Abstract
The main target of this paper is to discuss operator Hermite–Hadamard inequality for convex functions, without appealing to operator convexity. Several forms of this inequality will be presented and some applications including norm and mean inequalities will be shown too.
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1 Introduction and Preliminaries
Let \(\mathcal {B}\left( \mathcal {H} \right) \) be the \(C^*\)–algebra of all bounded linear operators on a Hilbert space \(\mathcal {H}\). As usual, we reserve m, M for scalars and \({{\mathbf {1}}_{\mathcal {H}}}\) for the identity operator on \(\mathcal {H}\). A self adjoint operator A is said to be positive (written \(A\ge 0\)) if \(\left\langle Ax,x \right\rangle \ge 0\) for all \(x\in \mathcal {H}\), while it is said to be strictly positive (written \(A>0\)) if A is positive and invertible. If A and B are self adjoint, we write \(B\ge A\) in case \(B-A\ge 0\).
The Gelfand map \(f\left( t \right) \mapsto f\left( A \right) \) is an isometrical \(*\)–isomorphism between the \({{C}^{*}}\)–algebra \(C\left( {\text {sp}}\left( A \right) \right) \) of continuous functions on the spectrum \({\text {sp}}\left( A \right) \) of a self adjoint operator A and the \({{C}^{*}}\)–algebra generated by A and the identity operator \({{\mathbf {1}}_{\mathcal {H}}}\). If \(f,g\in C\left( {\text {sp}}\left( A \right) \right) \), then \(f\left( t \right) \ge g\left( t \right) \) (\(t\in {\text {sp}}\left( A \right) \)) implies that \(f\left( A \right) \ge g\left( A \right) \). This is called the functional calculus for the operator A.
A real valued continuous function f defined on the interval J is said to be operator convex if \(f\left( \left( 1-v \right) A+vB \right) \le \left( 1-v \right) f\left( A \right) +vf\left( B \right) \) for every \(0<v<1\) and for every pair of bounded self adjoint operators A and B whose spectra are both in J. One of the most important examples is the power function \(t\mapsto {{t}^{p}}\) for \(1\le p\le 2\).
The Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard, states that if a function \(f:J\rightarrow \mathbb {R}\) is convex, then the following chain of inequalities hold:
Since (see, e.g. [4] [Lemma 2.1])
we can rewrite (1.1) in the following form
The Hermite–Hadamard inequality plays an essential role in research on inequalities and has quite a sizeable technical literature; as one can see in [1, 2, 5, 8,9,10,11].
Obtaining operator inequalities corresponding to certain scalar inequalities have been an active research area in operator theory. Dragomir [3] gave an operator version of Hermite–Hadamard inequality and proved that
whenever \(f:J\rightarrow \mathbb {R}\) is an operator convex and A, B are two self adjoint operators with spectra in J.
We emphasize here that the assumption operator convexity is essential to obtain (1.3). For example, if
then simple computations show that
It is easily seen that
So, even though \(f(t)=t^3\) is convex (not operator convex), (1.3) does not hold; showing that operator convexity cannot be dropped.
It is then natural to ask about which conditions one should have so that the inequalities in (1.3) are valid for any convex function.
In [7], it is shown that convex functions satisfy (1.3) if some empty intersection conditions are imposed on the spectra of A, B. We also refer the reader to [12]. In this article, we present several forms of (1.3) using the Mond–Pečarić method for convex functions. For example, we show that for appropriate constants \(\alpha , \beta ,\)
when \(m{{\mathbf {1}}_{\mathcal {H}}}\le A,B\le M{{\mathbf {1}}_{\mathcal {H}}}\) and f, g are certain functions. Then several converses and variants of (1.4) are presented. See Theorem 2.1 and the results that follow for the details.
In the end, we present other forms using properties of inner product; without appealing to the Mond–Pečarić method. Our results generalize some known inequalities presented in [3, 9].
In our proofs, we will frequently use the basic inequality [6, Theorem 1.2]
valid for the convex function \(f:J\rightarrow \mathbb {R}\), the self adjoint operator A with spectrum in J and the unit vector \(x\in \mathcal {H}.\)
2 Main Results
We present our main results in this section; where the Mond–Pečarić method is discussed first. Throughout this section, we use the following two standard notations for the function \(f:[m,M]\rightarrow {\mathbb {R}}\);
2.1 Hermite–Hadamard Inequalities Using the Mond–Pe\(\check{\mathrm{c}}\)arić Method
Our first convex (not operator convex) version of (1.3) reads as follows.
