Overview and refinements of the results are given for discrete,
integral, functional and operator variants of inequalities for quasiarithmetic
means. The general results are applied to further refinements of the power
means. Jensen's inequalities have been systematically presented, from the
older variants, to the most recent ones for the operators without operator
convexity.
1. Introduction
Quasiarithmetic means are very important because they are general and unavoidable in applications. This paper begins with the quasiarithmetic means of points, continues with the quasiarithmetic means of measurable function, through the quasiarithmetic means of functions with respect to linear functionals, and ends with the quasiarithmetic means of operators with respect to linear mappings. Conclusion of the paper is dedicated to the applications of operator quasiarithmetic means on power means with strictly positive operators. At this point, it should be emphasized that in all four of the next sections the basic and initial inequality was precisely the Jensen inequality (see Figure 1).
Figure 1: Graphic concept of Jensen’s inequality.
The applications of convexity often used strictly monotone continuous functions and such that is convex with respect to ( is convex); that is, is convex by [1, Definition 1.19]. Similar notation is used for concavity. We observe a monotonicity of quasiarithmetic means with these functions and . Good results for the monotonicity of quasiarithmetic means are obtained in [2] for the basic and integral case. The first results for the operator case without operator convexity are obtained in [3, 4]. Among other things, the paper gives some generalizations of the mentioned results.
Through this paper, we suppose that is a nondegenerate interval, and are strictly monotone continuous functions. It is assumed that the integer , wherever it appears in inequalities.
2. Results for Basic Case
For -tuple with numbers , sometimes we will write if all , and if for some .
Below is a discrete basic form of Jensen’s inequality for a convex function with respect to convex combinations points in interval.
Theorem A. Let be a function. Let be -tuple with points , and be -tuple with numbers such that .
A function is convex if and only if the following inequality
holds for all above -tuples and .
Consequently, if , not necessarily equals , then is convex if and only if
A function is concave if and only if the reverse inequality is valid in (2.1) and (2.2).
A function is strictly convex if and only if the inequality in (2.1) and (2.2) is strict for all and .
Let be a strictly monotone continuous function. Let be -tuple with points , and be -tuple with numbers such that . The discrete basic -quasiarithmetic mean of points (particles) with coefficients (weights) is a number
We understand that . The -quasiarithmetic mean resulting, first by moving the convex combination into convex combination , then its return using back in the interval . So, the number is in the interval , in fact in the closed interval . If is an identity function on , that is, for , then the -quasiarithmetic mean is just a convex combination as follows:
Basic quasiarithmetic means have the property
for every pair of real numbers and with .
Suppose that all coefficients . If we take , then is the arithmetic mean of numbers . If all and we take , then is the geometric mean of numbers . If all and we take , then is the harmonic mean of numbers .
Corollary 2.1. Let be strictly monotone continuous functions.
A function is either -convex and increasing or -concave and decreasing if and only if following the inequality:
holds for all -tuples and as in (2.3).
A function is either -concave and increasing or -convex and decreasing if and only if the reverse inequality is valid in (2.6).
A function is strictly -convex if and only if the inequality in (2.6) is strict for all and (see Figure 2).
Figure 2: The -order among quasiarithmetic means and for -convex and increasing .
Suppose that all . If we apply Corollary 2.1 on three strictly monotone functions , and (two by two in pairs), then we get the weighted harmonic-geometric-arithmetic inequality
Recall that a function is convex if and only if the following inequality:
holds for all triples such that . A function is strictly convex if and only if the above inequality is strict. So, the function is -convex if and only if
Let , where , be nonnegative continuous functions so that is a strictly monotone increasing positive function on an open interval , with boundary conditions and . Let both and be strictly monotone increasing or decreasing. For any , we define a strictly monotone continuous function
For example, we can take and for , and for , and for .
Lemma 2.2. Let , , , be real numbers.
If and , then
provided that denominators are the same sign. The inequality in (2.11) is strict if and .
If either and or and , then the reverse inequality is valid in (2.11).
Proposition 2.3. Let for be functions as in (2.10). Let such that .
If is -convex (resp. -concave), then is -convex (resp. -concave).
Proof. Suppose that is -convex. Show that the function is -convex. If , then , so we can suppose that . Let such that . Let, with respect to (2.9) and definition of functions and ,
Note that numbers and are positive because both and are strictly monotone increasing or decreasing. Applying Lemma 2.2 with , , , and , we obtain that
which shows the required convexity by (2.9). Case of the concavity can be proved in a similar way.
If is strictly -convex (resp. -concave), then is strictly -convex (resp. -concave).
