On the local and global comparison of generalized Bajraktarevi\'c means
Zsolt P\'ales, Amr Zakaria
TL;DR
This paper investigates conditions under which generalized Bajraktarević means, defined via continuous functions and measures, can be compared globally and locally, extending the understanding of their ordering properties.
Contribution
It provides new criteria for comparing these generalized means based on their generating functions, families of means, and measures, advancing the theoretical framework of mean inequalities.
Findings
Derived conditions for mean comparison inequalities
Established relationships between generating functions and mean orderings
Extended classical mean comparison results to generalized Bajraktarević means
Abstract
Given two continuous functions such that is positive and is strictly monotone, a measurable space , a measurable family of -variable means , and a probability measure on the measurable sets , the -variable mean is defined by The aim of this paper is to study the local and global comparison problem of these means, i.e., to find conditions for the generating functions and , for the families of means and , and for the measures such that the comparison inequality …
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On the local and global comparison of generalized Bajraktarević means
Zsolt Páles
Institute of Mathematics, University of Debrecen, H-4002 Debrecen, Pf. 400, Hungary
and
Amr Zakaria
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo 11341, Egypt
Abstract.
Given two continuous functions such that is positive and is strictly monotone, a measurable space , a measurable family of -variable means , and a probability measure on the measurable sets , the -variable mean is defined by
[TABLE]
The aim of this paper is to study the local and global comparison problem of these means, i.e., to find conditions for the generating functions and , for the families of means and , and for the measures such that the comparison inequality
[TABLE]
be satisfied.
Key words and phrases:
Generalized integral mean; quasi-arithmetic mean; Bajraktarević mean; Gini mean; comparison problem; Chebyshev system
2000 Mathematics Subject Classification:
Primary 26D10, 26D15, Secondary 26B25, 39B72, 41A50
This research has been supported by the Hungarian Scientific Research Fund (OKTA) Grants K-111651.
1. Introduction
In a recent paper [20], Losonczi and Páles investigated a general class of two-variable means given by the formula
[TABLE]
where are continuous functions such that is positive and is strictly monotone and is a probability measure on the Borel measurable subsets of . This definition includes many former classical and important settings. In [20] local and global comparison theorems (that provided necessary and in some cases also sufficient conditions) have been established for the comparison of two two-variable means from this general class.
The purpose of this note is to extend the results of [20] in several ways. In our approach we will use Chebyshev systems, measurable families of means and measures for the definition of a general class of -variable means.
Throughout this paper, the symbols , , and will stand for the sets of natural, real, and positive real numbers, respectively, and will always denote a nonempty open real interval. The classes of continuous strictly monotone and continuous positive real-valued functions defined on will be denoted by and , respectively.
In the sequel, a function is called a -variable mean on if the following so-called mean value property
[TABLE]
holds. Also, if both of the inequalities in (1) are strict for all with for some , then we say that is a strict mean on . The arithmetic and geometric means are well known instances for strict means on . More generally, if is a real number, then the -variable Hölder mean is defined as
[TABLE]
Obviously, and equal the arithmetic and geometric mean, respectively. It is easy to see that Hölder means are strict means. The -variable minimum and maximum functions are instances for non-strict means.
A classical generalization of Hölder means is the notion of -variable quasi-arithmetic mean (cf. [12]), which is introduced as follows: For define
[TABLE]
More generally, if denotes the -dimensional simplex given by
[TABLE]
then we can also define
[TABLE]
which is called the weighted -variable quasi-arithmetic mean on .
In this paper, we consider a much more general class of means. For their definition, we recall the notion of Chebyshev system. Let be continuous function. We say that the pair forms a (two-dimensional) Chebyshev system on if, for any distinct elements of , the determinant
[TABLE]
is different from zero. If, for , this determinant is positive, then is called a positive system, otherwise we call a negative system. Due to the connectedness of the triangle , it follows that every Chebyshev system is either positive or negative. Obviously, if is a positive Chebyshev system, then is a negative one.