Theorem 2.1
Let \(A,B\in \mathcal {B}\left( \mathcal {H} \right) \) be two self adjoint operators satisfying \(m{{\mathbf {1}}_{\mathcal {H}}}\le A,B\le M{{\mathbf {1}}_{\mathcal {H}}}\) and let \(f,g:\left[ m,M \right] \rightarrow \mathbb {R}\) be two continuous functions. If f and g are both convex functions, then for a given \(\alpha \ge 0,\)
where \(\beta =\underset{m\le x\le M}{\mathop {\max }}\,\left\{ {{a}_{f}}x+{{b}_{f}}-\alpha g\left( x \right) \right\} .\)
Proof
It follows from the convexity of \(f:\left[ m,M \right] \rightarrow \mathbb {R}\) that
for any \(m\le x\le M\). Since \(m{{\mathbf {1}}_{\mathcal {H}}}\le A,B\le M{{\mathbf {1}}_{\mathcal {H}}}\), then \(m{{\mathbf {1}}_{\mathcal {H}}}\le \left( 1-t \right) A+tB\le M{{\mathbf {1}}_{\mathcal {H}}}\). Applying functional calculus for the operator \(T=\left( 1-t \right) A+tB\) in (2.2) implies
Integrating the inequality over \(t\in \left[ 0,1 \right] \), we get
Now, let \(x\in \mathcal {H}\) be a unit vector. One can write
where in (2.3) we used (1.5), and (2.4) follows directly from convexity of g.
Consequently,
for any unit vector \(x\in \mathcal {H}\). This completes the proof of inequality (2.1). \(\square \)
Now we present some applications of Theorem 2.1.
Corollary 2.1
Let \(A,B\in \mathcal {B}\left( \mathcal {H} \right) \) be two self adjoint operators satisfying \(m{{\mathbf {1}}_{\mathcal {H}}}\le A,B\le M{{\mathbf {1}}_{\mathcal {H}}}\) and let \(f,g:\left[ m,M \right] \rightarrow \mathbb {R}\) be two continuous functions. If f and \(g>0\) are convex, then
where \(\alpha =\underset{m\le x\le M}{\mathop {\max }}\,\left\{ \frac{{{a}_{f}}x+{{b}_{f}}}{g\left( x \right) } \right\} \).
Further,
where \(\beta =\underset{m\le x\le M}{\mathop {\max }}\,\left\{ {{a}_{f}}x+{{b}_{f}}-g\left( x \right) \right\} \)
Proof
Notice that when \(\alpha =\underset{m\le x\le M}{\mathop {\max }}\,\left\{ \frac{{{a}_{f}}x+{{b}_{f}}}{g\left( x \right) } \right\} \), then \(a_fx+b_f-\alpha g(x)\le 0.\) Therefore, from Theorem 2.1, \(\beta \le 0\) and (2.1) implies (2.5). The other inequality follows similarly from Theorem 2.1. \(\square \)
Remark 2.1
Setting \(f=g>0\) the inequality (2.5) implies
where \(\alpha =\underset{m\le x\le M}{\mathop {\max }}\,\left\{ \frac{{{a}_{f}}x+{{b}_{f}}}{f\left( x \right) } \right\} \). We remark that a similar result as in (2.6) was shown in [9, Theorem 3.9]. Therefore, Theorem 2.1 can be considered as an extension of [9, Theorem 3.9].
Notice that Theorem 2.1 and its consequences above present operator order inequalities. In the next result, we obtain operator norm inequalities. Here, \(|A|=(A^*A)^{1/2},\) where \(A^*\) is the adjoint operator of A.
Proposition 2.1
Let \(A,B\in \mathcal {B}\left( \mathcal {H} \right) \) be two self adjoint operators satisfying \(m{{\mathbf {1}}_{\mathcal {H}}}\le \left| A \right| ,\left| B \right| \le M{{\mathbf {1}}_{\mathcal {H}}}\) and let \(f:\left[ m,M \right] \rightarrow \mathbb {R}\) be a nonnegative continuous increasing convex function. Then for a given \(\alpha \ge 0,\)
where \(\beta =\underset{m\le x\le M}{\mathop {\max }}\,\left\{ {{a}_{f}}x+{{b}_{f}}-\alpha f\left( x \right) \right\} \).
Proof
Recall that if \(T\in \mathcal {B}\left( \mathcal {H} \right) \) is a self adjoint operator, then \(\left\| T \right\| =\underset{\left\| x \right\| =1}{\mathop {\sup }}\,\left| \left\langle Tx,x \right\rangle \right| \). Let \(x\in \mathcal {H}\) be a unit vector. Then
Now, by taking supremum over \(x\in \mathcal {H}\) with \(\left\| x \right\| =1\) in (2.7) and noting that f is increasing,
thanks to (2.1). This completes the proof. \(\square \)
We end this section by giving the weighted generalization of operator Hermite–Hadamard inequality. For convenience, we use \(A{{\nabla }_{\lambda }B}\) to denote \(\left( 1-\lambda \right) A+\lambda B\). We then show that Theorem 2.2 is a generalization of (1.3).