According to Proposition 2.3, we can express refinements of the basic quasiarithmetic means.
Theorem 2.4. Let for be functions as in (2.10). Let such that .
If either is -convex with both and increasing or -concave with both and decreasing, then the following inequality:
holds for all -tuples and as in (2.3).
If either is -concave with both and increasing or -convex with both and decreasing, then the reverse inequality is valid in (2.14).
Proof. If is -convex with both and increasing, then the function is increasing, and -convex by Proposition 2.3, and according to Corollary 2.1 the inequality in (2.14) is valid. In the same way, we prove the concavity case.
In other words, the above theorem says that a function
is monotone increasing for any fixed and as in (2.3). In the case it is proved in [2, Lemma 4] for functions and with .
We emphasize that the inequality in (2.14) is strict for if is strictly -convex or -concave, and .
Let us take strictly monotone decreasing functions and with . Then , so is strictly -concave. Let
If we apply the inequality in (2.14) with on , we get
Let us take strictly monotone increasing functions and with . Then , so is strictly -convex. Let
If we apply the inequality in (2.14) with on , we get
Connecting two above inequalities results in
The inequality in (2.20) is strict for if all and for some , so in this case, we have refinements of the weighted harmonic-geometric-arithmetic inequality.
The weighted harmonic-geometric-arithmetic inequality is only the special case of a whole collection of inequalities which can be derived by applying of Corollary 2.1 on power means. As a special case of the basic quasiarithmetic mean in (2.3) with , for and , we can observe the discrete basic power mean
Very useful consequence of Corollary 2.1 is a well-known property of monotonicity of basic power means.
Corollary 2.5. If and are real numbers such that , then the following inequality:
holds for all -tuples and as in (2.3) with .
The inequality in (2.22) is strict for if and .
Let functions for be specially defined by
Then the functions with are monotone increasing in the next four cases.
Case .
Functions and are strictly monotone decreasing with strictly concave because .
Case .
Functions and are strictly monotone decreasing with strictly concave because .
Case .
Functions and are strictly monotone increasing with strictly convex .
Case .
Functions and are strictly monotone increasing with strictly convex because .
Given traditional signs of power means, we will mark with . The inequality in (2.22) can be refined using Theorem 2.4 with functions . The following are refinements of power means.
Corollary 2.6. Let such that . Let for be functions as in (2.23). Let such that .
If or or or , then the inequality
holds for all -tuples and as in (2.3) with .
If , then we can take the series of inequalities
The inequalities in (2.24)-(2.25) are strict for if and .
The inequality in (2.20) is a special case of the collection of inequalities in (2.24).
3. Applications on Integral Case
In this section, is a probability measure space. It is assumed that every weighted function is nonnegative almost everywhere on , that is, for almost all .
For -tuple with functions , sometimes we will write if all almost everywhere on , and if almost everywhere on for some .
Here is an integral form of Jensen’s inequality for a convex function with respect to measurable functions with weighted functions on the probability measure space.
Theorem B. Let be a function. Let be a probability measure space, be a measurable function, and be a weighted function with such that .
If a function is convex, then the inequality
holds for all above , and .
Consequently, if , not necessarily equals , then
If a function is concave, then the reverse inequality is valid in (3.1) and (3.2).
The assumption with nonnegative almost everywhere on , for the inequality in (3.1), assures that
Remark 3.1. The reverse of Theorem B is valid if for any a measurable set exists so that . In this case, we can determine a simple measurable function
where is a characteristic set function, for every and . If we take at the same, then
If we include these integrals in the inequality in (3.1), we have the convexity of the function .
Theorem B can be generalized to probability measures and measurable functions with weighted functions . The following is a discrete integral form of Jensen’s inequality.
Theorem 3.2. Let be a function. Let be -tuple with probability measures on , be -tuple with measurable functions , and be -tuple with weighted functions with such that .
A function is convex if and only if the inequality
holds for all above -tuples , and .
Consequently, if , not necessarily equals , then is convex if and only if
A function is concave if and only if the reverse inequality is valid in (3.6) and (3.7).
In the proof of sufficiency theorem, we simply take and in which case the inequality in (3.6) and (3.7) becomes the basic inequality of convexity. The fact that is coming to the fore.
A function is strictly convex if and only if the inequality in (3.6) and (3.7) is strict for all and .
Let be a strictly monotone continuous function. Let be -tuple with probability measures on , be -tuple with measurable functions , and be -tuple with weighted functions with such that . The discrete integral -quasiarithmetic mean of measurable functions with weighted functions with respect to measures (namely, with respect to integrals ) is a number
This number belongs to because the integral convex combination . Integral quasiarithmetic means also satisfy the property
for every pair of real numbers and with .