The most standard positive Chebyshev system on is given by and . More generally, if are continuous functions with , , then is a Chebyshev system. Indeed, we have
[TABLE]
From, here it is obvious that vanishes if and only if . Moreover, if is decreasing (resp. increasing), then, for , we have that (resp. ), i.e., is a positive (resp. negative) Chebyshev system. By symmetry, analogous properties can be established if is positive and strictly monotone.
For the sake of convenience and brevity, now we make the following hypotheses. We say that is a measurable family of -variable means on if
- (H1)
is a nonvoid open real interval, 2. (H2)
is a measurable space, where is the -algebra of measurable sets of , 3. (H3)
for all , is a -variable mean on , 4. (H4)
for all , the function is measurable over .
If, instead of (H2) and H(4), we have that
- (H2+)
is a topological space and equals the -algebra of the Borel sets of , 2. (H4+)
for all , the function is continuous over ,
then will be called a continuous family of -variable means on .
For a measurable family of -variable means , we introduce the notations:
[TABLE]
Obviously, by property (H3), for all , we have that . Provided that is a compact and connected topological space and is a continuous family of -variable means on , we have that
[TABLE]
For the construction of a mean in terms of a Chebyshev system, a measurable family of means, and a probability measure, we need the following basic lemma.
Lemma 1**.**
Let be a measurable family of -variable means, let be a probability measure on and let be a Chebyshev system on . Then, for all , there exists a unique element such that
[TABLE]
Furthermore, if is a positive Chebyshev system, then, for all ,
[TABLE]
In addition, if is positive and is strictly monotone, then
[TABLE]
Proof.
Without loosing the generality, we may assume that is a positive Chebyshev system throughout this proof.
For fixed , consider now the following function
[TABLE]
By the continuity of and , we have that is continuous on . If , then, for all we have that , hence . This implies that is positive for all with . Similarly, for all with , we have that . Therefore, by the intermediate value property of continuous functions, must have a zero between and .
To prove the uniqueness, assume that and are distinct zeros of between and . Then we have that
[TABLE]
This means that the two unknowns and are solutions of the following system of linear equations:
[TABLE]
Because and are distinct, the determinant of this system is nonzero, hence . In this case, for all , which contradicts the property that for . The contradiction obtained shows that , which proves the uniqueness of the solution of equation (7). The uniqueness also implies that for and for .
Finally, assume that is positive and is strictly monotone (then is a Chebyshev system). In this case, the equation can be rewritten as
[TABLE]
By applying the inverse function of to this equation side by side, we obtain that is of the form (8). ∎
The above lemma allows us to define a -variable mean . Given , let denote the unique solution of equation (7). In the particular case when is positive and is strictly monotone, we have that
[TABLE]
This mean will be called a -variable generalized Bajraktarević mean in the sequel. When , then
[TABLE]
which will be termed a -variable generalized quasi-arithmetic mean. If , (where denotes the Dirac measure concentrated at ) and , then
[TABLE]
which was introduced and studied by Bajraktarević [1], [2]. When , then , which is the -variable quasi-arithmetic mean introduced in (2).
To define -variable generalized Gini means, let and assume that . By taking
[TABLE]
we can define in the following manner:
[TABLE]
In the particular case when , and , the above formula reduces to the so-called -variable Gini mean (cf. [11]):
[TABLE]
Obviously, , i.e., Hölder means are particular Gini means.
In what follows, we describe further interesting particular cases of formula (9). If , , (where with ), and (where stands for the -th Hölder mean), then
[TABLE]
In the next example we use the notations introduced in (3) and (4). If , is the -dimensional Lebesgue measure on , and , where , then
[TABLE]
If is the Lebesgue measure on and are continuously differentiable functions such that and is strictly monotone, and , then, by the Fundamental Theorem of Calculus, one can easily see that
[TABLE]
which is called a Cauchy or difference mean in the literature. Their equality problem was solved by Losonczi [17].
