The Inequalities for Quasiarithmetic Means

and Applied Analysis 3 ∑n i 1 pi 1. The discrete basic φ-quasiarithmetic mean of points particles xi with coefficients weights pi is a number Mφ px φ−1 ( n ∑


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 ϕ and ψ such that ψ is convex with respect to ϕ ψ is ϕ convex ; that is, f ψ • ϕ −1 is convex by 1, Definition 1.19 . Similar notation is used for concavity. We observe a monotonicity of quasiarithmetic means with these functions ϕ and ψ. 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 I ⊆ R is a nondegenerate interval, and ϕ, ψ : I → R are strictly monotone continuous functions. It is assumed that the integer n ≥ 2, wherever it appears in inequalities.

Results for Basic Case
For n-tuple x x 1 , . . . , x n with numbers x i ∈ R, sometimes we will write x > 0 if all x i > 0, and x / c if x i / x j for some i / j.
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 f : I → R be a function. Let x x 1 , . . . , x n be n-tuple with points x i ∈ I, and p p 1 , . . . , p n be n-tuple with numbers p i ∈ 0, 1 such that n i 1 p i 1.
We understand that px n i 1 p i x i . The ϕ-quasiarithmetic mean resulting, first by moving the convex combination px ∈ I into convex combination pϕ x ∈ ϕ I , then its return using ϕ −1 back in the interval I. So, the number M ϕ px is in the interval I, in fact in the closed interval min{x i }, max{x i } . If ϕ is an identity function on I, that is, ϕ x id x x for x ∈ I, then the ϕ-quasiarithmetic mean is just a convex combination as follows: Basic quasiarithmetic means have the property for every pair of real numbers a and b with a / 0. Suppose that all coefficients p i 1/n. If we take ϕ 1 x x, then M ϕ 1 px is the arithmetic mean of numbers x i . If all x i > 0 and we take ϕ 0 x ln x, then M ϕ 0 px is the geometric mean of numbers x i . If all x i > 0 and we take ϕ −1 x 1/x, then M ϕ −1 px is the harmonic mean of numbers x i . A function ψ is strictly ϕ-convex if and only if the inequality in 2.6 is strict for all p > 0 and x / c see Figure 2 .
Suppose that all x i > 0. If we apply Corollary 2.1 on three strictly monotone functions ϕ −1 x 1/x, ϕ 0 x ln x and ϕ 1 x x two by two in pairs , then we get the weighted harmonic-geometric-arithmetic inequality Recall that a function f : I → R is convex if and only if the following inequality: holds for all triples x, y, z ∈ I such that x < y < z. A function f is strictly convex if and only if the above inequality is strict. So, the function ψ is ϕ-convex if and only if Let u, v : a 0 , a 1 → R, where a 0 < a 1 , be nonnegative continuous functions so that v/u is a strictly monotone increasing positive function on an open interval a 0 , a 1 , with boundary conditions u a 0 v a 1 1 and u a 1 v a 0 0. Let both ϕ and ψ be strictly monotone increasing or decreasing. For any t ∈ a 0 , a 1 , we define a strictly monotone continuous function For example, we can take u t 1 − t and v t t for t ∈ 0, 1 , u t 1 − √ t and v t t 2 for t ∈ 0, 1 , u t cos t and v t sin t for t ∈ 0, π/2 .
provided that denominators are the same sign. The inequality in 2.11 is strict if a < b and α < β.
If either a ≥ b and α ≤ β or a ≤ b and α ≥ β, then the reverse inequality is valid in 2.11 . Proposition 2.3. Let φ t u t ϕ v t ψ : I → R for t ∈ a 0 , a 1 be functions as in 2.10 . Let Proof. Suppose that ψ is ϕ-convex. Show that the function φ t 1 is φ t 0 -convex. If t 0 a 1 , then t 0 t 1 , so we can suppose that t 0 < a 1 . Let x, y, z ∈ I such that x < y < z. Let, with respect to 2.9 and definition of functions u and v, Note that numbers a and b are positive because both ϕ and ψ are strictly monotone increasing or decreasing. Applying Lemma 2.2 with a, b, α, and β, we obtain that which shows the required convexity by 2.9 . Case of the concavity can be proved in a similar way.
According to Proposition 2.3, we can express refinements of the basic quasiarithmetic means.
Theorem 2.4. Let φ t u t ϕ v t ψ : I → R for t ∈ a 0 , a 1 be functions as in 2.10 . Let t 0 , t 1 ∈ a 0 , a 1 such that t 0 ≤ t 1 .
If either ψ is ϕ-convex with both ϕ and ψ increasing or ϕ-concave with both ϕ and ψ decreasing, then the following inequality: 14 holds for all n-tuples p and x as in 2.3 . If either ψ is ϕ-concave with both ϕ and ψ increasing or ϕ-convex with both ϕ and ψ decreasing, then the reverse inequality is valid in 2.14 .
Proof. If ψ is ϕ-convex with both ϕ and ψ increasing, then the function φ t 1 is increasing, and φ t 0 -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.

