The Inequalities for Quasiarithmetic Means

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).

Figure 1 

Graphic concept of Jensen’s inequality.

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.

2. Results for Basic Case

For 𝑛-tuple 𝐱=(𝑥₁,…,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=(𝑥₁,…,𝑥_𝑛) be 𝑛-tuple with points 𝑥_𝑖∈𝐼, and 𝐩=(𝑝₁,…,𝑝_𝑛) be 𝑛-tuple with numbers 𝑝_𝑖∈[0,1] such that ∑^𝑛_𝑖=1𝑝_𝑖=1.
A function 𝑓 is convex if and only if the following inequality 𝑓^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖≤^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖, (2.1) holds for all above 𝑛-tuples 𝐩 and 𝐱.
Consequently, if ∑^𝑛_𝑖=1𝑝_𝑖=𝑝>0, not necessarily equals 1, then 𝑓 is convex if and only if 𝑓1𝑝^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖≤1𝑝^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖. (2.2)
A function 𝑓 is concave if and only if the reverse inequality is valid in (2.1) and (2.2).

A function 𝑓 is strictly convex if and only if the inequality in (2.1) and (2.2) is strict for all 𝐩>𝟎 and 𝐱≠𝐜.

Let 𝜑∶𝐼→ℝ be a strictly monotone continuous function. Let 𝐱=(𝑥₁,…,𝑥_𝑛) be 𝑛-tuple with points 𝑥_𝑖∈𝐼, and 𝐩=(𝑝₁,…,𝑝_𝑛) be 𝑛-tuple with numbers 𝑝_𝑖∈[0,1] such that ∑^𝑛_𝑖=1𝑝_𝑖=1. The discrete basic 𝜑-quasiarithmetic mean of points (particles) 𝑥_𝑖 with coefficients (weights) 𝑝_𝑖 is a number ℳ_𝜑(𝐩𝐱)=𝜑^−1𝑛_𝑖=1𝑝_𝑖𝜑𝑥_𝑖. (2.3) We understand that 𝐩𝐱=∑^𝑛_𝑖=1𝑝_𝑖𝑥_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: ℳ_id(𝐩𝐱)=^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖. (2.4)

Basic quasiarithmetic means have the property ℳ_{𝑎𝜑+𝑏}(𝐩𝐱)=ℳ_𝜑(𝐩𝐱), (2.5) for every pair of real numbers 𝑎 and 𝑏 with 𝑎≠0.

Suppose that all coefficients 𝑝_𝑖=1/𝑛. If we take 𝜑₁(𝑥)=𝑥, then ℳ_𝜑1(𝐩𝐱) is the arithmetic mean of numbers 𝑥_𝑖. If all 𝑥_𝑖>0 and we take 𝜑₀(𝑥)=ln𝑥, then ℳ_𝜑0(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.

Corollary 2.1. Let 𝜑,𝜓∶𝐼→ℝ be strictly monotone continuous functions.
A function 𝜓 is either 𝜑-convex and increasing or 𝜑-concave and decreasing if and only if following the inequality: ℳ_𝜑(𝐩𝐱)≤ℳ_𝜓(𝐩𝐱), (2.6) holds for all 𝑛-tuples 𝐩 and 𝐱 as in (2.3).
A function 𝜓 is either 𝜑-concave and increasing or 𝜑-convex and decreasing if and only if the reverse inequality is valid in (2.6).

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).

Figure 2 

The 𝜓-order among quasiarithmetic means ℳ𝜑 and ℳ𝜓 for 𝜑-convex and increasing 𝜓.

Suppose that all 𝑥_𝑖>0. If we apply Corollary 2.1 on three strictly monotone functions 𝜑₋₁(𝑥)=1/𝑥, 𝜑₀(𝑥)=ln𝑥 and 𝜑₁(𝑥)=𝑥 (two by two in pairs), then we get the weighted harmonic-geometric-arithmetic inequality 1𝑝₁/𝑥₁+⋯+𝑝_𝑛/𝑥_𝑛≤𝑥^𝑝1₁⋅…⋅𝑥^𝑝𝑛_𝑛≤𝑝₁𝑥₁+⋯+𝑝_𝑛𝑥_𝑛. (2.7)

Recall that a function 𝑓∶𝐼→ℝ is convex if and only if the following inequality: 𝑓𝑦−𝑓(𝑥)≤𝑓(𝑧)𝑦𝑦−𝑥−𝑓,𝑧−𝑦 (2.8) 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 𝜓𝑦−𝜓(𝑥)𝜑𝑦(𝑥)≤𝜓(𝑧)𝑦−𝜑−𝜓𝜑(𝑧)−𝜑𝑦. (2.9)

Let 𝑢,𝑣∶[𝑎₀,𝑎₁]→ℝ, where 𝑎₀<𝑎₁, be nonnegative continuous functions so that 𝑣/𝑢 is a strictly monotone increasing positive function on an open interval ⟨𝑎₀,𝑎₁⟩, with boundary conditions 𝑢(𝑎₀)=𝑣(𝑎₁)=1 and 𝑢(𝑎₁)=𝑣(𝑎₀)=0. Let both 𝜑 and 𝜓 be strictly monotone increasing or decreasing. For any 𝑡∈[𝑎₀,𝑎₁], we define a strictly monotone continuous function 𝜙_𝑡(𝑥)=𝑢(𝑡)𝜑(𝑥)+𝑣(𝑡)𝜓(𝑥)with𝑥∈𝐼. (2.10) For example, we can take 𝑢(𝑡)=1−𝑡 and 𝑣(𝑡)=𝑡 for 𝑡∈[0,1], 𝑢(𝑡)=1−√𝑡 and 𝑣(𝑡)=𝑡² for 𝑡∈[0,1], 𝑢(𝑡)=cos𝑡 and 𝑣(𝑡)=sin𝑡 for 𝑡∈[0,𝜋/2].

Lemma 2.2. Let 𝑎, 𝑏, 𝛼, 𝛽 be real numbers.
If 𝑎≤𝑏 and 𝛼≤𝛽, then 1+𝛽𝑎≤1+𝛼𝑎1+𝛽𝑏,1+𝛼𝑏 (2.11) 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∈[𝑎₀,𝑎₁] be functions as in (2.10). Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
If 𝜓 is 𝜑-convex (resp. 𝜑-concave), then 𝜙_𝑡1 is 𝜙_𝑡0-convex (resp. 𝜙_𝑡0-concave).