By taking and given in (10), the mean so obtained is the so-called Stolarsky mean, which was introduced in the papers [26] and [13]. Their comparison problem was solved by Leach and Sholander [14] on unbounded intervals and by Páles [22], [25] and by Czinder and Páles [9] on bounded intervals.
The aim of this paper is to study the global comparison problem
[TABLE]
and also its local analogue. In terms of the Chebyshev systems and , the measurable families of -variable means and , and the measures , we give necessary conditions (which, in general, are not sufficient) and also sufficient conditions (that are also necessary in a certain sense) for (12) to hold. Our main results generalize that of the paper by Losonczi and Páles [20] and also many former results obtained in various particular cases of this problem, cf. [7], [8], [10], [18], [19], [21], [23].
2. Invariants with respect to equality of means
In order to describe the regularity conditions related to the two generating functions of the mean , we introduce some regularity classes. The class consists of all those pairs of continuous functions that form a Chebyshev system over .
If , then we say that the pair is in the class if are -times continuously differentiable functions such that and the Wronski determinant
[TABLE]
does not vanish on . Provided that is positive, then we have that
[TABLE]
hence condition implies that is strictly monotone, whence it follows that . Obviously, .
It is easy to see that if and
[TABLE]
where the constants satisfy , then
[TABLE]
This implies that the identity
[TABLE]
also holds for any measurable family and probability measure .
If (15) holds for some constants , then we say that the pairs and are equivalent. It is obvious that any necessary and/or sufficient condition for (12) has to be invariant with respect to the equivalence of the generating functions.
The following result, which is based on [5, Theorem 3], allows us to assume more regularity on Chebyshev systems.
Lemma 2**.**
Let and . Then there exist with and such that (15) holds and is positive and is strictly monotone. Furthermore, if , then the derivative of does not vanish on .
Proof.
In the case , the statement is a direct consequence of the result [5, Theorem 3] obtained by Bessenyei and Páles.
Assume now that . Then, means that and are -times continuously differentiable and does not vanish for . Using what we have established in the case , there exist constants with and such that (15) holds and is positive and is strictly monotone. The condition and (15) imply that there exist with such that
[TABLE]
Hence and are also -times continuously differentiable. This immediately implies that
[TABLE]
which shows that does not vanish for . Therefore, holds, too.
Applying the identity (14) for and instead of and , we can also see that the derivative of does not vanish on . ∎
For the computation of the first- and second-order partial derivatives of the mean at the diagonal of , we will establish a result below. For brevity, we introduce the following notation: If and then let stand for the ball . Furthermore, if is a probability measure on the measurable space and , then the space of measurable functions such that is -integrable will be denoted by or shortly by .
If are measurable functions such that is -integrable (for instance, ), then we set
[TABLE]
More generally, if , and for some , the map is -integrable, then we write
[TABLE]
Given a number , a function is said to be of -type at , if is measurable, furthermore, there exist and a function such that
[TABLE]
Let denote the class of measurable families of -variable means with the following two additional properties:
- (H5)
For every , the function is continuously partially differentiable over such that, for all , , the function is of -type at .
Analogously, we define to be the following subclass of :
- (H6)
For every , the function is twice continuously partially differentiable over such that, for all and , the function is of -type and is of -type at .
Lemma 3**.**
Let and let be a -times continuously differentiable function and . Then the function defined by
[TABLE]
is -times continuously differentiable on . Furthermore, for ,
[TABLE]
and, for ,
[TABLE]
provided that .
Proof.
If is measurable family of -variable means, then, by the continuity of and the mean value property of , it easily follows that is well-defined for all . Due to the continuity of and the assumption , it easily follows that the integral on the right hand side of (19) is well-defined. Furthermore, if is continuous and , then also the right hand side of (20) exists.
First we elaborate the case . We need to show that, for every , the function is partially differentiable at with formula (19) and that the partial derivatives are continuous at .
