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Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means
Journal of Inequalities and Applications volume 2011, Article number: 350973 (2011)
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 Cauchytype 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
by
Further, introduce the function
via
For each , let
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.
(H_{1}) The following considerations concern a subset of satisfying
where and are fixed integers.
Next, we proceed inductively to define the sets by
By (1.6), and this implies that for . From (1.6), again, we have .
For every and for any , let
Using these sets we define the functions inductively by
Let be an interval in , let , let such that and , and let be a convex function. For any , set
and associate to each the number
We need the following hypotheses.
(H_{2})Let and be positive tuples such that .
(H_{3}) Let be an interval, let , let be a positive tuples such that , and let , be continuous and strictly monotone functions.
(H_{4}) Let be an interval, let , and let be positive tuples such that . Further, let be a convex function.
Assume (H_{1}) and (H_{2}). The power means of order corresponding to are given as
We also use the means
For , , we introduce the mixed symmetric means with positive weights as follows:
and, for ,
We deduce the monotonicity of these means from the following refinement of the discrete Jensen inequality in [1].
Theorem 1.1.
Assume (H_{1}) and (H_{4}). Then,
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,
Corollary 1.2.
Assume (H_{1}) and (H_{2}). Let , such that , then
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 (H_{1}) and (H_{3}). Then, we define the quasiarithmetic means with respect to (1.10) and (1.11) as follows:
and, for ,
The monotonicity of these generalized means is obtained in the next corollary.
Corollary 1.3.
Assume (H_{1}) and (H_{3}). For a continuous and strictly monotone function , one defines
Then,
if either is convex and is increasing or is concave and is decreasing,
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 1.41.5, 1.9–1.12, which are based on examples in [1], the conditions (H_{2}), in the mixed symmetric means, and (H_{3}), in the quasiarithmetic means, will be assumed.
Example 1.4.
Suppose
where means that divides . Since , therefore (1.6) holds. We note that
where is the largest positive integer not greater than , and means the number of positive divisors of . Then, (1.14) gives for ,
while (1.20) gives
Assume (H_{4}) holds, and consider the set in Example 1.4. Then, Theorem 1.1 implies that
and thus
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
Moreover, for
Under the above settings, (1.15) can be written as
while (1.21) becomes
Assume (H_{4}) holds, and consider the set in Example 1.5. Then, Theorem 1.1 yields that
This shows that
The following result is also given in [1].
Theorem 1.6.
Assume (H_{1}) and (H_{4}), and suppose for any . Then,
and thus
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
Assume (H_{1}) and (H_{2}), and suppose for any . In this case, the power means of order corresponding to has the form
Now, for , and , we introduce the mixed symmetric means with positive weights related to (1.37) as follows:
Corollary 1.7.
Assume (H_{1}) and (H_{2}), and suppose for any . Let , such that . Then,
Proof.
The proof comes from Corollary 1.2.
Assume (H_{1}) and (H_{3}), and suppose for any . We define for the quasiarithmetic means with respect to (1.37) as follows:
Corollary 1.8.
Assume (H_{1}) and (H_{3}), and suppose for any . Then,
where either is convex and is increasing or is concave and is decreasing,
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
then , that is, (1.6) is satisfied for . It comes easily that , , and for each
In this case, (1.41) becomes for
and (1.43) has the form
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 (H_{4}) holds, and consider the set in Example 1.9. Then, Theorem 1.6 shows that
Thus, we have
Example 1.10.
If we set
then and thus (1.6) is satisfied. It is easy to see that , , and for each
Under these settings (1.41) becomes
and (1.43) has the form
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 (H_{4}) holds, and consider the set in Example 1.10. Then, it follows from Theorem 1.6 that
where
This yields that
Example 1.11.
We set
Then, , hence (1.6) holds. It is not hard to see that , and for each ,
Therefore, under these settings, for , (1.41) leads to
and (1.43) gives
respectively.
Assume (H_{4}) holds, and consider the set in Example 1.11. Then, Theorem 1.6 implies that
where
Therefore, we have
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 ,
Therefore under these settings, for , (1.41) gives
and (1.43) has the form
respectively.
Assume (H_{4}) holds, and consider the set in Example 1.12. Then, Theorem 1.6 yields that
where for ,
Therefore, we have
2. Main Results
We have seen that
From now on, (H_{1}) and (H_{4}) are assumed if we consider , further, the hypothesis for any is also assumed if we consider . The numbers are generated by concrete examples, and (H_{4}) 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
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
for every and every .
Proposition 2.3.
If is an exponentially convex function, then is logconvex which means that for every , and all
First, we introduce a special class of functions.
(H_{5}) 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
and defined by
It is easy to see that the sets of functions and satisfy (H_{5}).
Assume (H_{5}). If is replaced by in (2.1), we obtain
Especially,
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 (H_{5}), and suppose that the functions are continuous. The following statements hold for .
(a)For every and , the matrix is positive semidefinite. Particularly,
(b)The function is exponentially convex.
Proof.
Fix .
(a)Let , and define the functions by for , where . Then is a convex function since
By taking in (2.1), we have
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
Theorem 3.2.
Assume , . Then, there exists such that
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
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

(b)
By choosing and , we can see that the expression between and in (3.4) defines a mean.
Corollary 3.5.
Assume (H_{1}) and (H_{2}), 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
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
(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:
Moreover, we have continuous extensions of these means in other cases. By taking the limit, we have
Now, we deduce the monotonicity of these means in the form of Dresher's inequality as follows.
Theorem 3.7.
Let such that . Then
Proof.
Fix . Corollary 2.5 shows that the function is exponentially convex, and hence, by Proposition 2.3, it is logconvex. Therefore, the function is convex, which implies (see [4]) that
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:
where . By taking the limit, we have
where .
Now, we give the monotonicity of these new means.
Theorem 3.8.
Let such that . Then,
Proof.
Suppose , 2 is fixed. The function is exponentially convex, and hence, by Proposition 2.3, it is logconvex. Therefore, the function is convex, which implies (as in the proof of Theorem 3.7) that
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.
References
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Horváth L, Pečarić J: A refinement of the discrete Jensen's inequality. to appear in Mathematical Inequalities and Applications
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Khan KA, Pečarić J, Perić I: Differences of weighted mixed symmetric means and related results. journal of Inequalities and Applications 2010, 2010:16.
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Anwar M, Jakšetić J, Pečarić J, Ur Rehman A: Exponential convexity, positive semidefinite matrices and fundamental inequalities. Journal of Mathematical Inequalities 2010,4(2):171–189.
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Pečarić JE, Proschan F, Tong YL: Convex Functions, Partial Orderings and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Boston, Mass, USA; 1992:xiv+467.
Acknowledgments
The research of L. Horváth was supported by the Hungarian National Foundations for Scientific Research, Grant no. K73274, and that of J. Pecărić was supported by the Croatian Ministry of Science, Education, and Sports under the research Grant 11711708890888.
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Horváth, L., Khan, K. & Pečarić, J. Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means. J Inequal Appl 2011, 350973 (2011). https://doi.org/10.1155/2011/350973
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Keywords
 Convex Function
 Previous Theorem
 Positive Semidefinite
 Positive Weight
 Continuous Extension