Proof. Suppose that 𝜓 is 𝜑-convex. Show that the function 𝜙_𝑡1 is 𝜙_𝑡0-convex. If 𝑡₀=𝑎₁, then 𝑡₀=𝑡₁, so we can suppose that 𝑡₀<𝑎₁. Let 𝑥,𝑦,𝑧∈𝐼 such that 𝑥<𝑦<𝑧. Let, with respect to (2.9) and definition of functions 𝑢 and 𝑣, 𝜓𝑦(𝑥)𝑎=−𝜓𝜑𝑦(𝑥)≤𝜓(𝑧)𝑦−𝜑−𝜓𝜑(𝑧)𝑦𝑣𝑡−𝜑=𝑏,𝛼=⁰𝑢𝑡₀≤𝑣𝑡₁𝑢𝑡₁=𝛽. (2.12) 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 𝜙_𝑡1𝑦−𝜙_𝑡1(𝑥)𝜙_𝑡0𝑦−𝜙_𝑡0(𝑥)=𝑢𝑡₁𝜑𝑦−𝜑(𝑥)+𝑣𝑡₁𝜓𝑦−𝜓(𝑥)𝑢𝑡₀𝜑𝑦−𝜑(𝑥)+𝑣𝑡₀𝜓𝑦(𝑥=𝑢𝑡−𝜓)₁𝑢𝑡₀1+𝛽𝑎≤𝑢𝑡1+𝛼𝑎𝑢𝑡¹₀1+𝛽𝑏=𝑢𝑡1+𝛼𝑏₁𝜑(𝑧)−𝜑𝑦+𝑣𝑡₁𝜓(𝑧)−𝜓𝑦𝑢𝑡₀𝜑(𝑧)−𝜑𝑦+𝑣𝑡₀𝜓(𝑧)𝑦=𝜙−𝜓_𝑡1(𝑧)−𝜙_𝑡1𝑦𝜙_𝑡0(𝑧)−𝜙_𝑡0𝑦, (2.13) which shows the required convexity by (2.9). Case of the concavity can be proved in a similar way.

If 𝜓 is strictly 𝜑-convex (resp. 𝜑-concave), then 𝜙_𝑡1 is strictly 𝜙_𝑡0-convex (resp. 𝜙_𝑡0-concave).

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∈[𝑎₀,𝑎₁] be functions as in (2.10). Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
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: ℳ_𝜑(𝐩𝐱)≤ℳ_𝜙𝑡0(𝐩𝐱)≤ℳ_𝜙𝑡1(𝐩𝐱)≤ℳ_𝜓(𝐩𝐱), (2.14) 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_𝑡1 is increasing, and 𝜙_𝑡0-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 𝑡↦ℳ_𝜙𝑡(𝐩𝐱)with𝑡∈𝑎₀,𝑎₁, (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 𝑎₀<𝑡₀<𝑡₁<𝑎₁ if 𝜓 is strictly 𝜑-convex or 𝜑-concave, 𝐩>𝟎 and 𝐱≠𝐜.

Let us take strictly monotone decreasing functions 𝜑(𝑥)=1/𝑥 and 𝜓(𝑥)=−ln𝑥 with 𝑥>0. Then (𝜓∘𝜑⁻¹)(𝑥)=ln𝑥, so 𝜓 is strictly 𝜑-concave. Let 𝜙_𝑡^ℎ𝑔(𝑥)(𝑡)1=𝑢𝑥(𝑡)𝑒−𝑣ln𝑥=ln^{𝑢(𝑡)/𝑥}𝑥^𝑣(𝑡). (2.16) If we apply the inequality in (2.14) with 𝑡=𝑡₀=𝑡₁ on 𝜙_𝑡^ℎ𝑔, we get 1∑^𝑛_𝑖=1𝑝_𝑖/𝑥_𝑖≤𝜙_𝑡^{ℎ𝑔−1𝑛}_𝑖=1𝑝_𝑖𝑒ln^{𝑢(𝑡)/𝑥𝑖}𝑥_𝑖^𝑣(𝑡)≤^𝑛_𝑖=1𝑥^𝑝𝑖_𝑖. (2.17)

Let us take strictly monotone increasing functions 𝜑(𝑥)=ln𝑥 and 𝜓(𝑥)=𝑥 with 𝑥>0. Then (𝜓∘𝜑⁻¹)(𝑥)=𝑒^𝑥, so 𝜓 is strictly 𝜑-convex. Let 𝜙_𝑡^𝑔𝑎(𝑥)=𝑢(𝑡)ln𝑥+𝑣(𝑡)𝑥=ln𝑥^𝑢(𝑡)𝑒^{𝑣(𝑡)𝑥}. (2.18) If we apply the inequality in (2.14) with 𝑡=𝑡₀=𝑡₁ on 𝜙_𝑡^𝑔𝑎, we get ^𝑛_𝑖=1𝑥^𝑝𝑖_𝑖≤𝜙_𝑡^{𝑔𝑎−1𝑛}_𝑖=1𝑝_𝑖ln𝑥_𝑖^𝑢(𝑡)𝑒^{𝑣(𝑡)𝑥𝑖}≤^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖. (2.19)

Connecting two above inequalities results in 1∑^𝑛_𝑖=1𝑝_𝑖/𝑥_𝑖≤𝜙_𝑡^{ℎ𝑔−1𝑛}_𝑖=1𝑝_𝑖𝑒ln^{𝑢(𝑡)/𝑥𝑖}𝑥_𝑖^𝑣(𝑡)≤^𝑛_𝑖=1𝑥^𝑝𝑖_𝑖≤𝜙_𝑡^{𝑔𝑎−1𝑛}_𝑖=1𝑝_𝑖ln𝑥_𝑖^𝑢(𝑡)𝑒^{𝑣(𝑡)𝑥𝑖}≤^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖. (2.20)

The inequality in (2.20) is strict for 𝑎₀<𝑡<𝑎₁ 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 𝜑₀(𝑥)=ln𝑥, we can observe the discrete basic power mean ℳ_𝑛^[𝑟](𝐩𝐱)=^𝑛𝑖=1𝑝_𝑖𝑥^𝑟_𝑖^1/𝑟for𝑟≠0exp^𝑛_𝑖=1𝑝_𝑖ln𝑥_𝑖for𝑟=0. (2.21)

Very useful consequence of Corollary 2.1 is a well-known property of monotonicity of basic power means.

Corollary 2.5. If 𝑟 and 𝑠 are real numbers such that 𝑟≤𝑠, then the following inequality: ℳ_𝑛^[𝑟](𝐩𝐱)≤ℳ_𝑛^[𝑠](𝐩𝐱), (2.22) holds for all 𝑛-tuples 𝐩 and 𝐱 as in (2.3) with 𝐼=⟨0,+∞⟩.

The inequality in (2.22) is strict for 𝑟<𝑠 if 𝐩>𝟎 and 𝐱≠𝐜.

Let functions 𝜙_𝑡^[𝑟,𝑠]∶⟨0,+∞⟩→ℝ for 𝑡∈[𝑎₀,𝑎₁] be specially defined by 𝜙_𝑡^[𝑟,𝑠](𝑥)=𝑢(𝑡)𝑥^𝑟+𝑣(𝑡)𝑥^𝑠𝑢(𝑡)𝑥for𝑟≠0,𝑠≠0_𝑟(𝑡)𝑢(𝑡)(𝑡)𝑥−𝑣ln𝑥for𝑟≠0,𝑠=0ln𝑥+𝑣^𝑠for𝑟=0,𝑠≠0. (2.23) Then the functions 𝑡↦ℳ_{𝜙𝑡[𝑟,𝑠]}(𝐩𝐱) with 𝐱>𝟎 are monotone increasing in the next four cases.

Case 𝑟<𝑠<0.
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.

Case 𝑟<0=𝑠.
Functions 𝜑(𝑥)=𝑥^𝑟 and 𝜓(𝑥)=−ln𝑥 are strictly monotone decreasing with strictly concave (𝜓∘𝜑⁻¹)(𝑥)=−(1/𝑟)ln𝑥 because −(1/𝑟)>0.

Case 𝑟=0<𝑠.
Functions 𝜑(𝑥)=ln𝑥 and 𝜓(𝑥)=𝑥^𝑠 are strictly monotone increasing with strictly convex (𝜓∘𝜑⁻¹)(𝑥)=𝑒^𝑠𝑥.