Abstract and Applied Analysis
In other words, the above theorem says that a function t → M φ t px with t ∈ a 0 , a 1 , 2.15 is monotone increasing for any fixed p and x as in 2.3 . In the case n 2 it is proved in 2, Lemma 4 for functions u t 1 − t and v t t with t ∈ 0, 1 . We emphasize that the inequality in 2.14 is strict for a 0 < t 0 < t 1 < a 1 if ψ is strictly ϕ-convex or ϕ-concave, p > 0 and x / c.
Let us take strictly monotone decreasing functions ϕ x If we apply the inequality in 2.14 with t t 0 t 1 on φ hg t , we get Let us take strictly monotone increasing functions ϕ x ln x and ψ x x with x > 0.

2.18
If we apply the inequality in 2.14 with t t 0 t 1 on φ ga t , we get n i 1 Connecting two above inequalities results in

2.20
The inequality in 2.20 is strict for a 0 < t < a 1 if all p i > 0 and x i / x j for some i / j, 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 I 0, ∞ , ϕ r x x r for r / 0 and ϕ 0 x ln x, we can observe the discrete basic power mean

2.21
Very useful consequence of Corollary 2.1 is a well-known property of monotonicity of basic power means. The inequality in 2.22 is strict for r < s if p > 0 and x / c.

2.23
Then the functions t → M φ r,s t px with x > 0 are monotone increasing in the next four cases.
Functions ϕ x x r and ψ x x s are strictly monotone decreasing with strictly concave ψ • ϕ −1 x x s/r because 0 < s/r < 1.
Functions ϕ x x r and ψ x − ln x are strictly monotone decreasing with strictly concave Case r 0 < s.
Functions ϕ x ln x and ψ x x s are strictly monotone increasing with strictly convex Case 0 < r < s.
Functions ϕ x x r and ψ x x s are strictly monotone increasing with strictly convex ψ • ϕ −1 x x s/r because s/r > 1.
If r < s < 0 or r < 0 s or r 0 < s or 0 < r < s, then the inequality holds for all n-tuples p and x as in 2.3 with I 0, ∞ . If r < 0 < s, then we can take the series of inequalities

2.25
The inequalities in 2.24 -2.25 are strict for a 0 < t 0 < t 1 < a 1 if p > 0 and x / c. The inequality in 2.20 is a special case of the collection of inequalities in 2.24 .

Applications on Integral Case
In this section, Ω, μ is a probability measure space. It is assumed that every weighted function w : Ω → R is nonnegative almost everywhere on Ω, that is, w ω ≥ 0 for almost all ω ∈ Ω.
For n-tuple g g 1 , . . . , g n with functions g i : Ω → R, sometimes we will write g > 0 if all g i > 0 almost everywhere on Ω, and g / c if g i / g j almost everywhere on Ω for some i / j.
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 f : I → R be a function. Let Ω, μ be a probability measure space, g : Ω → I be a measurable function, and w ∈ L 1 Ω, μ be a weighted function with Ω w dμ 1 such that If a function f is convex, then the inequality holds for all above w, g and μ.
Consequently, if Ω w dμ p > 0, not necessarily equals 1, then If a function f is concave, then the reverse inequality is valid in 3.1 and 3.2 .
Abstract and Applied Analysis 9 The assumption Ω w dμ 1 with nonnegative w almost everywhere on Ω, for the inequality in 3.1 , assures that

3.3
Remark 3.1. The reverse of Theorem B is valid if for any p ∈ 0, 1 a measurable set Ω p ⊆ Ω exists so that μ Ω p p. In this case, we can determine a simple measurable function where χ is a characteristic set function, for every x, y ∈ I and p ∈ 0, 1 . If we take w 1 at the same, then

3.5
If we include these integrals in the inequality in 3.1 , we have the convexity of the function f.
Theorem B can be generalized to n probability measures μ i and n measurable functions g i with weighted functions w i . The following is a discrete integral form of Jensen's inequality. Theorem 3.2. Let f : I → R be a function. Let µ μ 1 , . . . , μ n be n-tuple with probability measures μ i on Ω, g g 1 , . . . , g n be n-tuple with measurable functions g i : Ω → I, and w w 1 , . . . , w n be n-tuple with weighted functions holds for all above n-tuples w, g and µ.