Theorem 2.2
Let \(A,B\in \mathcal {B}\left( \mathcal {H} \right) \) be two self adjoint operators satisfying \(m{{\mathbf {1}}_{\mathcal {H}}}\le A,B\le M{{\mathbf {1}}_{\mathcal {H}}}\) and let \(f:\left[ m,M \right] \rightarrow \mathbb {R}\) be an operator convex function. Then for any \(0\le \lambda \le 1\),
Proof
Since for \(0\le \lambda ,v\le 1\),
holds, we infer from the operator convexity of f that
Integrating the inequality over \(v\in \left[ 0,1 \right] \), we get
which is the statement of the theorem. \(\square \)
Remark 2.2
To show that Theorem 2.2 is a generalization of (1.3), put \(\lambda ={1}/{2}\;\). Thus
On making use of the change of variable \(v=1-2t\) we have
and by the change of variable \(v=2t-1\),
Relations (2.9) and (2.10), gives
and the assertion follows by combining (2.8) and (2.11).
2.2 Reverse Hermite–Hadamard Inequalities Using the Mond–Pe\(\check{\mathrm{c}}\)arić Method
In the forthcoming theorem, we give additive, and multiplicative type reverses for the first and the second inequalities in (1.3).
Theorem 2.3
Let \(A,B\in \mathcal {B}\left( \mathcal {H} \right) \) be two self adjoint operators satisfying \(m{{\mathbf {1}}_{\mathcal {H}}}\le A,B\le M{{\mathbf {1}}_{\mathcal {H}}}\) and let \(f,g:\left[ m,M \right] \rightarrow \mathbb {R}\) be two continuous functions. If f is a convex function, then for a given \(\alpha \ge 0\)
and
where \(\beta =\underset{m\le x\le M}{\mathop {\max }}\,\left\{ {{a}_{f}}x+{{b}_{f}}-\alpha g\left( x \right) \right\} \).
Proof
From (2.2) and by applying functional calculus for the operator \(T=\left( 1-t \right) A+tB\), we have
Integrating both sides of the above inequality over \(t\in \left[ 0,1 \right] \), we have
Therefore,
Consequently,
which proves (2.12). To prove (2.13), notice that (2.2) implies, for \(0\le t\le 1,\)
From (2.14) and (2.15) we infer that
Therefore
Thus,
Integrating both sides of (2.16) over \(\left[ 0,1 \right] \) we get (2.13) and the proof is complete. \(\square \)
2.3 Operator Hermite–Hadamard Inequality Using the Gradient Inequality
In this subsection, we present versions of the operator Hermite–Hadamard inequality using the gradient inequality
where \(f:J\rightarrow \mathbb {R}\) is convex differentiable and \(s,t\in J.\)
Theorem 2.4
Let \(A,B\in \mathcal {B}(\mathcal {H})\) be self adjoint operators with spectra in the interval J and let \(f:J\rightarrow \mathbb {R}\) be a differentiable convex function. Then
where
Proof
Since f is convex differentiable, (2.17) applies. By applying functional calculus for the operator \(s=\frac{A+B}{2}\) we get
So, for any unit vector \(x\in \mathcal {H}\),
Again, by applying functional calculus for the operator \(t=\left( 1-v \right) A+vB\) we get
Integrating both sides over \(t\in \left[ 0,1 \right] \) implies
Whence, for any unit vector \(x\in \mathcal {H}\),
Thus,
where
Therefore,
which completes the proof. \(\square \)
Our last result in this direction is as follows.
Theorem 2.5
Let \(A,B\in \mathcal {B}(\mathcal {H})\) be self adjoint operators with spectra in the interval J and let \(f:J\rightarrow \mathbb {R}\) be a differentiable convex function. Then
where
Proof
By applying functional calculus for the operator \(T=\left( 1-v \right) A+vB\) in (2.17), we have
Hence for any unit vector \(x\in \mathcal {H}\),
Again, it follows from the functional calculus for \(t=A\) and \(t=B\), respectively
and
By combining (2.20) and (2.21) we obtain
This implies
for any unit vector \(x\in \mathcal {H}\). Integrating both sides over \(v\in \left[ 0,1 \right] \) we get
where
Consequently,
as desired. \(\square \)
Remark 2.3
Notice that in both Theorems 2.4 and 2.5, a quantity of the form
has been found as a refining term, for some self adjoint operator A. We show here that this quantity is always non-negative, when f is such a convex function.
Applying functional calculus for \(s=A\) in (2.17), we obtain
which implies
Now replacing t by \(\langle Ax,x\rangle \) and noting (1.5), we obtain
as desired.
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Communicated by Izchak Lewkowicz.
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Moradi, H.R., Sababheh, M. & Furuichi, S. On the Operator Hermite–Hadamard Inequality. Complex Anal. Oper. Theory 15, 122 (2021). https://doi.org/10.1007/s11785-021-01172-w
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DOI: https://doi.org/10.1007/s11785-021-01172-w