied. They are simultaneously satisfied if and only if both (34) and (35) hold true, i.e., (and consequently , ). □
Consequence 13
Let the real numbers
and
be such that
Then we have , and .
Proof Since by hypothesis holds, we can apply Lemma 11 to deduce and .
On the other hand, by virtue of the inverse inequality and Lemma 11, we obtain and . In conclusion, we get , and . □
Proof of Theorem 10 In order to prove (13), we define the real numbers
(36)
Then we have
(37)
If we apply the Consequence 12 for the numbers and , then we obtain that
(38)
In what follows, let us show that
(39)
Using the notations and
we have and . With the help of the function h defined in (27), we can write the inequality (39) in the form
(40)
The relation (40) asserts that the function h defined in (27) is increasing with respect to the first variable . To show this, we compute the derivative
(41)
By virtue of the Chebyshev’s sum inequality, we deduce from (41) that
Indeed, the Chebyshev’s sum inequality [[6], 2.17] asserts that: if and then
In our case, we derive the following result: for any real numbers x, y, z such that , the inequality
(43)
holds true, with equality if and only if .
Applying the result (43) to the function (41), we deduce the relation (42). This means that is an increasing function of r, i.e. the inequality (40) holds, and hence, we have proved (39).
One can show analogously that the inequality
(44)
is also valid. From (38), (39) and (44), it follows that the assertion (13) holds true. Thus, the proof of Theorem 10 is complete. □
Since the statements of the Theorems 8 and 10 are equivalent, we have proved also the inequality (8).
Remark 14 The inequality (8) becomes an equality if and only if , .
Proof Indeed, assume that . Then we can apply the Consequence 12 and we deduce that
(45)
Taking into account (7)1,2 in conjunction with (45), we find
(46)
By virtue of (46), we can apply the Consequence 13 to derive , and consequently , . □
Let us prove the following version of the inequality (6) for two pairs of numbers , and , :
Remark 15 If the real numbers and are such that
(47)
then the inequality
(48)
holds true. Note that the additional condition
is automatically fulfilled.
Proof Since and , , we have , and
so that the inequality (48) is equivalent to , i.e., we have to show that .
Indeed, if we insert and into the inequality (47)1 then we find
which means that since the function is increasing for . This completes the proof. □
Alternative proof of Remark 15 Let . Then (47) implies and as well as
(49)
because , and Theorem 6 provides the assertion. □