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.

Given traditional signs of power means, we will mark ℳ_{𝜙𝑡[𝑟,𝑠]}(𝐩𝐱) with ℳ^{𝜙𝑡[𝑟,𝑠]}_𝑛(𝐩𝐱). The inequality in (2.22) can be refined using Theorem 2.4 with functions 𝜙_𝑡^[𝑟,𝑠]. The following are refinements of power means.

Corollary 2.6. Let 𝑟,𝑠∈ℝ such that 𝑟<𝑠. Let 𝜙_𝑡^[𝑟,𝑠]∶⟨0,+∞⟩→ℝ for 𝑡∈[𝑎₀,𝑎₁] be functions as in (2.23). Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
If 𝑟<𝑠<0 or 𝑟<0=𝑠 or 𝑟=0<𝑠 or 0<𝑟<𝑠, then the inequality ℳ_𝑛^[𝑟](𝐩𝐱)≤ℳ^{𝜙𝑡0[𝑟,𝑠]}_𝑛(𝐩𝐱)≤ℳ^{𝜙𝑡1[𝑟,𝑠]}_𝑛(𝐩𝐱)≤ℳ_𝑛^[𝑠](𝐩𝐱), (2.24) 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 ℳ_𝑛^[𝑟](𝐩𝐱)≤ℳ^{𝜙𝑡0[𝑟,0]}_𝑛(𝐩𝐱)≤ℳ^{𝜙𝑡1[𝑟,0]}_𝑛(𝐩𝐱)≤ℳ_𝑛^[0](𝐩𝐱)≤ℳ^{𝜙𝑡0[0,𝑠]}_𝑛(𝐩𝐱)≤ℳ^{𝜙𝑡1[0,𝑠]}_𝑛(𝐩𝐱)≤ℳ_𝑛^[𝑠](𝐩𝐱). (2.25)

The inequalities in (2.24)-(2.25) are strict for 𝑎₀<𝑡₀<𝑡₁<𝑎₁ if 𝐩>𝟎 and 𝐱≠𝐜.

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, (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=(𝑔₁,…,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∈𝐿¹(Ω,𝜇) be a weighted function with ∫_Ω𝑤𝑑𝜇=1 such that 𝑤⋅𝑔,𝑤⋅(𝑓∘𝑔)∈𝐿¹(Ω,𝜇).
If a function 𝑓 is convex, then the inequality 𝑓_Ω𝑤⋅𝑔𝑑𝜇≤_Ω𝑤⋅𝑓∘𝑔𝑑𝜇, (3.1) holds for all above 𝑤, 𝑔 and 𝜇.
Consequently, if ∫_Ω𝑤𝑑𝜇=𝑝>0, not necessarily equals 1, then 𝑓1𝑝_Ω≤1𝑤⋅𝑔𝑑𝜇𝑝_Ω𝑤⋅𝑓∘𝑔𝑑𝜇. (3.2)
If a function 𝑓 is concave, then the reverse inequality is valid in (3.1) and (3.2).

The assumption ∫_Ω𝑤𝑑𝜇=1 with nonnegative 𝑤 almost everywhere on Ω, for the inequality in (3.1), assures that _Ω𝑤⋅𝑔𝑑𝜇∈𝐼,_Ω𝑤⋅𝑓∘𝑔𝑑𝜇∈𝑓(𝐼). (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 𝑔=𝑥𝜒_Ω𝑝+𝑦𝜒_Ω⧵Ω𝑝, (3.4) 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 _Ω𝑤⋅𝑔𝑑𝜇=𝑝𝑥+1−𝑝𝑦,_Ω𝑤⋅𝑓∘𝑔𝑑𝜇=_Ω𝑝𝑓(𝑥)𝑑𝜇+_Ω⧵Ω𝑝𝑓𝑦𝑑𝜇=𝑝𝑓(𝑥)+1−𝑝𝑓𝑦. (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=(𝜇₁,…,𝜇_𝑛) be 𝑛-tuple with probability measures 𝜇_𝑖 on Ω, 𝐠=(𝑔₁,…,𝑔_𝑛) be 𝑛-tuple with measurable functions 𝑔_𝑖∶Ω→𝐼, and 𝐰=(𝑤₁,…,𝑤_𝑛) be 𝑛-tuple with weighted functions 𝑤_𝑖∈𝐿¹(Ω,𝜇_𝑖) with ∑^𝑛_𝑖=1∫_Ω𝑤_𝑖𝑑𝜇_𝑖=1 such that 𝑤_𝑖⋅𝑔_𝑖,𝑤_𝑖⋅(𝑓∘𝑔_𝑖)∈𝐿¹(Ω,𝜇_𝑖).
A function 𝑓 is convex if and only if the inequality 𝑓^𝑛_𝑖=1Ω𝑤_𝑖⋅𝑔_𝑖𝑑𝜇_𝑖≤^𝑛_𝑖=1Ω𝑤_𝑖⋅𝑓∘𝑔_𝑖𝑑𝜇_𝑖, (3.6) holds for all above 𝑛-tuples 𝐰, 𝐠 and 𝝁.
Consequently, if ∑^𝑛_𝑖=1∫_Ω𝑤_𝑖𝑑𝜇_𝑖=𝑝>0, not necessarily equals 1, then 𝑓 is convex if and only if 𝑓1𝑝^𝑛_𝑖=1Ω𝑤_𝑖⋅𝑔_𝑖𝑑𝜇_𝑖≤1𝑝^𝑛_𝑖=1Ω𝑤_𝑖⋅𝑓∘𝑔_𝑖𝑑𝜇_𝑖. (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=(𝜇₁,…,𝜇_𝑛) be 𝑛-tuple with probability measures 𝜇_𝑖 on Ω, 𝐠=(𝑔₁,…,𝑔_𝑛) be 𝑛-tuple with measurable functions 𝑔_𝑖∶Ω→𝐼, and 𝐰=(𝑤₁,…,𝑤_𝑛) be 𝑛-tuple with weighted functions 𝑤_𝑖∈𝐿¹(Ω,𝜇_𝑖) with ∑^𝑛_𝑖=1∫_Ω𝑤_𝑖𝑑𝜇_𝑖=1 such that 𝑤_𝑖⋅(𝜑∘𝑔_𝑖)∈𝐿¹(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 ℳ_𝜑(𝐰𝐠,𝝁)=𝜑^−1𝑛_𝑖=1Ω𝑤_𝑖⋅𝜑∘𝑔_𝑖𝑑𝜇_𝑖. (3.8) This number belongs to 𝐼 because the integral convex combination ∑^𝑛_𝑖=1∫_Ω𝑤_𝑖⋅(𝜑∘𝑔_𝑖)𝑑𝜇_𝑖∈𝜑(𝐼). Integral quasiarithmetic means also satisfy the property ℳ_{𝑎𝜑+𝑏}(𝐰𝐠,𝝁)=ℳ_𝜑(𝐰𝐠,𝝁), (3.9) for every pair of real numbers 𝑎 and 𝑏 with 𝑎≠0.

Bearing in mind Theorem 3.2, the following corollary is valid.