3.7
A function f 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 w i p i and g i x i in which case the inequality in 3.6 and 3.7 becomes the basic inequality of convexity. The fact that n ≥ 2 is coming to the fore.
A function f is strictly convex if and only if the inequality in 3.6 and 3.7 is strict for all w > 0 and g / c.
Let ϕ : I → R be a strictly monotone continuous function. Let µ μ 1 , . . . , μ n be n-tuple with probability measures μ i on Ω, g g 1 , . . . , g n be n-tuple with measurable functions g i : Ω → I, and w w 1 , . . . , w n be n-tuple with weighted functions w i ∈ L 1 Ω, μ i with n i 1 Ω w i dμ i 1 such that w i · ϕ•g i ∈ L 1 Ω, μ i . The discrete integral ϕ-quasiarithmetic mean of measurable functions g i with weighted functions w i with respect to measures μ i namely, with respect to integrals Ω · dμ i is a number This number belongs to I because the integral convex combination n i 1 Ω w i · ϕ • g i dμ i ∈ ϕ I . Integral quasiarithmetic means also satisfy the property for every pair of real numbers a and b with a / 0. Bearing in mind Theorem 3.2, the following corollary is valid.  holds for all n-tuples w, g, and µ as in 3.8 .

Abstract and Applied Analysis 11
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 g.
The following integral analogy of Theorem 2.4.
Theorem 3.5. Let φ t u t ϕ v t ψ : I → R for t ∈ a 0 , a 1 be functions as in 2.10 . Let t 0 , t 1 ∈ a 0 , a 1 such that t 0 ≤ t 1 .
If either ψ is ϕ-convex with both ϕ and ψ increasing or ϕ-concave with both ϕ and ψ decreasing, then the inequality holds for all n-tuples w, g, and µ as in 3.8 . If either ψ is ϕ-concave with both ϕ and ψ increasing or ϕ-convex with both ϕ and ψ decreasing, then the reverse inequality is valid in 3.13 .
The inequality in 3.13 is strict for a 0 < t 0 < t 1 < a 1 if ψ is strictly ϕ-convex, w > 0 and g / c.
An integral version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality 3.14 holds for all n-tuples w w 1 , . . . , w n , g g 1 , . . . , g n and µ μ 1 , . . . , μ n as in 3.8 with I 0, ∞ . The above inequality is strict for a 0 < t < a 1 if all w i > 0 almost everywhere on Ω and g i / g j almost everywhere on Ω for some i / j.
As a special case of the integral quasiarithmetic mean in 3.8 with I 0, ∞ , ϕ r x x r for r / 0, and ϕ 0 x ln x, we can observe the integral power mean

3.15
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 r, s ∈ R such that r < s. Let φ r,s t : 0, ∞ → R for t ∈ a 0 , a 1 be functions as in 2.23 . Let t 0 , t 1 ∈ a 0 , a 1 such that t 0 ≤ t 1 .
If r < s < 0 or r < 0 s or r 0 < s or 0 < r < s, then the inequality

3.17
The inequalities in 3.16 -3.17 are strict for a 0 < t 0 < t 1 < a 1 if w > 0 and g / c. All the observed integral cases are reduced to the corresponding basic cases when we take constants g i x i and w i p i .

Applications on Functional Case
Let S be a nonempty set and S be a vector space of real-valued functions g : S → R. Linear functional P : S → R is positive nonnegative or monotone if P g ≥ 0 for every nonnegative function g ∈ S. If a space S contains a unit function 1, by definition 1 s 1 for every s ∈ S, and P 1 1, we say that functional P is unital or normalized. In this section, it is assumed that every weighted function w : S → R is nonnegative, that is, w s ≥ 0 for every s ∈ S.
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 f : I → R be a continuous function where I is the closed interval. Let P : S → R be a positive linear functional, g : S → I be a function, and w ∈ S be a weighted function with P w 1 such that w · g, w · f • g ∈ S. If a function f is convex, then the inequality holds for all above w, g, and P. Consequently, if P w p > 0, not necessarily equals 1, then If a function f is concave, then the reverse inequality is valid in 4.1 and 4.2 .
The inequality in 4.1 with w 1 assuming 1 ∈ S and P 1 1 is usually called the Jessen functional form of Jensen's inequality.
The interval I must be closed; otherwise, it could happen that P w · g / ∈ I or P w · f • g / ∈ f I . The following example shows such an undesirable situation. If P : S → R is defined by P g lim then P is positive linear functional. In that way, functional P is also unital because 1 ∈ S and P 1 1. If we take g x x for x ∈ I, then g ∈ S and its image in I, but P g 0 / ∈ I.

