#### Abstract

We study one class of Ky Fan-type inequalities, which has ties with the original Ky Fan inequality. Our result extends the known ones.

#### 1. Introduction

Let be the generalized weighted means: , where denotes the limit of as . Here , with satisfying . In this paper, we always assume . To any given and , we set .

We define , and we shall write for , for , and for if and similarly for other means when there is no risk of confusion. We further denote .

When , we define

where we set and we shall write for and for . In order to include the case of equality for various inequalities in our discussions, for any given inequality, we define to be the number which makes the inequality an equality. The author [1, Theorem ] has shown the following (in fact, only the case is shown there but one can easily extend the result to all following the method there).

Theorem 1.1. *For and , the following inequalities are equivalent:
**
where in (1.3) one requires .*

In fact, one can further show that (see [2]) the two inequalities in Theorem 1.1 are equivalent to

being valid for all . We point out here that when inequality (1.2) holds for some , one can often expect for a better result than (1.4), namely,

We note that inequality (1.2) does not hold for all pairs (see [1]). Cartwright and Field [3] first proved the validity of (1.2) for . For other extensions and refinements of (1.2), see [2, 4–8]. When , inequality (1.3) is commonly referred as the additive Ky Fan’s inequality. We refer the reader to the survey article [9] and the references therein for an account of Ky Fan's inequality.

In this paper, we will focus on the special case of (1.2), which has ties with the following result of Ky Fan that initiated the study of the whole subject.

Theorem 1.2 (see [10,page 5]). * For , , with equality holding if and only if .*

A nice result of Wang and Chen [11] determines all the pairs with such that is satisfied when . Their result is contained in the following.

Theorem 1.3. *For , holds if and only if when , when .*

We note here that Theorem 1.2 follows from the left-hand side inequality of (1.3) for the case , which in turn is a consequence of the above mentioned result of Cartwright and Field. In fact, we have the following result which is contained implicitly in [12].

Theorem 1.4. *If either side of inequality (1.2) holds for , then the same side inequality of (1.2) also holds for and any . Moreover, the above assertion also holds when applied to (1.3) or (1.4).*

On combining the above result with the result of Cartwright and Field we see that (1.2) holds for and consequently (1.3) holds for in virtue of Theorem 1.1.

Now, it is natural to be motivated by the result of Wang and Chen, in view of the discussions above, to ask whether one can determine all the pairs with such that either one of the inequalities (1.2)–(1.4) holds for . It is our goal in this paper to investigate such a problem. Before we proceed, we would like to summarize the known results in this area. On taking in [5, Proposition ], we deduce with the help of Theorem 1.4 that (1.2) holds for . On the other hand, [5, Corollary ] combined with Theorem 1.4 implies that (1.3) holds for and . We also observe that if (1.2) holds for and , then it also holds for . As (1.2) and (1.3) are equivalent, we conclude that when , (1.2) holds for any .

#### 2. The Main Theorem

Lemma 2.1. *Let , and let denote the region . Define
**
Then for , holds for all if and only if and and holds for all if and only if and . **For , if , then holds for all if and only if and and holds for all if and only if and . **For , holds for all if and only if and or and .*

*Proof. *When , in order for to hold for all , one just needs to check the case . In this case we can rewrite as
Note that ; hence in order for to hold for all , it is necessary that . Note that and from this one checks easily that is equivalent to . On the other hand, on taking , we see that one needs to have in order for to hold for all . Now, it also follows from that . Hence one deduces via Taylor expansion of at that for all .

Similarly, when , in order for to hold for all , one just needs to check for . As , certainly it is necessary to have and . These imply that and and one checks easily that these conditions are also sufficient.

As a consequence of the above discussion, one can deduce the assertion of the lemma for the case and by noting that .

It remains to treat the case . We let , and note that . It follows from this that in order for to hold for , it suffices to check the cases . When , we only need to check the case and in this case one can discuss similarly to the case above to conclude the assertion of the lemma. We just point out here that as , we have . When , it suffices to check the case and in this case one uses the relation to convert this to the previous case that has been discussed.

Theorem 2.2. *Let . The right-hand side inequality of (1.2) holds for when or and . The left-hand side inequality of (1.2) holds for when .*

*Proof. *To prove the first assertion of the theorem, we may assume or in view of our discussion in the last paragraph of Section 1 and for the case , we define
Similar to the proof of Theorem [2], it suffices to show that . Calculation shows that
We now show by induction on that . When , there is nothing to prove. When , this becomes
by Lemma 2.1.

Suppose now ; in order to show , we may assume that are being fixed and it suffices to show that the maximum value of is non-positive on the region , where and .

Let be a point of in which the absolute maximum of is reached. If for some , by combining with and with , we are back to the case of variables with different weights. If for some , then we have
by Lemma 2.1. If for some , we are back to the case of variables. If , then we may assume that and note that we have and that
and we are again back to the case . If , then similarly we may assume that and if we can show that (again with and here)
then we are back to the case of variables. Note that the above inequality will follow if the function
is an increasing function for (in fact, one only needs this for ) and its derivative is
with the inequality holding for the case (note that together with , this implies that ). It also follows from that has no root in . One then deduces from and that for .