Before proceeding to the proof, we shall establish, for every , the following equality
[TABLE]
First choose so that and be elements of . Let be the supremum of over the compact interval . The continuity of implies that is finite. By the mean value property of , for every and , we have that . Hence, for every and , the inequality holds.
Using the assumption that is of -type at , we can find and a function such that
[TABLE]
Let be an arbitrary sequence converging to [math] with for all . By the Lagrange Mean Value Theorem, for every and for every , there exists such that
[TABLE]
Using the continuity of at , it immediately follows that the sequence of measurable functions defined by
[TABLE]
converges to zero for every . By the continuity of the partial derivative at , we also have that the sequence of measurable functions defined by
[TABLE]
converges to zero for every . Using the above estimations, we can now obtain that
[TABLE]
The expression on the right hand side of this inequality converges to zero for each , and these functions are dominated by the integrable function . Hence, by Lebesgue’s Dominated Convergence Theorem, it follows that
[TABLE]
Because the sequence converging to [math] was arbitrary, it follows that (21) holds.
Let be fixed and let denote the th vector of the standard basis on . For the proof that the th partial derivative of at is given by (19), consider the following estimation for , :
[TABLE]
Applying the Lagrange Mean Value Theorem for the function
[TABLE]
for every , we can find an element between [math] and such that , that is
[TABLE]
Using this formula, inequality (22) and the equality (21), for , it follows that
[TABLE]
Thus, we have proved that tends to zero as . This completes the proof of the partial differentiability of with respect to the th variable at and also the validity of formula (19).
Finally, we show that the function is continuous at every . Let be an arbitrary sequence in converging to and denote . Then is a null sequence and we have that
[TABLE]
Due to the equality (21), the right hand side in the above inequality tends to zero as , whence it follows that converges to , which proves the continuity of at .
Analogously, using a similar argument as in the proof of (21), for the case , the reader can show that the following two equalities hold:
[TABLE]
[TABLE]
Let be fixed. To prove equality (20) which establishes the formula for the th partial derivative of at , consider the following estimation for , :
[TABLE]
Applying, for every , the Lagrange Mean Value Theorem for the function
[TABLE]
we can find an element between [math] and such that
[TABLE]
Now, by using equality (26), inequality (25) and equalities (23) and (24), respectively, with an analogous argument that we applied in the case , we get that tends to zero as , proving the partial differentiability of at with respect to the th variable and formula (20). On the other hand, again by a similar train of thoughts, it easily follows from (23) and (24) that the function is continuous on . This completes the proof of the lemma. ∎
Theorem 4**.**
Let , let be a measurable family of means, and let be a probability measure on the measurable space . Then is continuously differentiable on and, for all and ,
[TABLE]
If, in addition, , let , then is twice continuously differentiable on and, for all and ,
[TABLE]
Proof.
Let and assume that , . In view of Lemma 2, we may assume that is positive, is strictly monotone with a non-vanishing first-order derivative. Then , and the inverse of are -times continuously differentiable and, by Lemma 3, we also have that the mappings
[TABLE]
are -times continuously differentiable on . On the other hand, we now also have formula (9) for the -variable mean . Thus, using the standard calculus rules, it follows that is -times continuously differentiable on
To prove the first formula stated in (27), let us consider the case . Differentiating the identity (7) with respect to the variable once, we get
[TABLE]
Now taking and substituting , the above equation simplifies to
[TABLE]
Observe that for all , hence the former equation yields the desired equality (27).
Now consider the case . Differentiating the identity that we obtained in the first lines of the proof with respect to the variable , we obtain
[TABLE]
respectively. Using the identities , , and (that are consequences of the asymmetry property ), and substituting , we get that
[TABLE]
Dividing both sides of this equation by and using (27), for the second-order partial derivative , we obtain the formula stated in (28). ∎
One of the most important particular case of the above theorem is when the -variable family of means is a family of weighted -variable arithmetic means.
