Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means

Abstract

A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means.

1. Introduction and Preliminary Results

A new refinement of the discrete Jensen inequality is given in [1]. The following notations are also introduced in [1].

Let be a set, its power set, and denotes the number of elements in . Let and be fixed integers. Define the functions

(1.1)

by

(1.2)

Further, introduce the function

(1.3)

via

(1.4)

For each , let

(1.5)

It is easy to observe from the construction of the functions and that they do not depend essentially on , so we can write for short for , and so on.

(H₁) The following considerations concern a subset of satisfying

(1.6)

where and are fixed integers.

Next, we proceed inductively to define the sets by

(1.7)

By (1.6), and this implies that for . From (1.6), again, we have .

For every and for any , let

(1.8)

Using these sets we define the functions inductively by

(1.9)

Let be an interval in , let , let such that and , and let be a convex function. For any , set

(1.10)

and associate to each the number

(1.11)

We need the following hypotheses.

(H₂)Let and be positive -tuples such that .

(H₃) Let be an interval, let , let be a positive -tuples such that , and let , be continuous and strictly monotone functions.

(H₄) Let be an interval, let , and let be positive -tuples such that . Further, let be a convex function.

Assume (H₁) and (H₂). The power means of order corresponding to are given as

(1.12)

We also use the means

(1.13)

For , , we introduce the mixed symmetric means with positive weights as follows:

(1.14)

and, for ,

(1.15)

We deduce the monotonicity of these means from the following refinement of the discrete Jensen inequality in [1].

Theorem 1.1.

Assume (H₁) and (H₄). Then,

(1.16)

where the numbers are defined in (1.10) and (1.11). If is a concave function, then the inequalities in (1.16) are reversed.

Under the conditions of the previous theorem,

(1.17)

Corollary 1.2.

Assume (H₁) and (H₂). Let , such that , then

(1.18)

(1.19)

Proof.

Assume . To obtain (1.18), we can apply Theorem 1.1 to the function and the -tuples to get the analogue of (1.16) and to raise the power . Equation (1.19) can be proved in a similar way by using and and raising the power .

When or , we get the required results by taking limit.

Assume (H₁) and (H₃). Then, we define the quasiarithmetic means with respect to (1.10) and (1.11) as follows:

(1.20)

and, for ,

(1.21)

The monotonicity of these generalized means is obtained in the next corollary.

Corollary 1.3.

Assume (H₁) and (H₃). For a continuous and strictly monotone function , one defines

(1.22)

Then,

(1.23)

if either is convex and is increasing or is concave and is decreasing,

(1.24)

if either is convex and is decreasing or is concave and is increasing.

Proof.

First, we can apply Theorem 1.1 to the function and the -tuples , then we can apply to the inequality coming from (1.16). This gives (1.23). A similar argument gives (1.24): and can be used.

Throughout Examples 1.4-1.5, 1.9–1.12, which are based on examples in [1], the conditions (H₂), in the mixed symmetric means, and (H₃), in the quasiarithmetic means, will be assumed.

Example 1.4.

Suppose

(1.25)

where means that divides . Since , therefore (1.6) holds. We note that

(1.26)

where is the largest positive integer not greater than , and means the number of positive divisors of . Then, (1.14) gives for ,

(1.27)

while (1.20) gives

(1.28)

Assume (H₄) holds, and consider the set in Example 1.4. Then, Theorem 1.1 implies that

(1.29)

and thus

(1.30)

Example 1.5.

Let be an integer , let , and also let consist of all sequences in which the number of occurrences of is . Obviously, (1.6) holds, and, by simple calculations, we have

(1.31)

Moreover, for

(1.32)

Under the above settings, (1.15) can be written as

(1.33)

while (1.21) becomes

(1.34)

Assume (H₄) holds, and consider the set in Example 1.5. Then, Theorem 1.1 yields that

(1.35)

This shows that

(1.36)

The following result is also given in [1].

Theorem 1.6.

Assume (H₁) and (H₄), and suppose for any . Then,

(1.37)

and thus

(1.38)

If is a concave function then the inequalities (1.38) are reversed.

Under the conditions of the previous theorem, we have, from (1.38), that

(1.39)

Assume (H₁) and (H₂), and suppose for any . In this case, the power means of order corresponding to has the form

(1.40)

Now, for , and , we introduce the mixed symmetric means with positive weights related to (1.37) as follows:

(1.41)

Corollary 1.7.

Assume (H₁) and (H₂), and suppose for any . Let , such that . Then,

(1.42)

Proof.

The proof comes from Corollary 1.2.

Assume (H₁) and (H₃), and suppose for any . We define for the quasiarithmetic means with respect to (1.37) as follows:

(1.43)

Corollary 1.8.

Assume (H₁) and (H₃), and suppose for any . Then,

(1.44)

where either is convex and is increasing or is concave and is decreasing,

(1.45)

where either is convex and is decreasing or is concave and is increasing.

Proof.

The proof is a consequence of Corollary 1.3.

Example 1.9.

If we set

(1.46)

then , that is, (1.6) is satisfied for . It comes easily that , , and for each

(1.47)

In this case, (1.41) becomes for

(1.48)

and (1.43) has the form

(1.49)

Equation (1.48) is a weighted mixed symmetric mean and (1.49) is a generalized mean, as given in [2]. Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries  1.2 and 1.3 given in [2].

Assume (H₄) holds, and consider the set in Example 1.9. Then, Theorem 1.6 shows that

(1.50)

Thus, we have

(1.51)

Example 1.10.

If we set

(1.52)

then and thus (1.6) is satisfied. It is easy to see that , , and for each

(1.53)

Under these settings (1.41) becomes

(1.54)

and (1.43) has the form

(1.55)

Equation (1.54) represents weighted mixed symmetric means, and (1.55) defines generalized means, as given in [2]. Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries  1.9 and 1.10 given in [2].