Corollary 3.3. Let 𝜑,𝜓∶𝐼→ℝ be strictly monotone continuous functions.
A function 𝜓 is either 𝜑-convex and increasing or 𝜑-concave and decreasing if and only if the inequality ℳ_𝜑(𝐰𝐠,𝝁)≤ℳ_𝜓(𝐰𝐠,𝝁), (3.10) holds for all 𝑛-tuples 𝐰, 𝐠, and 𝝁 as in (3.8).
A function 𝜓 is either 𝜑-concave and increasing or 𝜑-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 𝜑,𝜓∶𝐼→ℝ be strictly monotone continuous functions. Then the inequality ℳ_𝜑(𝐩𝐱)≤ℳ_𝜓(𝐩𝐱), (3.11) holds for all 𝑛-tuples 𝐩 and 𝐱 as in (2.3) if and only if the inequality ℳ_𝜑(𝐰𝐠,𝝁)≤ℳ_𝜓(𝐰𝐠,𝝁), (3.12) holds for all 𝑛-tuples 𝐰, 𝐠, and 𝝁 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 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∈[𝑎₀,𝑎₁] be functions as in (2.10). Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
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,𝝁)≤ℳ_𝜙𝑡0(𝐰𝐠,𝝁)≤ℳ_𝜙𝑡1(𝐰𝐠,𝝁)≤ℳ_𝜓(𝐰𝐠,𝝁), (3.13) 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 𝑎₀<𝑡₀<𝑡₁<𝑎₁ 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 1∑^𝑛_𝑖=1∫_Ω𝑤_𝑖/𝑔_𝑖𝑑𝜇_𝑖≤𝜙_𝑡^{ℎ𝑔−1𝑛}_𝑖=1Ω𝑤_𝑖𝑒⋅ln^{𝑢(𝑡)/𝑔𝑖}𝑔_𝑖^𝑣(𝑡)𝑑𝜇_𝑖≤^𝑛_𝑖=1Ω𝑔^𝑤𝑖_𝑖𝑑𝜇_𝑖≤𝜙_𝑡^{𝑔𝑎−1𝑛}_𝑖=1Ω𝑤_𝑖⋅ln𝑔_𝑖^𝑢(𝑡)𝑒^{𝑣(𝑡)𝑔𝑖}𝑑𝜇_𝑖≤^𝑛_𝑖=1Ω𝑤_𝑖⋅𝑔_𝑖𝑑𝜇_𝑖, (3.14) holds for all 𝑛-tuples 𝐰=(𝑤₁,…,𝑤_𝑛), 𝐠=(𝑔₁,…,𝑔_𝑛) and 𝝁=(𝜇₁,…,𝜇_𝑛) as in (3.8) with 𝐼=⟨0,+∞⟩. The above inequality is strict for 𝑎₀<𝑡<𝑎₁ 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 𝜑₀(𝑥)=ln𝑥, we can observe the integral power mean ℳ_𝑛^[𝑟](𝐰𝐠,𝝁)=^𝑛𝑖=1Ω𝑤_𝑖⋅𝑔^𝑟_𝑖𝑑𝜇_𝑖^1/𝑟for𝑟≠0exp^𝑛_𝑖=1Ω𝑤_𝑖⋅ln𝑔_𝑖𝑑𝜇_𝑖for𝑟=0. (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_𝑡^[𝑟,𝑠]∶⟨0,+∞⟩→ℝ for 𝑡∈[𝑎₀,𝑎₁] be functions as in (2.23). Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
If 𝑟<𝑠<0 or 𝑟<0=𝑠 or 𝑟=0<𝑠 or 0<𝑟<𝑠, then the inequality ℳ_𝑛^[𝑟](𝐰𝐠,𝝁)≤ℳ^{𝜙𝑡0[𝑟,𝑠]}_𝑛(𝐰𝐠,𝝁)≤ℳ^{𝜙𝑡1[𝑟,𝑠]}_𝑛(𝐰𝐠,𝝁)≤ℳ_𝑛^[𝑠](𝐰𝐠,𝝁), (3.16) holds for all 𝑛-tuples 𝐰, 𝐠, and 𝝁 as in (3.8) with 𝐼=⟨0,+∞⟩.
If 𝑟<0<𝑠, then we can take the series of inequalities ℳ_𝑛^[𝑟](𝐰𝐠,𝝁)≤ℳ^{𝜙𝑡0[𝑟,0]}_𝑛(𝐰𝐠,𝝁)≤ℳ^{𝜙𝑡1[𝑟,0]}_𝑛(𝐰𝐠,𝝁)≤ℳ_𝑛^[0](𝐰𝐠)≤ℳ^{𝜙𝑡0[0,𝑠]}_𝑛(𝐰𝐠,𝝁)≤ℳ^{𝜙𝑡1[0,𝑠]}_𝑛(𝐰𝐠,𝝁)≤ℳ_𝑛^[𝑠](𝐰𝐠,𝝁). (3.17)

The inequalities in (3.16)-(3.17) are strict for 𝑎₀<𝑡₀<𝑡₁<𝑎₁ 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.

4. 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 𝑓P𝑤⋅𝑔≤P𝑤⋅𝑓∘𝑔, (4.1) holds for all above 𝑤, 𝑔, and P.
Consequently, if P(𝑤)=𝑝>0, not necessarily equals 1, then 𝑓1𝑝P≤1𝑤⋅𝑔𝑝P𝑤⋅𝑓∘𝑔. (4.2)
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.

Example 4.1. Let 𝑆=𝐼=⟨0,1] and 𝒮=𝑔∶𝐼⟶ℝ∣lim_𝑥→0+𝑔(𝑥)finite. (4.3) If P∶𝒮→ℝ is defined by P𝑔=lim_𝑥→0+𝑔(𝑥), (4.4) 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_𝑆𝑝∈𝒮 and P(𝜒_𝑆𝑝)=𝑝. If we take 𝑔=𝑥𝜒_𝑆𝑝+𝑦𝜒_{𝑆⧵𝑆𝑝} and 𝑤=1, then it follows that PP𝑓(𝑥)𝜒𝑤⋅𝑔=𝑝𝑥+1−𝑝𝑦,𝑤⋅𝑓∘𝑔=P_𝑆𝑝+𝑓𝑦𝜒_{𝑆⧵𝑆𝑝}=𝑝𝑓(𝑥)+1−𝑝𝑓𝑦, (4.5) 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.