Corollary 5**.**
Let , let be a probability measure on the measurable space and let be a measurable family of -variable means given by
[TABLE]
where are measurable functions with . Then is continuously differentiable on and, for all and ,
[TABLE]
If, in addition, , let , then is twice continuously differentiable on and, for all and ,
[TABLE]
Proof.
Observe that we have and . By the boundedness of the measurable function, it follows that is - and -type, and is -type at every point of . Therefore, Theorem 4 applies, and formulas (27) and (28) reduce to (29) and (30), respectively. ∎
The particular case when , , and was considered by Losonczi and Páles in the paper [20], where also the related local and global comparison problems were investigated. The above Theorem 4 and Corollary 5 generalize the result of [20, Lemma 4].
The following lemma, which is an extension of [20, Lemma 3], will play an important role in establishing the necessary conditions for the (global) comparison of means. We recall that a sequence of probability measures on is said to converge weakly to a measure if, for all bounded Borel measurable functions , we have
[TABLE]
Lemma 6**.**
Let and let be a sequence of probability measures on weakly converging to a measure , let be a null sequence of positive numbers in and let . Set for . Then
[TABLE]
Proof.
Let be a fixed vector. By the assumptions of the lemma, we have that converges to weakly. More generally, for an arbitrary bounded sequence of Borel measurable functions , which converges uniformly to as , we get
[TABLE]
First, we are going to show that the sequence converges to . We have that for all . Hence it is sufficient to prove that every convergent subsequence of converges to the same limit point. To show this, let be any convergent subsequence of such that as . Then, the sequence of Borel measurable functions tends uniformly to the limit function . Thus, in view of formula (32), we get
[TABLE]
On the other hand, for all , we have that
[TABLE]
which implies that is zero, i.e., . Hence as .
Moreover, as , we similarly obtain
[TABLE]
Taking and applying the Lagrange mean value theorem for the differentiable function , for every , we can find a number between and such that
[TABLE]
Since , therefore it follows that
[TABLE]
Thus,
[TABLE]
Then, obviously, converges to . By taking the limit of both sides of (36) as and using (33) and (32), we get
[TABLE]
By dividing both sides by , we get
[TABLE]
This completes the proof of the lemma. ∎
3. Necessary conditions, sufficient conditions for local comparison of means
Our first result offers a necessary as well as a sufficient condition for the local comparison of means. Given two -variable means , we say that is locally smaller than at if there exists a neighborhood of such that
[TABLE]
holds for all . The case being trivial, we always assume that holds in the subsequent considerations.
Theorem 7**.**
Let be -variable means such that is locally smaller than at a point . Assume that and are partially differentiable at the diagonal point . Then, for and for all ,
[TABLE]
If, in addition, and are twice differentiable at , then the symmetric -matrix
[TABLE]
*is positive semidefinite.
On the other hand, if, for some , the equality (40) holds for all and for all in a neighborhood of , furthermore, and are twice continuously differentiable at and the symmetric -matrix given by (41) is positive definite, then is locally smaller than at .*
Proof.
Assume that is locally smaller than at , i.e., (39) holds for all in a neighborhood of . Assume that and are partially differentiable at the diagonal point . Define the function by
[TABLE]
Then is nonnegative by inequality (39) and attains its minimum (which equals zero) at . Therefore for all , which yields (40).
If, in addition, and are twice differentiable at . Then D^{\prime\prime}(x_{0},\dots,x_{0})=\big{(}\partial_{i}\partial_{j}D(x_{0},\dots,x_{0})\big{)}_{i,j=1}^{d} is a positive semidefinite symmetric -matrix. By the well-known necessary and sufficient conditions of positive semidefiniteness (cf. [6]), this implies that the symmetric -matrix \big{(}\partial_{i}\partial_{j}D(x_{0},\dots,x_{0})\big{)}_{i,j=1}^{d-1} is also positive semidefinite.
