Abstract

The paper offers generalizations of the Jensen-Mercer inequality for self-adjoint operators and generally convex functions. The obtained results are applied to define the quasi-arithmetic operator means without using operator convexity. The version of the harmonic-geometric-arithmetic operator mean inequality is derived as an example.

1. Introduction

Throughout the paper we will use a real interval with the nonempty interior and real segments and with .

We briefly summarize a development path of the operator form of Jensen’s inequality. Let and be Hilbert spaces, let and be associated -algebras of bounded linear operators, and let and be their identity operators.

Combining the results from [1, 2], it follows that every operator convex function satisfies the Schwarz inequality where is a positive linear mapping such that and is a self-adjoint operator with the spectrum . The above inequality was extended in [3] to the inequality where are positive linear mappings such that and are self-adjoint operators with spectra . The operator inequality of (2) was formulated for convex (without operator) continuous functions in [4] assuming the spectral conditions: and for all , where .

Including positive operators satisfying , we have that every convex continuous function satisfies the inequality if provided the spectral conditions: and for all self-adjoint operators , and the operator sum . The inequality in (3) is possible because the operators and are self-adjoint.

2. Discrete and Operator Inequalities for Convex Functions and Trinomial Affine Combinations

2.1. Discrete Variants

Every number can be uniquely presented as the binomial affine combination which is convex if and only if the number belongs to the interval . Given the function , let be the function of the chord line passing through the points and of the graph of . Applying the affinity of to the combination in (4), we get If the function is convex, then we have the inequality and the reverse inequality

Let be coefficients such that . Let be points where . We consider the affine combination . Inserting the affine combination assuming that , we get the binomial form

Lemma 1. Let be coefficients such that . Let be points such that and .
Then the affine combination and every convex function satisfies the inequality

Proof. The condition involves . Then the binomial combination of the right-hand side in (8) is convex since its coefficients and also . So, the combination belongs to . Applying the inequality in (6) and the affinity of , we get because .

Lemma 1 is trivially true if . It is also valid for because then the observed affine combinations with become convex, and associated inequalities follow from Jensen’s inequality. The similar combinations including were observed in [5, Corollary 11 and Theorem 12] additionally using a monotone function . If , then the inequality in (10) is reduced to simple Mercer’s variant of Jensen’s inequality obtained in [6].

Lemma 2. Let be coefficients such that . Let be points such that and .
Then the affine combination and every convex function , where satisfies the inequality

Proof. The condition entails or , and the coefficients of the binomial form of (8) satisfy if , or if . So, the combination does not belong to . Applying the inequality in (7), we get the series of inequalities as in (11) but with the reverse inequality signs.

It is not necessary to require in Lemma 2, because it follows from the other coefficient conditions.

2.2. Operator Variants

We write for self-adjoint operators if the inner product inequality holds for every vector . A self-adjoint operator is positive (nonnegative) if it is greater than or equal to null operator (). If and are continuous functions such that for every , then the operator inequality is valid. The bounds of a self-adjoint operator are defined with and its spectrum is contained in wherein we have the operator inequality

More details on the theory of bounded operators and their inequalities can be found in [7]. The operator versions of Lemmas 1 and 2 follow.

Corollary 3. Let be coefficients such that . Let be a self-adjoint operator such that .
Then and every convex continuous function satisfies the inequality

Proof. The spectral inclusion in (16) follows from the inclusion in (9). Using the affinity of the function and the operator inequalities , we can replace the discrete inequalities in (11) with the operator inequalities.

Corollary 4. Let be coefficients such that . Let be a self-adjoint operator such that .
Then and every convex continuous function , where contains and , satisfies the inequality

3. Main Results

We want to extend and generalize the inequalities in (17) and (19) including positive operators and positive linear mappings. The main results are Theorems 8 and 9.