Theorem 4.3. Let 𝑓∶𝐼→ℝ be a continuous function where 𝐼 is the closed interval. Let 𝐏=(P₁,…,P_𝑛) be 𝑛-tuple with positive linear functionals P_𝑖∶𝒮→ℝ, 𝐠=(𝑔₁,…,𝑔_𝑛) be 𝑛-tuple with functions 𝑔_𝑖∶𝑆→𝐼, and 𝐰=(𝑤₁,…,𝑤_𝑛) be 𝑛-tuple with weighted functions 𝑤_𝑖∈𝒮 with ∑^𝑛_𝑖=1P_𝑖(𝑤_𝑖)=1 such that 𝑤_𝑖⋅𝑔_𝑖,𝑤_𝑖⋅(𝑓∘𝑔_𝑖)∈𝒮.
If a function 𝑓 is convex, then the inequality 𝑓^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝑔_𝑖≤^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝑓∘𝑔_𝑖, (4.6) holds for all above 𝑛-tuples 𝐰, 𝐠, and 𝐏.
Consequently, if ∑^𝑛_𝑖=1P_𝑖(𝑤_𝑖)=𝑝>0, not necessarily equals 1, then 1𝑝^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝑔_𝑖≤1𝑝^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝑓∘𝑔_𝑖. (4.7)
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 𝑓^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝑔_𝑖=𝑓^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖≤^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖=^𝑛_𝑖=1P_𝑖𝑤_𝑖𝑓1P_𝑖𝑤_𝑖P_𝑖𝑤_𝑖⋅𝑔_𝑖≤^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝑓∘𝑔_𝑖. (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=(P₁,…,P_𝑛) be 𝑛-tuple with positive linear functionals P_𝑖∶𝒮→ℝ, 𝐠=(𝑔₁,…,𝑔_𝑛) be 𝑛-tuple with functions 𝑔_𝑖∶𝑆→𝐼, and 𝐰=(𝑤₁,…,𝑤_𝑛) be 𝑛-tuple with weighted functions 𝑤_𝑖∈𝒮 with ∑^𝑛_𝑖=1P_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 ℳ_𝜑(𝐰𝐠,𝐏)=𝜑^−1𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝜑∘𝑔_𝑖. (4.9) This number belongs to 𝐼 because the functional convex combination ∑^𝑛_𝑖=1P_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. Indeed, if 𝜙(𝑥)=𝑎𝜑(𝑥)+𝑏, then 𝜙⁻¹(𝑥)=𝜑⁻¹(𝑥−𝑏/𝑎), and we have ℳ_𝜙(𝐰𝐠,𝐏)=𝜙^−1𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝜙∘𝑔_𝑖=𝜑⁻¹∑^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝑎𝜑∘𝑔_𝑖+𝑏−𝑏𝑎=𝜑^−1𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝜑∘𝑔_𝑖=ℳ_𝜑(𝐰𝐠,𝐏). (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 𝜓∘𝜓^−1𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝜑∘𝑔_𝑖≤^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝜓∘𝑔_𝑖. (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.

Proposition 4.6. Let 𝜑,𝜓∶𝐼→ℝ be strictly monotone continuous functions where 𝐼 is the closed interval. Then the inequality ℳ_𝜑(𝐩𝐱)≤ℳ_𝜓(𝐩𝐱), (4.15) holds for all 𝐩 and 𝐱 as in (2.3) if and only if the inequality ℳ_𝜑(𝐰𝐠,𝐏)≤ℳ_𝜓(𝐰𝐠,𝐏), (4.16) 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∈[𝑎₀,𝑎₁] be functions as in (2.10) where 𝐼 is the closed interval. Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
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_𝜙𝑡0(𝐰𝐠,𝐏)≤ℳ_𝜙𝑡1(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 𝑎₀<𝑡₀<𝑡₁<𝑎₁ 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 1∑^𝑛_𝑖=1P_𝑖𝑤_𝑖/𝑔_𝑖≤𝜙_𝑡^{ℎ𝑔−1𝑛}_𝑖=1P_𝑖𝑤_𝑖𝑒⋅ln^{𝑢(𝑡)/𝑔𝑖}𝑔_𝑖^𝑣(𝑡)≤^𝑛_𝑖=1P_𝑖𝑔^𝑤𝑖_𝑖≤𝜙_𝑡^{𝑔𝑎−1𝑛}_𝑖=1P_𝑖𝑤_𝑖⋅ln𝑔_𝑖^𝑢(𝑡)𝑒^{𝑣(𝑡)𝑔𝑖}≤^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅𝑔_𝑖, (4.18) holds for all 𝑛-tuples 𝐰=(𝑤₁,…,𝑤_𝑛), 𝐠=(𝑔₁,…,𝑔_𝑛) and, 𝐏=(P₁,…,P_𝑛) as in (4.9) with 𝐼=[𝑎,+∞⟩ where 𝑎>0. The above inequality is strict for 𝑎₀<𝑡<a₁ 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 𝜑₀(𝑥)=ln𝑥, we can observe the functional power mean ℳ_𝑛^[𝑟](𝐰𝐠,𝐏)=^𝑛𝑖=1P_𝑖𝑤_𝑖⋅𝑔^𝑟_𝑖^1/𝑟for𝑟≠0exp^𝑛_𝑖=1P_𝑖𝑤_𝑖⋅ln𝑔_𝑖for𝑟=0. (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_𝑡^[𝑟,𝑠]∶[𝑎,+∞⟩→ℝ for 𝑡∈[𝑎₀,𝑎₁] be functions as in (2.23) where 𝑎>0. Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
If 𝑟<𝑠<0 or 𝑟<0=𝑠 or 𝑟=0<𝑠 or 0<𝑟<𝑠, then the inequality ℳ_𝑛^[𝑟](𝐰𝐠,𝐏)≤ℳ^{𝜙𝑡0[𝑟,𝑠]}_𝑛(𝐰𝐠,𝐏)≤ℳ^{𝜙𝑡1[𝑟,𝑠]}_𝑛(𝐰𝐠,𝐏)≤ℳ_𝑛^[𝑠](𝐰𝐠,𝐏), (4.20) 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 ℳ_𝑛^[𝑟](𝐰𝐠,𝐏)≤ℳ^{𝜙𝑡0[𝑟,0]}_𝑛(𝐰𝐠,𝐏)≤ℳ^{𝜙𝑡1[𝑟,0]}_𝑛(𝐰𝐠,𝐏)≤ℳ_𝑛^[0](𝐰𝐠,𝐏)≤ℳ^{𝜙𝑡0[0,𝑠]}_𝑛(𝐰𝐠,𝐏)≤ℳ^{𝜙𝑡1[0,𝑠]}_𝑛(𝐰𝐠,𝐏)≤ℳ_𝑛^[𝑠](𝐰𝐠,𝐏). (4.21)

The inequalities in (4.20)-(4.21) are strict for 𝑎₀<𝑡₀<𝑡₁<𝑎₁ if 𝐰>𝟎 and 𝐠≠𝐜.

All the observed functional cases are reduced to the corresponding integral cases when we take P_𝑖𝑔_𝑖=_Ω𝑔_𝑖𝑑𝜇_𝑖, (4.22) for all functions 𝑔_𝑖∈𝐿¹(Ω,𝜇_𝑖) such that 𝑔_𝑖(𝜔)∈𝐼 for almost all 𝜔∈Ω.

5. Results for Operator Case

We recall some notations and definitions. Let 𝐻 be a Hilbert space. We define the bounds of linear operator 𝐴∶𝐻→𝐻 with 𝑚_𝐴=inf_‖𝑥‖=1⟨𝐴𝑥,𝑥⟩,𝑀_𝐴=sup_‖𝑥‖=1⟨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: 𝑚_𝐴1_𝐻≤𝐴≤𝑀_𝐴1_𝐻,‖𝐴‖=sup_‖𝑥‖=1|⟨𝐴𝑥,𝑥⟩|=max𝑚_𝐴,𝑀_𝐴. (5.2)

A continuous function 𝑓∶𝐼→ℝ is said to be operator increasing on 𝐼 if 𝐴≤𝐵implies𝑓(𝐴)≤𝑓(𝐵), (5.3) for every pair of self-adjoint operators 𝐴,𝐵 on 𝐻 with spectra in 𝐼. A function 𝑓 is said to be operator decreasing if—𝑓 is operator increasing. A function 𝑓 is operator monotone if it is operator increasing or decreasing.

For 𝑛-tuple 𝐀=(𝐴₁,…,𝐴_𝑛) with operators 𝐴_𝑖∈ℬ(𝐻) sometimes, we will write 𝐀>𝟎 if all 𝐴_𝑖>0, and 𝐀≠C if 𝐴_𝑖≠𝐴_𝑗 for some 𝑖≠𝑗.