Bearing in mind Theorem 3.2, the following corollary is valid.
Corollary 3.3. Let be strictly monotone continuous functions.
A function is either -convex and increasing or -concave and decreasing if and only if the inequality
holds for all -tuples , , and as in (3.8).
A function is either -concave and increasing or -convex and decreasing if and only if the reverse inequality is valid in (3.10).
Combining basic and integral case by Corollaries 2.1 and 3.3, we get the following.
Proposition 3.4. Let be strictly monotone continuous functions. Then the inequality
holds for all -tuples and as in (2.3) if and only if the inequality
holds for all -tuples , , and as in (3.8).
The one direction of Proposition 3.4 is proved in [2, Theorem 1]. It is proved that the inequality for basic case implies the inequality for integral case with one function .
The following integral analogy of Theorem 2.4.
Theorem 3.5. Let for be functions as in (2.10). Let such that .
If either is -convex with both and increasing or -concave with both and decreasing, then the inequality
holds for all -tuples , , and as in (3.8).
If either is -concave with both and increasing or -convex with both and decreasing, then the reverse inequality is valid in (3.13).
The inequality in (3.13) is strict for if is strictly -convex, and .
An integral version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality
holds for all -tuples , and as in (3.8) with . The above inequality is strict for if all almost everywhere on and almost everywhere on for some .
As a special case of the integral quasiarithmetic mean in (3.8) with , for , and , we can observe the integral power mean
We quote the integral analogy of Corollary 2.6. The following is the property of monotonicity, with refinements, of integral power means.
Corollary 3.6. Let such that . Let for be functions as in (2.23). Let such that .
If or or or , then the inequality
holds for all -tuples , , and as in (3.8) with .
If , then we can take the series of inequalities
The inequalities in (3.16)-(3.17) are strict for if and .
All the observed integral cases are reduced to the corresponding basic cases when we take constants and .
4. Applications on Functional Case
Let be a nonempty set and be a vector space of real-valued functions . Linear functional is positive (nonnegative) or monotone if for every nonnegative function . If a space contains a unit function , by definition for every , and , we say that functional is unital or normalized.
In this section, it is assumed that every weighted function is nonnegative, that is, for every .
Bellow is a functional form of Jensen’s inequality for a convex function with respect to real-valued functions with weighted functions on the vector space of real-valued functions.
Theorem C. Let be a continuous function where is the closed interval. Let be a positive linear functional, be a function, and be a weighted function with such that .
If a function is convex, then the inequality
holds for all above , , and .
Consequently, if , not necessarily equals , then
If a function is concave, then the reverse inequality is valid in (4.1) and (4.2).
The inequality in (4.1) with (assuming and ) is usually called the Jessen functional form of Jensen’s inequality.
The interval must be closed; otherwise, it could happen that or . The following example shows such an undesirable situation.
Example 4.1. Let and
If is defined by
then is positive linear functional. In that way, functional is also unital because and . If we take for , then and its image in , but .
Remark 4.2. Suppose that and functional is unital, that is, . Then the reverse of Theorem C is valid if for any a subset exists so that and . If we take and , then it follows that
for every and . If we include these expressions in the inequality in (4.1), we get the convexity of .
Theorem C can be generalized to linear functionals and functions with weighted functions . The following is a discrete functional form of Jensen’s inequality.
Theorem 4.3. Let be a continuous function where is the closed interval. Let be -tuple with positive linear functionals , be -tuple with functions , and be -tuple with weighted functions with such that .
If a function is convex, then the inequality
holds for all above -tuples , , and .
Consequently, if , not necessarily equals , then
If a function is concave, then the reverse inequality is valid in (4.6) and (4.7).
Proof. Let us prove the inequality in (4.6). If for some , then . Without loss of generality, suppose that all . Let . All numbers belong to . Then from the basic inequality in (2.1) and functional inequality in (4.2), it follows that
If is strictly convex, then the inequality in (4.6) and (4.7) is strict for all and .
Remark 4.4. Suppose that and all functionals are unital; that is, holds for all . Then it is and for every constant . With the above assumptions, the reverse of Theorem 4.3 follows trivially if we take and .
Let be a strictly monotone continuous function where is the closed interval. Let be -tuple with positive linear functionals , be -tuple with functions , and be -tuple with weighted functions with such that . The discrete functional -quasiarithmetic mean of functions with weighted functions with respect to functionals is a number
This number belongs to because the functional convex combination belongs to . Functional quasiarithmetic means also satisfy the property
for every pair of real numbers and with . Indeed, if , then , and we have
Corollary 4.5. Let be strictly monotone continuous functions where is the closed interval.