Remark 4.2.
Suppose that 1 ∈ S and functional P is unital, that is, P 1 1. Then the reverse of Theorem C is valid if for any p ∈ 0, 1 a subset S p ⊆ S exists so that χ S p ∈ S and P χ S p p. If we take g xχ S p yχ S\S p and w 1, then it follows that for every x, y ∈ I and p ∈ 0, 1 . If we include these expressions in the inequality in 4.1 , we get the convexity of f.
Theorem C can be generalized to n linear functionals P i and n functions g i with weighted functions w i . The following is a discrete functional form of Jensen's inequality.

If a function f is convex, then the inequality
holds for all above n-tuples w, g, and P. Consequently, if n i 1 P i w i p > 0, not necessarily equals 1, then If a function f is concave, then the reverse inequality is valid in 4.6 and 4.7 .
Proof. Let us prove the inequality in 4.6 . If P i w i 0 for some i, then P i w i · g i 0. Without loss of generality, suppose that all p i P i w i > 0. Let x i 1/p i P i w i · g i . All numbers x i belong to I. Then from the basic inequality in 2.1 and functional inequality in 4.2 , it follows that

4.8
If f is strictly convex, then the inequality in 4.6 and 4.7 is strict for all w > 0 and g / c. Remark 4.4. Suppose that 1 ∈ S and all functionals P i are unital; that is, P i 1 1 holds for all P i . Then it is c · 1 ∈ S and P i c · 1 cP i 1 c for every constant c ∈ R. With the above assumptions, the reverse of Theorem 4.3 follows trivially if we take w i p i and g i x i .
Let ϕ : I → R be a strictly monotone continuous function where I is the closed interval. Let P P 1 , . . . , P n be n-tuple with positive linear functionals P i : S → R, g g 1 , . . . , g n be n-tuple with functions g i : S → I, and w w 1 , . . . , w n be n-tuple with weighted functions w i ∈ S with n i 1 P i w i 1 such that w i · ϕ • g i ∈ S. The discrete functional ϕ-quasiarithmetic mean of functions g i with weighted functions w i with respect to functionals P i is a number This number belongs to I because the functional convex combination n i 1 P i w i · ϕ • g i belongs to ϕ I . Functional quasiarithmetic means also satisfy the property M aϕ b wg, P M ϕ wg, P , 4.10 for every pair of real numbers a and b with a / 0.

4.11
Corollary 4.5. Let ϕ, ψ : I → R be strictly monotone continuous functions where I is the closed interval. If a function ψ is either ϕ-convex and increasing or ϕ-concave and decreasing, then the inequality M ϕ wg, P ≤ M ψ wg, P , 4.12 holds for all n-tuples w, g, and P as in 4.9 . If a function ψ is either ϕ-concave and increasing or ϕ-convex and decreasing, then the reverse inequality is valid in 4.12 .
Proof. Suppose that ψ is ϕ-convex and increasing. If we apply the inequality in 4.6 with f ψ • ϕ −1 : f I → R, and ψ • g i : S → f I instead of g i , we get

4.13
After taking ψ −1 of the both sides, it follows that M ϕ wg, P ≤ M ψ wg, P .

4.14
In the same way, we can prove the case when ψ is ϕ-concave and decreasing.
According to Remark 4.2, the reverse of Corollary 4.5 is valid if 1 ∈ S and all functionals P i are unital. Then we connect the basic and functional case in the following proposition. holds for all n-tuples w, g, and P as in 4.9 with 1 ∈ S and unital functionals P i .
Next in line is a functional analogy of refinements.
Theorem 4.7. Let φ t u t ϕ v t ψ : I → R for t ∈ a 0 , a 1 be functions as in 2.10 where I is the closed interval. Let t 0 , t 1 ∈ a 0 , a 1 such that t 0 ≤ t 1 .
If either ψ is ϕ-convex with both ϕ and ψ increasing or ϕ-concave with both ϕ and ψ decreasing, then the inequality M ϕ wg, P ≤ M φ t 0 wg, P ≤ M φ t 1 wg, P ≤ M ψ wg, P , 4.17 holds for all n-tuples w, g, and P as in 4.9 with 1 ∈ S and unital functionals P i .
If either ψ is ϕ-concave with both ϕ and ψ increasing or ϕ-convex with both ϕ and ψ decreasing, then the reverse inequality is valid in 4.17 .
The inequality in 4.17 is strict for a 0 < t 0 < t 1 < a 1 if ψ is strictly ϕ-convex, w > 0 and g / c.
A functional version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality