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.

Theorem 5.1. Let 𝑓∶𝐼→ℝ be a continuous function. Let Φ=(Φ₁,…,Φ_𝑛) be 𝑛-tuple with positive linear mappings Φ_𝑖∶ℬ(𝐻)→ℬ(𝐾), 𝐀=(𝐴₁,…,𝐴_𝑛) be 𝑛-tuple with self-adjoint operators 𝐴_𝑖∈ℬ(𝐻) with bounds 𝑚_𝑖≤𝑀_𝑖 from 𝐼, and 𝐖=(𝑊₁,…,𝑊_𝑛) be 𝑛-tuple with weighted operators 𝑊_𝑖∈ℬ(𝐻) with ∑^𝑛_𝑖=1Φ_𝑖(𝑊_𝑖)=1_𝐾. Let 𝑚_𝐵≤𝑀_𝐵 be bounds of an operator 𝐵=∑^𝑛_𝑖=1Φ_𝑖(𝑊_𝑖𝐴_𝑖).
If a function 𝑓 is convex, then the inequality 𝑓^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝐴_𝑖≤^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝑓𝐴_𝑖, (5.4) holds for all above 𝑛-tuples 𝐖, 𝐀, and Φ provided spectral conditions 𝑚_𝐵,𝑀_𝐵∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.5)
Consequently, if ∑^𝑛_𝑖=1Φ_𝑖(𝑊_𝑖)=𝑊_Φ is strictly positive, not necessarily equals 1_𝐾, then 𝑓𝑊_Φ^𝑛−1_𝑖=1Φ_𝑖𝑊_𝑖𝐴_𝑖≤𝑊_Φ^𝑛−1_𝑖=1Φ𝑊_𝑖𝑓𝐴_𝑖. (5.6)
If a function 𝑓 is concave, then the reverse inequality is valid in (5.4) and (5.6).

Proof. If 𝑚_𝐵<𝑀_𝐵, then we take the chord line 𝑓^cho_{[𝑚𝐵,𝑀𝐵]}(𝑥)=𝑘𝑥+𝑙 through the points 𝑇₁(𝑚_𝐵,𝑓(𝑚_𝐵)), and 𝑇₂(𝑀_𝐵,𝑓(𝑀_𝐵)). It follows: 𝑓^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝐴_𝑖≤𝑓^cho_{𝑚𝐵,𝑀𝐵}^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝐴_𝑖=𝑘^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝐴_𝑖+𝑙1_𝐾=^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝑘𝐴_𝑖+𝑙1_𝐻=^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝑓^cho_{𝑚𝐵,𝑀𝐵}𝐴_𝑖≤^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝑓𝐴_𝑖. (5.7)
If 𝑚_𝐵=𝑀_𝐵, then we take any support line 𝑓^sup_[𝑚𝐵](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.

Remark 5.2. The reverse of Theorem 5.1 is trivially valid if all Φ_𝑖 are unital. With this assumption, we can take 𝑊_𝑖=𝑝_𝑖1_𝐻 and 𝐴_𝑖=𝑥_𝑖1_𝐻.

Let 𝜑∶𝐼→ℝ be a strictly monotone continuous function. Let Φ=(Φ₁,…,Φ_𝑛) be 𝑛-tuple with positive linear mappings Φ_𝑖∶ℬ(𝐻)→ℬ(𝐾), 𝐀=(𝐴₁,…,𝐴_𝑛) be 𝑛-tuple with self-adjoint operators 𝐴_𝑖∈ℬ(𝐻) with spectra in 𝐼, and 𝐖=(𝑊₁,…,𝑊_𝑛) be 𝑛-tuple with weighted operators 𝑊_𝑖∈ℬ(𝐻) with ∑^𝑛_𝑖=1Φ_𝑖(𝑊_𝑖)=1_𝐾. The discrete operator 𝜑-quasiarithmetic mean of operators 𝐴_𝑖 with weighted operators 𝑊_𝑖 with respect to mappings Φ_𝑖 is an operator ℳ_𝜑(𝐖𝐀,𝚽)=𝜑^−1𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝜑𝐴_𝑖. (5.8) The spectrum of operator ℳ_𝜑(𝐖𝐀,Φ) is contained in 𝐼 because the spectrum of operator ∑^𝑛_𝑖=1Φ_𝑖(𝑊_𝑖𝜑(𝐴_𝑖)) is contained in 𝜑(𝐼). Operator quasiarithmetic means also have the property ℳ_{𝑎𝜑+𝑏}(𝐖𝐀,𝚽)=ℳ_𝜑(𝐖𝐀,𝚽), (5.9) 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 ℳ_𝜙(𝐖𝐀,𝚽)=𝜙^−1𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝜙𝐴_𝑖=𝜑⁻¹∑^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝑎𝜑𝐴_𝑖+𝑏1_𝐻−𝑏1_𝐾𝑎=𝜑^−1𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝜑𝐴_𝑖=ℳ_𝜑(𝐖𝐀,𝚽). (5.10)

Corollary 5.3. Let 𝜑,𝜓∶𝐼→ℝ be strictly monotone continuous functions with operator monotone 𝜓⁻¹.
Let 𝐖, 𝐀 and Φ be as in (5.8). Let 𝑚_𝑖≤𝑀_𝑖 and 𝑚_𝜑≤𝑀_𝜑 be bounds of operators 𝐴_𝑖 and ℳ_𝜑(𝐖𝐀,Φ), respectively.
If a function 𝜓 is either 𝜑-convex with operator increasing 𝜓⁻¹ or 𝜑-concave with operator decreasing 𝜓⁻¹, then the inequality ℳ_𝜑(𝐖𝐀,𝚽)≤ℳ_𝜓(𝐖𝐀,𝚽), (5.11) holds for all above 𝑛-tuples 𝐖, 𝐀, and Φ provided spectral conditions 𝑚_𝜑,𝑀_𝜑∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (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).

The following is operator analogy of Theorem 2.4.

Theorem 5.4. Let 𝜙_𝑡=𝑢(𝑡)𝜑+𝑣(𝑡)𝜓∶𝐼→ℝ for 𝑡∈[𝑎₀,𝑎₁] be functions as in (2.10) with operator monotone 𝜓⁻¹. Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
If either 𝜓 is 𝜑-convex with operator increasing 𝜙_𝑡⁻¹₀, 𝜙_𝑡⁻¹₁, and 𝜓⁻¹ or 𝜑-concave with operator decreasing 𝜙_𝑡⁻¹₀, 𝜙_𝑡⁻¹₁, and 𝜓⁻¹, then the inequality ℳ_𝜑(𝐖𝐀,𝚽)≤ℳ_𝜙𝑡0(𝐖𝐀,𝚽)≤ℳ_𝜙𝑡1(𝐖𝐀,𝚽)≤ℳ_𝜓(𝐖𝐀,𝚽), (5.13) holds for all 𝑛-tuples 𝐖, 𝐀, and Φ as in (5.8) that provided the following spectral conditions: 𝑚_𝜑,𝑀_𝜑∩𝑚_𝑖,𝑀_𝑖𝑚=∅orendpointfor𝑖=1,…,𝑛_𝜙𝑡0,𝑀_𝜙𝑡0∩𝑚_𝑖,𝑀_𝑖𝑚=∅orendpointfor𝑖=1,…,𝑛_𝜙𝑡1,𝑀_𝜙𝑡1∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.14)
If either 𝜓 is 𝜑-concave with operator increasing 𝜙_𝑡⁻¹₀, 𝜙_𝑡⁻¹₁, and 𝜓⁻¹ or 𝜑-convex with operator decreasing 𝜙_𝑡⁻¹₀, 𝜙_𝑡⁻¹₁, and 𝜓⁻¹, 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 𝜙_𝑡0 and 𝜙_𝑡1. If 𝜓 is 𝜑-convex with both 𝜑 and 𝜓 increasing, then 𝜙_𝑡1 is 𝜙_𝑡0-convex by Proposition 2.3. If 𝜙_𝑡⁻¹₁ is operator increasing, then by Corollary 5.3 the inequality ℳ_𝜙𝑡0(𝐖𝐀,𝚽)≤ℳ_𝜙𝑡1(𝐖𝐀,𝚽), (5.15) is valid with spectral conditions 𝑚_𝜙𝑡0,𝑀_𝜙𝑡0∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.16) Any part of the series of inequalities in (5.13) is proved similarly.