If a function is either -convex and increasing or -concave and decreasing, then the inequality
holds for all -tuples , , and as in (4.9).
If a function is either -concave and increasing or -convex and decreasing, then the reverse inequality is valid in (4.12).
Proof. Suppose that is -convex and increasing. If we apply the inequality in (4.6) with , and instead of , we get
After taking of the both sides, it follows that
In the same way, we can prove the case when is -concave and decreasing.
According to Remark 4.2, the reverse of Corollary 4.5 is valid if and all functionals are unital. Then we connect the basic and functional case in the following proposition.
Proposition 4.6. Let be strictly monotone continuous functions where is the closed interval. Then the inequality
holds for all and as in (2.3) if and only if the inequality
holds for all -tuples , , and as in (4.9) with and unital functionals .
Next in line is a functional analogy of refinements.
Theorem 4.7. Let for be functions as in (2.10) where is the closed interval. Let such that .
If either is -convex with both and increasing or -concave with both and decreasing, then the inequality
holds for all -tuples , , and as in (4.9) with and unital functionals .
If either is -concave with both and increasing or -convex with both and decreasing, then the reverse inequality is valid in (4.17).
The inequality in (4.17) is strict for if is strictly -convex, and .
A functional version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality
holds for all -tuples , and, as in (4.9) with where . The above inequality is strict for if all and for some .
As a special case of the functional quasiarithmetic mean in (4.9) with where , for and , we can observe the functional power mean
The following is the property of monotonicity, with refinements, of functional power means.
Corollary 4.8. Let such that . Let for be functions as in (2.23) where . Let such that .
If or or or , then the inequality
holds for all -tuples , , and as in (4.9) with , and unital functionals .
If , then we can take the series of inequalities
The inequalities in (4.20)-(4.21) are strict for if and .
All the observed functional cases are reduced to the corresponding integral cases when we take
for all functions such that for almost all .
5. Results for Operator Case
We recall some notations and definitions. Let be a Hilbert space. We define the bounds of linear operator with
Let be the -algebra of all bounded linear operators . If denotes the spectrum of a self-adjoint operator , then it is well-known that is a subset of and . If denotes the identity operator on , then the following holds:
A continuous function is said to be operator increasing on if
for every pair of self-adjoint operators on with spectra in . A function is said to be operator decreasing if— is operator increasing. A function is operator monotone if it is operator increasing or decreasing.
For -tuple with operators sometimes, we will write if all , and if for some .
In this section, it is assumed that every weighted operator is positive.
From the second half of the last century, Jensen’s inequality was formulated for operator convex functions, self-adjoint operators, and positive linear mappings (see [5–8]). Very recently, Jensen’s inequality for operators without operator convexity is formulated in [3], and generalized in [4].
The following theorem essentially coincides with the main theorem in [3]. The only difference is that now we add the weighted operators. We also give a short proof of the theorem that relies on the geometric property of convexity and affinity of the chord line or support line. So, we start with an operator form of Jensen’s inequality for a convex function with respect to self-adjoint operators with weighted operators on the Hilbert space, and positive linear mappings.
Theorem 5.1. Let be a continuous function. Let be -tuple with positive linear mappings , be -tuple with self-adjoint operators with bounds from , and be -tuple with weighted operators with . Let be bounds of an operator .
If a function is convex, then the inequality
holds for all above -tuples , , and provided spectral conditions
Consequently, if is strictly positive, not necessarily equals , then
If a function is concave, then the reverse inequality is valid in (5.4) and (5.6).
Proof. If , then we take the chord line through the points , and . It follows:
If , then we take any support line instead of the chord line.
If is strictly convex, then the inequality in (5.4) and (5.6) is strict for all and .
Remark 5.2. The reverse of Theorem 5.1 is trivially valid if all are unital. With this assumption, we can take and .
Let be a strictly monotone continuous function. Let be -tuple with positive linear mappings , be -tuple with self-adjoint operators with spectra in , and be -tuple with weighted operators with . The discrete operator -quasiarithmetic mean of operators with weighted operators with respect to mappings is an operator
The spectrum of operator is contained in because the spectrum of operator is contained in . Operator quasiarithmetic means also have the property
for every pair of real numbers and with . To verify this equality, let us take , so if , and we get
Corollary 5.3. Let be strictly monotone continuous functions with operator monotone .
Let , and be as in (5.8). Let and be bounds of operators and , respectively.
