Some complementary inequalities to Jensen’s operator inequality

In this paper, we study some complementary inequalities to Jensen’s inequality for self-adjoint operators, unital positive linear mappings, and real valued twice differentiable functions. New improved complementary inequalities are presented by using an improvement of the Mond-Pečarić method. These results are applied to obtain some inequalities with quasi-arithmetic means.

Jensen's inequality is one of the most important inequalities. It has many applications in mathematics and statistics, and some other well-known inequalities are its particular cases. There is an extensive literature devoted to Jensen's operator inequality regarding its variuous generalizations, refinements, and extensions; see, for example, [3][4][5][6][7][8][9].
Using the Mond-Pečarić method, a generalized complementary inequality of Jensen's operator inequality is presented in [14]. A continuous version of this inequality is presented in [15]. Also, Mićić, Pavić, and Pečarić [16] obtained a better bound than that in [15] under the assumptions that (x t ) t∈T is a bounded continuous field of self-adjoint elements in a unital C * -algebra A with spectra in [m, M], m < M, defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ, and (φ t ) t∈T is a unital field of positive linear mappings φ t : A → B from A to another unital C * -algebra B. If m x where , Moreover, Mićić, Pečarić, and Perić [7] obtained a refinement of (2). For convenience, we introduce abbreviationsx and δ f as follows: where m, M, m < M, are scalars such that the spectra of x t are in [m, M], t ∈ T; where f : [m, M] → R is a continuous function. Obviously,x ≥ 0, and δ f ≥ 0 for convex f or δ f ≤ 0 for concave f . Under the above assumptions, they proved in [7] that where mx is the lower bound of the operatorx. More precisely, they obtained the following improved difference-and radio-type inequalities (for a convex function f ): and where (7) and (8).
In this paper, we obtain some complementary inequalities to Jensen's operator inequality for twice differentiable functions. We obtain new inequalities improving the same type inequalities given in [17]. In particular, we improve inequalities (5), (7), and (8). Applying the obtained results, we give some inequalities for quasi-arithmetic means.

Some auxiliary results without convexity
In this section, we give some results, which we will use in the next sections.
To prove our next result, we need the following lemma.
Lemma A ( [7]) Let f be a convex function on an interval I, m, M ∈ I, and p 1 , p 2 ∈ [0, 1] such that p 1 + p 2 = 1. Then where Proof These results follow from [18, Theorem 1, p. 717] for n = 2. For the reader's convenience, we give an elementary proof.
Since f is convex, we have for all x, y ∈ I and positive weights α, β. Suppose that 0 < p 1 < p 2 < 1, p 1 + p 2 = 1. Then, applying (10), we obtain It follows that which gives the second inequality in (9).
Applying Lemma A, we obtain the following inequalities for twice differentiable functions. If α ≤ f on [m, M] for some α ∈ R, then and where k f , l f are defined by (1), δ f is defined by (4), and If f ≤ β on [m, M] for some β ∈ R, then the reverse inequalities are valid in (11) and (12) with β instead of α.

Main results
In this section, we generalize or improve some inequalities in Section 1 and [17].
Applying Lemma (1) and using the Mond-Pečarić method, we present versions of inequalities (2) and (5) without convexity and for one operator. We omit the proof.
where M 1 is the upper bound of the operator α (A 2 ), M 1 ≤ max{αm 2 , αM 2 }, and m A is the lower bound of the operator A. Further, where m 1 is the lower bound of the operator α ( for some β ∈ R, then the reverse inequalities are valid in (20) and (21) (6) is not in general valid: but applying the first inequality in (20), we have: It suffices to put Then (22) and (23)  Applying Lemma 2 to a strictly convex function f , we improve inequalities (2) and (5). ≤ sup Proof Since F(·, v) is operator monotone in the first variable and m (A) ≤ (A) ≤ M (A) , we obtain (24).

Remark 2
We can easily generalize the above results to a bounded continuous field of self-adjoint elements in a unital C * -algebra A. Indeed, replacing A with x t and with φ t in (11), integrating, and using the equality T φ t (1 H ) dμ(t) = 1 K , we get the following inequality: Next, using the operator monotonicity of F(·, v) in the first variable, we obtain the desired inequalities.

Difference-type inequalities
Applying Lemma 2 to the function F(u, v) = uv, we can obtain complementary inequalities to Jensen's operator inequality for neither a convex nor a concave function f . These are versions of the corresponding inequalities for one operator given in [16] and [7]. We omit the details. Next, applying this result to a convex function f , we obtain an improved inequality (6) and its complementary inequality for one operator. and Proof Using Theorem 3, we obtain (25). Next, (26) follows from the following inequalities:

Remark 3 (i) Using elementary calculus, we can precisely determine the values of the constants
provided that g is a convex or concave function: if g is concave, then and for some z ∈ (m (A) , M (A) ), if g is convex, then C is equal to RHS in (28) with reverse inequality signs, and c is equal to RHS in (27) with min instead of max. (ii) Using the same technique as in Remark 2, we can obtain generalizations of the above results for a bounded continuous field of self-adjoint elements in a unital C * -algebra. We omit the details.
(iii) If f ≡ g is strictly convex twice differentiable on [m, M] and 0 < α ≤ f on [m, M], then (25) improves inequality in [16,Theorem 3.4]. If f is operator convex, then Jensen's operator inequality holds, but if f is not operator convex, then (26) gives its complementary inequality.
Applying Lemma 2 and Theorem 4 to the functions f (z) = z p and g(z) = z q for selected integers p and q, we obtain the following example. These inequalities are generalizations of some inequalities in [7, Corollary 7] for nonpositive operators.  where α = p(p -1)m p-2 , + m), and l p := (Mm p -mM p )/(Mm). Also, (ii) Let p > 1 be an even number (see RHS of Figure 1), and let q > 0 be an integer. Then f (z) ≥ 0 is an even function. So we can put α = p(p-1)·min{m p-2 , M p-2 } > 0 in Theorem 4. Applying (25) and (26), we obtain

Example 2 Let
where C and C 1 are defined by (30) with k p := M p -m p M-m -α 2 (M + m) and k p := M p -m p M-m , respectively, and where c is defined by (32). (ii) If p ∈ (0, 1), then the reverse inequalities are valid in (33) and (34) with α = p(p -1)M p-2 . If q = p, then the inequality (A p ) ≤ (A) p is tighter than (33). In all these inequalities the constants C and c are determined as follows:
if q ∈ (0, 1), then C (or C 1 ) is equal to RHS in (32) with max instead of min and c is equal to the right side in (30) with reverse inequality signs.

Ratio-type converse inequalities
Applying Lemma 2, similarly to the previous subsection, we can obtain complementary inequalities to Jensen's operator inequality for neither a convex nor a concave function f .
These are versions of the corresponding inequalities for one operator given in [16] and [7]. We omit the details.
Moreover, applying this result to a convex function f , we improve inequality (7) and obtain its complementary inequality for one operator.  if g -(z) ≥ ag(z) az+b for every z ∈ (m (A) , M (A) ), if g > 0 is convex, then K is equal to RHS in (38) with reverse inequality signs, and k is equal to RHS in (37) with min instead of max.
First, we recall the operator order between quasi-arithmetic means (see e.g. provided that (i) ψ • ϕ -1 is operator convex, and ψ -1 is operator monotone, or (ii) ψ • ϕ -1 is operator concave, and -ψ -1 is operator monotone, or (iii) ϕ -1 is operator convex, and ψ -1 is operator concave. The order (41) without operator convexity or operator concavity is given in [27, Theorem 3.1]) under spectra conditions. In [30], some techniques are used while manipulating some inequalities related to continuous fields of operators.
Complementary inequalities to (41) are observed in [27]. We give a general result, which is tighter than that given in [ If (i) ψ • ϕ -1 is convex and ψ -1 is operator monotone, or (i ) ψ • ϕ -1 is concave and -ψ -1 is operator monotone, then where m ϕ and M ϕ , m ϕ < M ϕ , are bounds of the mean M ϕ (x, ), and Now we will study an extension and improvement of (42).
For convenience, we introduce some notation corresponding to δ f in (4) and A in (13): First, we give a version of Lemma 2 for means. This is an extension of (42) without convexity or concavity.
ψ -1 is operator monotone, and where M 1 is the upper bound of αϕ(M ϕ ) 2 , and mx ϕ is the lower bound ofx ϕ .
If, in addition, where m 1 is the lower bound of αϕ(M ϕ ) 2 .
Replacing A with ϕ(M ϕ ) and with the identical mapping in (11), we obtain Next, applying the operator monotonicity of ψ -1 and taking into account (45), we obtain we obtain the desired sequence of inequalities (46).

Results and discussion
In this paper, we obtain some complementary inequalities to Jensen's inequality for a realvalued twice differentiable functions f . We obtain a generalization of known inequalities for a wider class of twice differentiable functions. Also, we obtain a refinement of some known inequalities for a class of continuous concave or convex functions. Finally, we obtain some complementary inequalities to quasi-arithmetic means with weaker conditions.
Our results have enriched the theory for the complementary inequality to Jensen's operator inequality.

Conclusions
Jensen's inequality is one of the most important inequalities. It has many applications in mathematics and statistics and some other well-known inequalities are its particular cases.
This paper conducts a further study to the development of the existing theory of Jensen's inequality for self-adjoint operators in a Hilbert space. The main contribution is the obtained complementary to Jensen's inequality for general real-valued twice differentiable functions. The numerical examples confirm that the proposed method gives new inequalities for functions that are neither convex nor concave.
Moreover, our method gives improvements of inequalities given in [10][11][12][13] for convex or concave functions. The conditions in this paper are weaker than those in the previous research.
Finally, using the same method, we obtained new inequalities with quasi-arithmetic means. For further research, we should study improved inequalities given in [27].