The inequality in (5.13) is strict for 𝑎₀<𝑡₀<𝑡₁<𝑎₁ if 𝜓 is strictly 𝜑-convex, 𝜙_𝑡⁻¹₀,𝜙_𝑡⁻¹₁,  and𝜓⁻¹ are strictly operator increasing, 𝐖>𝟎 and 𝐀≠𝐂.

We are interested in sufficient conditions under which the functions 𝜙_𝑡 will be operator increasing.

Lemma 5.5. Let 𝜙=𝛼𝜑+𝛽𝜓 be a convex combination (𝛼,𝛽≥0,𝛼+𝛽=1) of strictly monotone increasing or decreasing continuous functions 𝜑,𝜓∶𝐼→ℝ such that 𝜑(𝐼)=𝜓(𝐼)=𝐽. Then 𝜙⁻¹=𝑢⋅𝜑⁻¹+𝑣⋅𝜓⁻¹, (5.17) where 𝑢,𝑣∶𝐽→ℝ are nonnegative continuous functions such that 𝑢(𝑦)+𝑣(𝑦)=1 for every 𝑦∈𝐽.

Proof. Take any 𝑥∈𝐼. If 𝑦=𝜙(𝑥)=𝛼𝜑(𝑥)+𝛽𝜓(𝑥), (5.18) then 𝑥=𝜙⁻¹𝑦=𝑢𝑦⋅𝜑⁻¹𝑦+𝑣𝑦⋅𝜓⁻¹𝑦, (5.19) 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 𝑢𝑦=𝜙⁻¹𝑦−𝜓⁻¹𝑦𝜑⁻¹𝑦−𝜓⁻¹𝑦for𝜑⁻¹𝑦≠𝜓⁻¹𝑦lim_𝑦→𝑦0𝜙⁻¹𝑦−𝜓⁻¹𝑦𝜑⁻¹𝑦−𝜓⁻¹𝑦for𝜑⁻¹𝑦₀=𝜓⁻¹𝑦₀. (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.

Figure 3 

Convex combination of strictly monotone increasing functions 𝜑 and 𝜓.

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∈[𝑎₀,𝑎₁] 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 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
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).

Proof. According to Lemma 5.5 continuous functions 𝑢_𝑡 and 𝑣_𝑡, with 𝑢_𝑡+𝑣_𝑡=1, exist for every 𝑡∈[𝑎₀,𝑎₁] so that 𝜙_𝑡⁻¹=𝑢_𝑡⋅𝜑⁻¹+𝑣_𝑡⋅𝜓⁻¹=𝑢_𝑡⋅𝜑⁻¹−𝜓⁻¹+𝜓⁻¹. (5.21) Let 𝑡>𝑎₀; otherwise, it is 𝜙_𝑎⁻¹₀=𝜑⁻¹. Then 𝑣_𝑡≠0 and 𝜙_𝑡⁻¹ is operator increasing (resp. decreasing) if 𝜓⁻¹ is operator increasing (resp. decreasing).

A special case of the operator quasiarithmetic mean in (5.8) with 𝐼=⟨0,+∞⟩,  𝜑_𝑟(𝑥)=𝑥^𝑟 for 𝑟≠0 and 𝜑₀(𝑥)=ln𝑥, we can observe the operator power mean ℳ_𝑛^[𝑟](𝐖𝐀,𝚽)=^𝑛𝑖=1Φ_𝑖𝑊_𝑖𝐴^𝑟_𝑖^1/𝑟for𝑟≠0exp^𝑛_𝑖=1Φ_𝑖𝑊_𝑖ln𝐴_𝑖for𝑟=0. (5.22)

The consequence of Corollary 5.3 for operator power means the following.

Corollary 5.7. Let 𝑟 and 𝑠 be real numbers such that 𝑟≤𝑠.
Let 𝐖, 𝐀, and Φ be as in (5.8) with strictly positive 𝐀. Let 𝑚_𝑖≤𝑀_𝑖, 𝑚^[𝑟]≤𝑀^[𝑟], and 𝑚^[𝑠]≤𝑀^[𝑠] be bounds of operators 𝐴_𝑖, ℳ_𝑛^[𝑟](𝐖𝐀,Φ), and ℳ_𝑛^[𝑠](𝐖𝐀,Φ), respectively.
If 𝑠≤−1 or 𝑠≥1, then the inequality ℳ_𝑛^[𝑟](𝐖𝐀,𝚽)≤ℳ_𝑛^[𝑠](𝐖𝐀,𝚽), (5.23) holds for all above 𝑛-tuples 𝐖, 𝐀, and Φ provided spectral conditions 𝑚^[𝑟],𝑀^[𝑟]∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.24)
If 𝑟≤−1 or 𝑟≥1, then the inequality in (5.23) holds provided spectral conditions 𝑚^[𝑠],𝑀^[𝑠]∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.25)

The inequality in (5.23) is strict for 𝑟<𝑠 if 𝐖>𝟎 and 𝐀≠𝐂.

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 ℳ_𝑛^[−1](𝐖𝐀,Φ)≤ℳ_𝑛^[0](𝐖𝐀,Φ)≤ℳ_𝑛^[1](𝐖𝐀,Φ), that is, ^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝐴_𝑖⁻¹⁻¹≤exp^𝑛_𝑖=1Φ_𝑖𝑊_𝑖ln𝐴_𝑖≤^𝑛_𝑖=1Φ_𝑖𝑊_𝑖𝐴_𝑖, (5.26) holds for all 𝑛-tuples 𝐖=(𝑊₁,…,𝑊_𝑛), 𝐀=(𝐴₁,…,𝐴_𝑛) and Φ=(Φ₁,…,Φ_𝑛) as in (5.8) with strictly positive 𝐴_𝑖 provided spectral conditions 𝑚^[0],𝑀^[0]∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.27)

There remain only the refinements of the operator power means by using Corollary 5.6.