Assume (H₄) holds, and consider the set in Example 1.10. Then, it follows from Theorem 1.6 that

(1.56)

where

(1.57)

This yields that

(1.58)

Example 1.11.

We set

(1.59)

Then, , hence (1.6) holds. It is not hard to see that , and for each ,

(1.60)

Therefore, under these settings, for , (1.41) leads to

(1.61)

and (1.43) gives

(1.62)

respectively.

Assume (H₄) holds, and consider the set in Example 1.11. Then, Theorem 1.6 implies that

(1.63)

where

(1.64)

Therefore, we have

(1.65)

Example 1.12.

Let and let consist of all sequences of distinct numbers from . Then, , hence (1.6) holds. It is immediate that , and for every ,

(1.66)

Therefore under these settings, for , (1.41) gives

(1.67)

and (1.43) has the form

(1.68)

respectively.

Assume (H₄) holds, and consider the set in Example 1.12. Then, Theorem 1.6 yields that

(1.69)

where for ,

(1.70)

Therefore, we have

(1.71)

2. Main Results

We have seen that

(2.1)

From now on, (H₁) and (H₄) are assumed if we consider , further, the hypothesis for any is also assumed if we consider . The numbers are generated by concrete examples, and (H₄) is assumed.

We need the following subclass of convex functions (see [3]).

Definition 2.1.

A function is exponentially convex if it is continuous and

(2.2)

for all and all choices and .

We quote here useful propositions from [3].

Proposition 2.2.

Let be a function. Then, the following statements are equivalent

(i) is exponentially convex.

(ii) is continuous and

(2.3)

for every and every .

Proposition 2.3.

If is an exponentially convex function, then is log-convex which means that for every , and all

(2.4)

First, we introduce a special class of functions.

(H₅) Let be a set of twice differentiable convex functions such that the function is exponentially convex for every fixed .

As examples, consider two classes of functions defined by

(2.5)

and defined by

(2.6)

It is easy to see that the sets of functions and satisfy (H₅).

Assume (H₅). If is replaced by in (2.1), we obtain

(2.7)

Especially,

(2.8)

In this paper we prove the exponential convexity of the functions , and we give mean value theorems for . We also define the respective means of Cauchy type and study their monotonicity. The results for are special cases of the results for , and the results for are special cases of results for . Especially, the results for are also given in [2].

Theorem 2.4.

Assume (H₅), and suppose that the functions are continuous. The following statements hold for .

(a)For every and , the matrix is positive semidefinite. Particularly,

(2.9)

(b)The function is exponentially convex.

Proof.

Fix .

(a)Let , and define the functions by for , where . Then is a convex function since

(2.10)

By taking in (2.1), we have

(2.11)

This means that the matrix is positive semidefinite, that is, (2.9) is valid.

(b)It is assumed that the function is continuous. By using Poposition 2.2 and (a), we get the exponential convexity of the function .

Since the functions are continuous , we have the following.

Corollary 2.5.

The function are exponentially convex. This remains valid if we replace by in (2.8).

3. Cauchy Means

In this section, first, we are interested in mean value theorems.

Theorem 3.1.

Assume and . Then, there exists such that

(3.1)

Theorem 3.2.

Assume , . Then, there exists such that

(3.2)

provided that the denominators are nonzero.

The idea of the proofs of Theorems 3.1 and 3.2 is the same as the proofs of Theorems  2.3 and 2.4 in [2].

Corollary 3.3.

Let , , and . Then, for distinct real numbers and , different from 0 and 1, there exists such that

(3.3)

Proof.

Theorem 3.2 can be applied.

Remark 3.4.

Suppose the conditions of Corollary 3.3 are satisfied.

(a)Since the function , is invertible, then we get, from (3.3), that for

(3.4)

• (b) By choosing and , we can see that the expression between and in (3.4) defines a mean.

Corollary 3.5.

Assume (H₁) and (H₂), and suppose . In (3.6), it is also supposed that for any . Then, for distinct real numbers , , and , all are different from 0 and 1, there exists , such that

(3.5)

(3.6)

Proof.

We can apply Theorem 3.2 to the functions , , , and , and the -tuples .

Remark 3.6.

Suppose the conditions of Corollary 3.5 are satisfied.

(a)Since the function is invertible, then we get, from (3.5) and (3.6) that

(3.7)

(b)As in Remark 3.4 (b), the expressions in (3.7) define means.

By Remark 3.4 (b), we can define Cauchy means for as follows:

(3.8)

Moreover, we have continuous extensions of these means in other cases. By taking the limit, we have

(3.9)

Now, we deduce the monotonicity of these means in the form of Dresher's inequality as follows.

Theorem 3.7.

Let such that . Then

(3.10)

Proof.

Fix . Corollary 2.5 shows that the function is exponentially convex, and hence, by Proposition 2.3, it is log-convex. Therefore, the function is convex, which implies (see [4]) that

(3.11)

This gives (3.10) if and . The other cases come from this by taking limit.

By Remark 3.6 (b), we can define Cauchy means in the following form:

(3.12)

where . By taking the limit, we have

(3.13)

where .

Now, we give the monotonicity of these new means.

Theorem 3.8.

Let such that . Then,

(3.14)

Proof.

Suppose , 2 is fixed. The function is exponentially convex, and hence, by Proposition 2.3, it is log-convex. Therefore, the function is convex, which implies (as in the proof of Theorem 3.7) that

(3.15)

If , set in (3.15) to obtain (3.14).

For , we get the required result by limit.

Funding

This research was partially funded by the Higher Education Commission, Pakistan.