#### Abstract

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).

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).

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:
*