Lemma 5. Let be linear mappings and let be positive linear operators so that . Let be self-adjoint operators.
Then every affine function , where and are real constants, satisfies the operator equality

Proof. Applying the affinity of the function and the assumption , it follows that achieving the equality in (20).

Lemma 6. Let be positive linear mappings and let be positive linear operators so that . Let be self-adjoint operators such that .
Then the spectrum of the operator sum is contained in .

Proof. Applying the positive operators and the positive mappings to the assumed spectral inequalities we get Summing the above inequalities and using the assumption , we have which provides that .

Corollary 7. Let be positive linear mappings and let be positive linear operators so that . Let be self-adjoint operators such that .
Then every convex continuous function satisfies the inequality

Proof. The inequality in (25) is the consequence of Lemmas 5 and 6, and the discrete inequality where are points and are coefficients of the sum equal to .

Theorem 8. Let be coefficients such that . Let be positive linear mappings and let be positive linear operators so that . Let be self-adjoint operators such that .
Then the spectrum of the operator is contained in , and every convex continuous function satisfies the inequality
If the function is concave, then the reverse inequality is valid in (28).

Proof. Taking the operator sum , the spectral inclusion follows from Lemma 6 and the inclusion in (16). Assuming and applying the convexity of and the affinity of according to Lemma 5, we get because .

The version of Theorem 8 for and all was obtained in [8] as the main result.

Theorem 9. Let be coefficients such that . Let be positive linear mappings and let be positive linear operators so that . Let be self-adjoint operators such that , and let be the operator sum such that .
Then the spectrum of the operator satisfies the relation , and every convex continuous function , where contains all spectra and , satisfies the inequality If the function is concave, then the reverse inequality is valid in (31).

Proof. The relation is the consequence of the relation in (18). Assuming and using the convexity of and the affinity of , as well as the inequalities , we get the series of inequalities as in (29) but with the reverse inequality signs.

4. Application to Quasi-Arithmetic Means

In applications of convexity to quasi-arithmetic means, we use strictly monotone continuous functions such that the function is convex, in which case we say that is -convex. A similar notation is used for concavity. This terminology is taken from [9, Definition 1.19].

A continuous function is said to be operator increasing on if implies for every pair of self-adjoint operators with spectra in . A function is said to be operator decreasing if the function is operator increasing.

Take an operator affine combination as in Theorem 8. If is a strictly monotone continuous function, we define the -quasi-arithmetic mean of the combination as the operator The spectrum of the operator is contained in because the spectrum of the operator is contained in . The quasi-arithmetic means defined in (33) are invariant with respect to the affinity; that is, the equality holds for all pairs of real numbers and . Indeed, if , then and therefore, it follows that

The order of the pair of quasi-arithmetic means and depends on convexity of the function and monotonicity of the function . Theorem 8 can be applied to operator means as follows.

Corollary 10. Let be an affine combination as in (32) satisfying the assumptions of Theorem 8. Let be strictly monotone continuous functions.
If is either -convex with operator increasing or -concave with operator decreasing , then one has the inequality
If is either -convex with operator decreasing or -concave with operator increasing , then one has the reverse inequality in (38).

Proof. Let us prove the case in which is -convex with operator increasing . Put . Applying the inequality in (28) of Theorem 8 to the affine combination of (34) with and the convex function , we get Assigning the increasing function to the above inequality, we attain which finishes the proof.

Using Corollary 10 we get the following version of the harmonic-geometric-arithmetic mean inequality for operators.

Corollary 11. If is an affine operator combination as in (32) satisfying the assumptions of Theorem 8 with the addition that , then one has the harmonic-geometric-arithmetic operator inequality

Proof. To prove the left-hand side of the inequality in (41) we use the functions and . Then and , so is -convex and is operator decreasing. Applying Corollary 10 to this case, we have
To prove the right-hand side we use the functions and . Then and , so is -convex and is operator increasing. Applying the inequality in (38), we get
The double inequality in (41) follows by connecting the inequalities in (42) and (43).