4.18
holds for all n-tuples w w 1 , . . . , w n , g g 1 , . . . , g n and, P P 1 , . . . , P n as in 4.9 with I a, ∞ where a > 0. The above inequality is strict for a 0 < t < a 1 if all w i > 0 and g i / g j for some i / j.
As a special case of the functional quasiarithmetic mean in 4.9 with I a, ∞ where a > 0, ϕ r x x r for r / 0 and ϕ 0 x ln x, we can observe the functional power mean

4.19
The following is the property of monotonicity, with refinements, of functional power means.
Corollary 4.8. Let r, s ∈ R such that r < s. Let φ r,s t : a, ∞ → R for t ∈ a 0 , a 1 be functions as in 2.23 where a > 0. Let t 0 , t 1 ∈ a 0 , a 1 such that t 0 ≤ t 1 .
If r < s < 0 or r < 0 s or r 0 < s or 0 < r < s, then the inequality holds for all n-tuples w, g, and P as in 4.9 with I a, ∞ , 1 ∈ S and unital functionals P i . If r < 0 < s, then we can take the series of inequalities

4.21
The inequalities in 4.20 -4.21 are strict for a 0 < t 0 < t 1 < a 1 if w > 0 and g / c.

Abstract and Applied Analysis 17
All the observed functional cases are reduced to the corresponding integral cases when we take for all functions g i ∈ L 1 Ω, μ i such that g i ω ∈ I for almost all ω ∈ Ω.

Results for Operator Case
We recall some notations and definitions. Let H be a Hilbert space. We define the bounds of linear operator A : Ax, x .

5.1
Let B H be the C * -algebra of all bounded linear operators A : H → H. If Sp A denotes the spectrum of a self-adjoint operator A ∈ B H , then it is well-known that Sp A is a subset of R and Sp A ⊆ m A , M A . If 1 H denotes the identity operator on H, then the following holds:

5.2
A continuous function f : I → R is said to be operator increasing on I if In this section, it is assumed that every weighted operator W ∈ B H 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.
If a function f is concave, then the reverse inequality is valid in 5.4 and 5.

5.7
If m B M B , then we take any support line f sup m B x kx l instead of the chord line.
If f is strictly convex, then the inequality in 5.4 and 5.6 is strict for all W > 0 and A / C.
The spectrum of operator M ϕ WA, Φ is contained in I because the spectrum of operator for every pair of real numbers a and b with a / 0. To verify this equality, let us take φ aϕ b, so φ −1 B ϕ −1 B − b1 K /a if B ∈ B K , and we get

5.12
If a function ψ is either ϕ-concave with operator increasing ψ −1 or ϕ-convex with operator decreasing ψ −1 , then the reverse inequality is valid in 5.11 . for some nonnegative numbers u y and v y such that u y v y 1 see Figure 3 . Now, first replace v y with 1 − u y in expression in 5.19 , and then express u y . Realizing u y as a function of the variable y, we obtain that

5.20
The above limit is onesided if ϕ −1 ψ −1 on some subinterval of an interval J. The functions ϕ −1 , ψ −1 , and φ −1 are continuous on J, and the same is true for the function u. Thus, the expression in 5.19 is the required presentation of function φ −1 as the convex combination of functions ϕ −1 and ψ −1 with coefficient functions u and v.
Theorem 5.4 can be simplified by using Lemma 5.5.
Corollary 5.6. Let φ t u t ϕ v t ψ : I → R for t ∈ a 0 , a 1 be functions as in 2.10 with the addition of u t v t 1, ϕ I ψ I and operator monotone ψ −1 . Let t 0 , t 1 ∈ a 0 , a 1 such that t 0 ≤ t 1 .
If either ψ is ϕ-convex with operator increasing ψ −1 or ϕ-concave with operator decreasing ψ −1 , then the inequality in 5.13 holds for all n-tuples W, A, and Φ as in 5.8 with spectral conditions as in Theorem 5.4. If either ψ is ϕ-concave with operator increasing ψ −1 or ϕ-convex with operator decreasing ψ −1 , then the reverse inequality is valid in 5.13 .