If a function is either -convex with operator increasing or -concave with operator decreasing , then the inequality
holds for all above -tuples , , and provided spectral conditions
If a function is either -concave with operator increasing or -convex with operator decreasing , then the reverse inequality is valid in (5.11).
The following is operator analogy of Theorem 2.4.
Theorem 5.4. Let for be functions as in (2.10) with operator monotone . Let such that .
If either is -convex with operator increasing , , and or -concave with operator decreasing , , and , then the inequality
holds for all -tuples , , and as in (5.8) that provided the following spectral conditions:
If either is -concave with operator increasing , , and or -convex with operator decreasing , , and , then the reverse inequality is valid in (5.13).
Proof. Let us prove the middle part of the inequality in (5.13), one that refers to and . If is -convex with both and increasing, then is -convex by Proposition 2.3. If is operator increasing, then by Corollary 5.3 the inequality
is valid with spectral conditions
Any part of the series of inequalities in (5.13) is proved similarly.
The inequality in (5.13) is strict for if is strictly -convex, , and are strictly operator increasing, and .
We are interested in sufficient conditions under which the functions will be operator increasing.
Lemma 5.5. Let be a convex combination , of strictly monotone increasing or decreasing continuous functions such that . Then
where are nonnegative continuous functions such that for every .
Proof. Take any . If
then
for some nonnegative numbers and such that (see Figure 3). Now, first replace with in expression in (5.19), and then express . Realizing as a function of the variable , we obtain that
The above limit is onesided if on some subinterval of an interval . The functions , and are continuous on , and the same is true for the function . Thus, the expression in (5.19) is the required presentation of function as the convex combination of functions and with coefficient functions and .
Figure 3: Convex combination of strictly monotone increasing functions and .
Theorem 5.4 can be simplified by using Lemma 5.5.
Corollary 5.6. Let for be functions as in (2.10) with the addition of , and operator monotone . Let such that .
If either is -convex with operator increasing or -concave with operator decreasing , then the inequality in (5.13) holds for all -tuples , , and as in (5.8) with spectral conditions as in Theorem 5.4.
If either is -concave with operator increasing or -convex with operator decreasing , then the reverse inequality is valid in (5.13).
Proof. According to Lemma 5.5 continuous functions and , with , exist for every so that
Let ; otherwise, it is . Then and is operator increasing (resp. decreasing) if is operator increasing (resp. decreasing).
A special case of the operator quasiarithmetic mean in (5.8) with , for and , we can observe the operator power mean
The consequence of Corollary 5.3 for operator power means the following.
Corollary 5.7. Let and be real numbers such that .
Let , , and be as in (5.8) with strictly positive . Let , , and be bounds of operators , , and , respectively.
If or , then the inequality
holds for all above -tuples , , and provided spectral conditions
If or , then the inequality in (5.23) holds provided spectral conditions
The inequality in (5.23) is strict for if and .
The proof of Corollary 5.7 is the same as the proof of [3, Corollary 7].
An operator version of the harmonic-geometric-arithmetic inequality is the consequence of Corollary 5.7. The inequality , that is,
holds for all -tuples , and as in (5.8) with strictly positive provided spectral conditions
There remain only the refinements of the operator power means by using Corollary 5.6.
Corollary 5.8. Let such that . Let for be functions as in (2.23) with the addition of . Let such that .
Let , , and be as in (5.8) with strictly positive . Let , , , , and be bounds of operators , , , , and , respectively.
If or , then the inequality
holds for all above -tuples , , and provided spectral conditions:
If , then the inequality
holds for all above -tuples , , and provided spectral conditions:
Proof. Recall that for . The next is and with , so .
Case .
We have with , and with . The function is strictly -convex with strictly operator increasing . Then the inequality in (5.28) is valid with associated spectral conditions by Corollary 5.6.
Case .
We have with , and . The function is strictly -concave with strictly operator decreasing . In this case, the inequality in (5.28) is also valid with associated spectral conditions by Corollary 5.6.
Case .
We use functions . In this case and , thus with , and . The function is strictly -concave with strictly operator increasing . Then the inequality in (5.30) is valid with associated spectral conditions by Corollary 5.6.
The inequalities in (5.28)–(5.30) are strict for if and .
Unfortunately, we cannot use a logarithmic function because .
Remark 5.9. Let , , , , ,, , and be as in Corollary 5.8.
If and , then the problem remains the inequality
valid for all above -tuples , , and provided spectral conditions:
The inequality in (5.32) is valid for with associated spectral conditions because with , that is, is -convex, and , that is, is operator decreasing.