Of course, we may assume without loss of generality that and (and the same for , , , ).
The proof begins with the crucial lemma.
Then the inequality
is satisfied if and only if the relation
, or equivalently
, if and only if
Let us denote by
. Then, from (19), it follows
and we find
In view of (19) and
, one can show that
Indeed, let us verify the relations (24). We have
which hold true since
. Similarly, we have
which holds true since
. 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
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
We have to show that
is decreasing with respect to
. We compute the first derivative
The function (28) has the same sign as the function
, the function
In order to show that
, we remark that
and we compute
since implies and .
Consequently, the function
is decreasing with respect to r
and for any
we have that
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. □
Let the real numbers
be such that
Then one of the following inequalities holds
The inequalities (32) and (33) are satisfied simultaneously if and only if , and .
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 , ). □
Let the real numbers
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
Then we have
If we apply the Consequence 12 for the numbers
, then we obtain that
In what follows, let us show that
Using the notations
. With the help of the function h
defined in (27), we can write the inequality (39) in the form
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
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
In our case, we derive the following result: for any real numbers x
, the inequality
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
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 , .
Indeed, assume that
. Then we can apply the Consequence 12 and we deduce that
Taking into account (7)1,2
in conjunction with (45), we find
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 , :
If the real numbers
are such that
then the inequality
holds true. Note that the additional condition
is automatically fulfilled.
, we have
so that the inequality (48) is equivalent to , i.e., we have to show that .
Indeed, if we insert
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
. Then (47) implies
as well as
because , and Theorem 6 provides the assertion. □