Now let and assume that there exists a neighborhood of such that the means and are twice differentiable on , their second-order partial derivatives are continuous at , the equality (40) holds for all and for all , and the symmetric -matrix-valued function A(\boldsymbol{x}):=\big{(}\partial_{i}\partial_{j}D(\boldsymbol{x})\big{)}_{i,j=1}^{d-1} is positive definite at .
By Sylvester’s criterion, is positive definite if and only if all of its leading principal minors are positive. By the continuity of the second-order partial derivatives, is continuous at , hence its leading principal minors are also continuous at . Therefore, there is a neighborhood of where these leading principal minors are positive and hence, at the points of , is also positive definite. By shrinking the neighborhood of if necessary, we may assume that is an interval and . Hence is positive definite for all .
In order to show that the inequality (39) holds for all , let be fixed and apply the Taylor Mean Value Theorem to the function
[TABLE]
at the base point . In view of this theorem, there exists , such that
[TABLE]
We have that , equation (40) applied for implies that for all . Finally, is positive definite at the point , hence the last term on the right hand side of (42) is nonnegative. Thus (42) shows that , which implies that is smaller than on . ∎
Remark 8**.**
We note that, for the sufficiency part of the theorem, the standard 2nd-order sufficient condition for the local minimum cannot be applied. The reason is that the matrix
[TABLE]
can never be positive definite. Indeed, if is locally smaller than at , then is locally smaller than at every in a neighborhood of and hence (40) holds for all and . Differentiating (40) with respect to at , we obtain, for all , that
[TABLE]
This shows that the sum of the columns of the matrix in (43) is the zero vector. Therefore, the determinant of this matrix is zero, showing that this matrix is not positive definite.
Corollary 9**.**
Let , let and be measurable families of means, and let and be probability measures on the measurable spaces and , respectively. Suppose that is locally smaller than at . Then, there exists a neighborhood of such that for and for all ,
[TABLE]
If, in addition, , , and , then the -matrix whose th entry is given by
[TABLE]
*for is positive semidefinite.
On the other hand, if , , , and (44) holds for all and for all in a neighborhood of and the -matrix whose th entry is given by (LABEL:E7) is positive definite, then is locally smaller than at .*
Proof.
If , , and , then Theorem 4 implies that and are -times continuously differentiable on in the cases .
Assume that is locally smaller than at . Then, by Theorem 7, there exists a neighborhood of such that for and for all ,
[TABLE]
Applying formula (27) of Theorem 4, the necessity of condition (44) follows.
In addition, if the second-order regularity assumptions are satisfied, then, by Theorem 7, the -matrix whose th entry is given by
[TABLE]
for is positive semidefinite. Now the application of formula (28) of Theorem 4 yields the necessity of condition (LABEL:E7).
Now, under the second-order regularity assumptions suppose that (44) holds for all and for all in a neighborhood of and the -matrix whose th entry is given by (LABEL:E7) is positive definite. Since and are twice continuously differentiable and by Theorem 7, we have is locally smaller than at . ∎
In the special setting when , , is given by , the above Corollary 9 simplifies to the result of [20, Theorem 5]. Now we consider the particular case when the families of means and as well as the measures and coincide.
Corollary 10**.**
Let , let be a measurable family of means, and let be a probability measure on the measurable space . Let and assume that there exists such that, the map is not -almost everywhere constant on . If is locally smaller than at , then
[TABLE]
On the other hand, if the functions
[TABLE]
are -linearly independent and (46) holds with strict inequality, then is locally smaller than at .
Proof.
Assume that is locally smaller than at . Then, by Corollary 9, the -matrix whose th entry is given by
[TABLE]
for is positive semidefinite at . This implies that all the diagonal elements of this matrix are nonnegative, i.e., for all ,
[TABLE]
If, for some , the map is not -almost everywhere constant, then
[TABLE]
whence
[TABLE]
Inequality (49), combined with (LABEL:E8i), implies that
[TABLE]
i.e, the inequality (46) holds.