So from now on it remains to consider the case for and this implies that is an interior point of . We will now show that this cannot happen.

We define
Note here in the definition of that are not functions of , they take values at some point to be specified, and is also a constant to be specified.

As is an interior point of , we may use the Lagrange multiplier method to obtain a real number so that at ,
for all and .

By (2.12), a computation shows that each () is a root of (where take their values at ) and each () is a root of . Now implies . As , it follows from Rolle’s Theorem that there must be two numbers such that . However, it is easy to see that has at most two positive roots and this contradiction implies the first assertion of the theorem for the case .

Now to show the right-hand side inequality hold of (1.2) for the case and , once again it suffices to show that the function defined above is nonnegative for any integer . We note that when , this is obvious and when , this follows again from by Lemma 2.1.

Suppose now ; in order to show , we may assume that are being fixed and it suffices to show the minimum value of is nonnegative on the region , where and are defined as above.

Let be a point of in which the absolute minimum of is reached. Note that ; thus if for some , by combining with and with , we are back to the case of variables with different weights. Similarly, if for some , then we are back to the case . If for some , we are back to the case of variables. If , then we may assume that and note that we have and that
and we are again back to the case of variables.

So from now on it remains to consider the case for , and this implies that is an interior point of . We will now show that this cannot happen.

We define
Here we define . Also note here in the definition of , that are not functions of , they take values at some point to be specified, and is also a constant to be specified.

As is an interior point of , we may use the Lagrange multiplier method to obtain a real number so that at ,
for all and .

By (2.15), a computation shows that each () is a root of (where take their values at ) and each () is a root of . Now implies . As , it follows from Rolle’s Theorem that there must be two numbers such that . However, we have
It is easy to see that has at most one positive root, which implies that has at most three positive roots. As , it follows from and that has even numbers of roots so that can have at most two positive roots. This contradiction now establishes the right-hand side inequality of (1.2) for the case , and .

One can show the second assertion of the theorem using an argument similar to the above and we shall leave this to the reader.

#### 3. Further Discussions

As we have pointed out in Section 1 that if either one of the inequalities (1.2)–(1.4) holds for some , then one often expects inequality (1.5) to hold as well for the same . In view of this, one may ask whether it is feasible to prove so for those pairs satisfying Theorem 2.2. We now prove a special case here.

Theorem 3.1. *Let , then the following inequality holds:
*

*Proof. *We first prove the theorem for the case . For this, we may assume that is fixed and replace with so that in what follows. We define
As in the proof of Theorem 2.2, it suffices to show that and calculation shows
where
It is easy to check that
In view of (3.5), the inequality will follow from
for . We may assume that here and it is easy to see that can have at most one root in between and . This combined with the observation that implies that reaches its local maximum at if it exists. Hence we are left to check that . In this case we note that and we rewrite as
where
In view of (3.5) again, we just need to show that is an increasing function for . Note that
and it is easy to see that the function is non-negative for when by considering its Taylor expansion at and this completes the proof for the assertion of the theorem for the case .

To prove the theorem for the case , we may again assume that is fixed and define
Again it suffices to show that and calculation shows
where
In view of (3.5), the inequality will follow from
for . We may assume that here and it is easy to see that can have at most one root in between and . This combined with the observation that implies that reaches its local maximum at if it exists. Hence we are left to check that . As , this completes the proof for the remaining case of the theorem.

Now we show that, in general, it is not true that for ,

To proceed, we first look at the following related inequalities (with here):

Let denote the right-hand side expression of (3.15); then (3.15) holds if and only if . As is arbitrary, we can recast this condition as

Similarly, (3.16) holds if and only if the following inequality holds:

As a first step towards establishing (3.17), we consider the case here; in this case we let and rewrite the left-hand side of (3.17) as

with being defined as in Lemma 2.1. Using the same notations as in Lemma 2.1, we see that in order for (3.17) to hold for , we need to have for . Similar treatment of (3.18) shows that in order for it to hold in the case , one needs to have for .

It follows from the proof of Lemma 2.1 that fails to hold for all when . In another words, there exists such that when ,

holds. Now we return to the inequality (3.14) and we take there. Just as in the discussion above, one sees that (3.14) is equivalent to

This combined with (3.20) now implies that for ,

However, on taking on (3.14), we get the above inequality reversed (with replaced by ) and this leads to a contradiction; hence (3.14) does not hold for in general.

To end this paper, we note that it is an open problem to determine all the triples so that inequality (1.2) holds. However, when with or , the result given in Theorem 2.2 is best possible. In this case Theorem 2.2 implies that for ,

We point out here that inequality (3.23) does not hold in general when . To see this, it suffices to consider the case and in this case we can set and consider more generally for , the function defined in the proof of Theorem 2.2, regarding it as a function of . It is easy to check that ; hence by the Taylor expansion of around , we need in order for to hold for any . Calculation shows that

On taking , one sees immediately that we must have here in order for for all . On taking , we see that one needs in order for (3.23) to hold for positive . Similarly, one checks easily that in the case , if inequality (3.23) holds for some , then it also holds for by a change of variables . Hence one needs in order for (3.23) to hold for any negative .