Corollary 5.8. Let 𝑟,𝑠∈ℝ⧵{0} such that 𝑟<𝑠. Let 𝜙_𝑡^[𝑟,𝑠]∶⟨0,+∞⟩→ℝ for 𝑡∈[𝑎₀,𝑎₁] be functions as in (2.23) with the addition of 𝑢(𝑡)+𝑣(𝑡)=1. Let 𝑡₀,𝑡₁∈[𝑎₀,𝑎₁] such that 𝑡₀≤𝑡₁.
Let 𝐖, 𝐀, and Φ be as in (5.8) with strictly positive 𝐀. Let 𝑚_𝑖≤𝑀_𝑖, 𝑚^[𝑟]≤𝑀^[𝑟], 𝑚^[𝑠]≤𝑀^[𝑠], 𝑚^{[𝜙𝑡0[𝑟,𝑠]]}≤𝑀^{[𝜙𝑡0[𝑟,𝑠]]}, and 𝑚^{[𝜙𝑡1[𝑟,𝑠]]}≤𝑀^{[𝜙𝑡1[𝑟,𝑠]]} be bounds of operators 𝐴_𝑖, ℳ_𝑛^[𝑟](𝐖𝐀,Φ), ℳ_𝑛^[𝑠](𝐖𝐀,Φ), ℳ^{𝜙𝑡0[𝑟,𝑠]}_𝑛(𝐖𝐀,Φ), and ℳ^{𝜙𝑡1[𝑟,𝑠]}_𝑛(𝐖𝐀,Φ), respectively.
If 𝑟>0,𝑠≥1 or 𝑠=−1, then the inequality ℳ_𝑛^[𝑟](𝐖𝐀,𝚽)≤ℳ^{𝜙𝑡0[𝑟,𝑠]}_𝑛(𝐖𝐀,𝚽)≤ℳ^{𝜙𝑡1[𝑟,𝑠]}_𝑛(𝐖𝐀,𝚽)≤ℳ_𝑛^[𝑠](𝐖𝐀,𝚽), (5.28) holds for all above 𝑛-tuples 𝐖, 𝐀, and Φ provided spectral conditions: 𝑚^[𝑟],𝑀^[𝑟]∩𝑚_𝑖,𝑀_𝑖𝑚=∅orendpointfor𝑖=1,…,𝑛,_{[𝜙𝑡0[𝑟,𝑠]]},𝑀^{[𝜙𝑡0[𝑟,𝑠]]}∩𝑚_𝑖,𝑀_𝑖𝑚=∅orendpointfor𝑖=1,…,𝑛,_{[𝜙𝑡1[𝑟,𝑠]]},𝑀^{[𝜙𝑡1[𝑟,𝑠]]}∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.29)
If 𝑟=1, then the inequality ℳ_𝑛^[1](𝐖𝐀,𝚽)≤ℳ^{𝜙𝑡0[𝑠,1]}_𝑛(𝐖𝐀,𝚽)≤ℳ^{𝜙𝑡1[𝑠,1]}_𝑛(𝐖𝐀,𝚽)≤ℳ_𝑛^[𝑠](𝐖𝐀,𝚽), (5.30) holds for all above 𝑛-tuples 𝐖, 𝐀, and Φ provided spectral conditions: 𝑚^[𝑠],𝑀^[𝑠]∩𝑚_𝑖,𝑀_𝑖𝑚=∅orendpointfor𝑖=1,…,𝑛,_{[𝜙𝑡0[𝑠,1]]},𝑀^{[𝜙𝑡0[𝑠,1]]}∩𝑚_𝑖,𝑀_𝑖𝑚=∅orendpointfor𝑖=1,…,𝑛,_{[𝜙𝑡1[𝑠,1]]},𝑀^{[𝜙𝑡1[𝑠,1]]}∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.31)

Proof. Recall that 𝜙_𝑡^[𝑟,𝑠](𝑥)=𝑢(𝑡)𝑥^𝑟+𝑣(𝑡)𝑥^𝑠 for 𝑟≠0,𝑠≠0. The next is 𝜑(𝑥)=𝑥^𝑟 and 𝜓(𝑥)=𝑥^𝑠 with 𝑥∈𝐼=⟨0,+∞⟩, so 𝜑(𝐼)=𝜓(𝐼)=𝐼.

Case 𝑟>0,𝑠≥1.
We have (𝜓∘𝜑⁻¹)(𝑥)=𝑥^𝑠/𝑟 with (𝑠/𝑟)>1, and 𝜓⁻¹(𝑥)=𝑥^1/𝑠 with 0<(1/𝑠)≤1. The function 𝜓 is strictly 𝜑-convex with strictly operator increasing 𝜓⁻¹. Then the inequality in (5.28) is valid with associated spectral conditions by Corollary 5.6.

Case 𝑠=−1.
We have (𝜓∘𝜑⁻¹)(𝑥)=𝑥^−1/𝑟 with 0<−(1/𝑟)<1, and 𝜓⁻¹(𝑥)=𝑥⁻¹. The function 𝜓 is strictly 𝜑-concave with strictly operator decreasing 𝜓⁻¹. In this case, the inequality in (5.28) is also valid with associated spectral conditions by Corollary 5.6.

Case 𝑟=1.
We use functions 𝜙_𝑡^[𝑠,1](𝑥)=𝑢(𝑡)𝑥^𝑠+𝑣(𝑡)𝑥. In this case 𝜑(𝑥)=𝑥^𝑠 and 𝜓(𝑥)=𝑥, thus (𝜓∘𝜑⁻¹)(𝑥)=𝑥^1/𝑠 with 0<(1/𝑠)<1, and 𝜓⁻¹(𝑥)=𝑥. The function 𝜓 is strictly 𝜑-concave with strictly operator increasing 𝜓⁻¹. 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 𝑎₀<𝑡₀<𝑡₁<𝑎₁ if 𝐖>𝟎 and 𝐀≠𝐂.

Unfortunately, we cannot use a logarithmic function because ln(𝐼)=ℝ≠𝐼.

Remark 5.9. Let 𝑟, 𝑠, 𝜙_𝑡, 𝑡₀, 𝑡₁,𝐖, 𝐀, and Φ be as in Corollary 5.8.
If 𝑟≤−1 and 𝑠<0, then the problem remains the inequality ℳ_𝑛^[𝑟](𝐖𝐀,𝚽)≤ℳ^{𝜙𝑡0[𝑠,𝑟]}_𝑛(𝐖𝐀,𝚽)≤ℳ^{𝜙𝑡1[𝑠,𝑟]}_𝑛(𝐖𝐀,𝚽)≤ℳ_𝑛^[𝑠](𝐖𝐀,𝚽), (5.32) valid for all above 𝑛-tuples 𝐖, 𝐀, and Φ provided spectral conditions: 𝑚^[𝑠],𝑀^[𝑠]∩𝑚_𝑖,𝑀_𝑖𝑚=∅orendpointfor𝑖=1,…,𝑛,_{[𝜙𝑡0[𝑠,𝑟]]},𝑀^{[𝜙𝑡0[𝑠,𝑟]]}∩𝑚_𝑖,𝑀_𝑖𝑚=∅orendpointfor𝑖=1,…,𝑛,_{[𝜙𝑡1[𝑠,𝑟]]},𝑀^{[𝜙𝑡1[𝑠,𝑟]]}∩𝑚_𝑖,𝑀_𝑖=∅orendpointfor𝑖=1,…,𝑛. (5.33)
The inequality in (5.32) is valid for 𝑟=−1 with associated spectral conditions because (𝜓∘𝜑⁻¹)(𝑥)=𝑥^−1/𝑠 with −(1/𝑠)>1, that is, 𝜓 is 𝜑-convex, and 𝜓⁻¹(𝑥)=𝑥⁻¹, that is, 𝜓⁻¹ is operator decreasing.