Now assume that the functions
[TABLE]
are -linearly independent and (46) holds with strict inequality. It is clear that the -matrix whose th entry is given by
[TABLE]
for is a so-called Gram matrix which is always positive semmidefinite (see [6]). Since the functions (50) are -linearly independent it follows that the Gram matrix with entries given by (51) is positive definite. This result, combined with the strict inequality (46), implies that the -matrix whose th entry is given by (LABEL:E8) is positive definite at . Hence, by Corollary 9, is locally smaller than at . ∎
Now we formulate a particular case concerning generalized Gini means when the partial derivatives can be calculated more explicitly. Indeed, if, for given , the functions and are given by equations (10), then
[TABLE]
where
[TABLE]
Then, one can easily get that and , whence
[TABLE]
Therefore,
[TABLE]
Corollary 11**.**
Assume that . Let , let and be measurable families of means, and let and be probability measures on the measurable spaces and , respectively. Suppose that is locally smaller than at . Then, there exists a neighborhood of such that for , for all , (44) holds. In addition, the -matrix whose th entry is given by
[TABLE]
*for is positive semidefinite.
On the other hand, if (44) holds for all and for all in a neighborhood of and the -matrix whose th entry is given by (LABEL:EG7) is positive definite, then is locally smaller than at .*
Proof.
The proof is a direct consequence of Corollary 9 and formula (54). ∎
Corollary 12**.**
Assume that . Let , let be a measurable family of means, and let be a probability measure on the measurable space . Let and assume that there exists such that, the map is not -almost everywhere constant on . If is locally smaller than at , then
[TABLE]
On the other hand, if , the functions
[TABLE]
are -linearly independent and (56) holds with strict inequality, then is locally smaller than at .
Proof.
Applying Corollary 10 and using formula (54), the result follows immediately. ∎
4. Necessary and sufficient conditions for global comparison of means
In the rest of the paper, we consider the case when and . In what follows, we give a condition containing two independent variables for (12) which does not involve the measure and assumes first-order continuous differentiability of the Chebyshev system. In the special setting when , , is given by , the following theorem simplifies to the result of [20, Theorem 6].
Theorem 13**.**
Let be Chebyshev systems, let be a compact and connected topological space and let be a continuous family of -variable means. Define the set by
[TABLE]
The following three assertions are equivalent:
- (i)
for all Borel probability measures on ,
[TABLE] 2. (ii)
there exists a nullsequence of positive numbers in such that, for all and for all ,
[TABLE] 3. (iii)
for all ,
[TABLE]
Proof.
The implication (i)(ii) is obvious.
To prove (ii)(iii), let . Then there exists such that . Due to the compactness and connectedness of , we have that (6) holds. Therefore, there exits such that and . Applying Lemma 6 twice with the measure sequence and using inequality (58), we get
[TABLE]
which proves (60).
For the proof of (iii)(i), let be arbitrarily fixed. In view of the inclusion and the equality (6), there exits such that
[TABLE]
Taking any and applying inequality (60) for and , we get
[TABLE]
Integrating this inequality with respect to the variable , we get
[TABLE]
By the definition of the value , the numerator of the right hand side of this inequality is zero, whence we obtain
[TABLE]
If for all , then
[TABLE]
and also is a positive Chebyshev system, hence, by Lemma 1, the above inequality implies that (58) holds. In the other possible case, i.e., when for all , then inequality (62) is reversed but is a negative Chebyshev system, thus by Lemma 1 again, inequality (58) follows. ∎
Having a look at the proof, one can see that the compactness and connectedness of was only used to prove implication (ii)(iii).
Corollary 14**.**
Assume that and . Let be a compact and connected topological space and let be a continuous family of means. Define the constant by
[TABLE]
The following three assertions are equivalent:
- (i)
for all Borel probability measures on ,
[TABLE] 2. (ii)
there exists a null sequence of positive numbers in such that, for all and for all ,
[TABLE] 3. (iii)
[TABLE] 4. (iv)
In the case ,
[TABLE]
while in the case ,
[TABLE]
Proof.
Applying Theorem 13 and using notations introduced in (52) and (53) imply that conditions (63) and (64) are equivalent to the inequality
[TABLE]
where the set is defined in (57). This inequality can be rewritten as
[TABLE]
Observe that
[TABLE]
Indeed, if t\in\big{]}(m^{*})^{-1},m^{*}\big{[} and , then , hence there exits such that . Then, with , , we have that and . Therefore is of the form for some . A similar argument yields for a similar representation. This proves the first inclusion.
For the second inclusion, observe that if , then, for some , we have . Hence
[TABLE]
Therefore, in view of the inclusions in (69), inequality (53) is equivalent to condition (65).
To show the equivalence of condition (iv) to the previous ones, we have to distinguish two cases. If , then , therefore (iii) can be rewritten as
[TABLE]
This inequality is known to be equivalent (cf. [10]) to the comparison inequality
[TABLE]
of Gini means (with arbitrary many variables over the interval ). In view of the result [10, Satz 5], the above inequality is characterized by the condition (66). Therefore (iii) is equivalent to (iv) in this case.
Now consider the case . Then the inequality in (iii) is equivalent to the comparison inequality
[TABLE]
of Gini means (with arbitrary many variables over the interval ). Using the results of the papers [15, Theorem 7] or [24], it follows that the above inequality is characterized by (67), which implies that (iii) is equivalent to (iv) also in this case.
This completes the proof of the corollary. ∎
5. Necessary and sufficient conditions for the local and global comparison of generalized
quasi-arithmetic means
In the next result we offer 6 equivalent conditions for the comparison of -variable generalized quasi-arithmetic means. The interesting feature of this result is the equivalence of the global and local comparability.
Theorem 15**.**
Let be twice continuously differentiable functions with non-vanishing first derivatives, and let be a measurable family of -variable means. Let be a probability measure such that, for all , there exists such that is not -almost everywhere constant on . The following assertions are equivalent:
- (i)
for all Borel probability measures on ,
[TABLE] 2. (ii)
[TABLE] 3. (iii)
for all , there exists a neighborhood of such that
[TABLE] 4. (iv)
for all ,
[TABLE] 5. (v)
the function is convex (concave) on provided that is increasing (decreasing); 6. (vi)
for all ,
[TABLE]
Proof.
The implications (i)(ii) and (ii)(iii) are trivial.
To prove (iii)(iv), will apply Corollary 10. Let be arbitrary. Then (iii) asserts that is locally smaller than at and we also have an index such that is not -almost everywhere constant on . Therefore, by Corollary 10, inequality (46) follows with the functions . It is immediate to see that (46) implies (71) for .
Now assume (iv) and that is increasing (the nondecreasing case is analogous). Then is twice differentiable on . By (71), the ratio is a nondecreasing function. Therefore, is also nondecreasing, which proves that . Hence must be convex on , i.e., (v) holds.
If (v) is valid and is increasing, then the convexity of implies that
[TABLE]
for all . With the substitution , , where , the above inequality reduces to (72), proving (vi).
Finally, assume that (vi) holds. Observe that then (60) is valid for all with the functions . Thus, the condition (iii) of Theorem 13 is satisfied, whence it follows that the mean is (globally) smaller than on , i.e., (70) holds. ∎
As an immediate consequence, we obtain the characterization of the comparison among generalized Hölder means.
Corollary 16**.**
Let , , and let be a measurable family of -variable means. Let be a probability measure such that, for all , there exists such that is not -almost everywhere constant on . The following assertions are equivalent:
- (i)
for all Borel probability measures on ,
[TABLE] 2. (ii)
[TABLE] 3. (iii)
for all , there exists a neighborhood of such that
[TABLE] 4. (iv)
.
Proof.
By taking if and if and if and if and applying Theorem 15 the result follows immediately because conditions (i), (ii) and (iii) are equivalent to the same conditions of Theorem 15, and is equivalent to condition (iv) of Theorem 15. ∎
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