Ostrowski-Type Inequalities in Complex Hilbert Spaces for Square Modulus Convex Functions of Operators

Abstract

The present work is devoted to deriving Ostrowski-type inequalities for square modulus convex mappings $\mathcal{K}:[\alpha ,\beta ]\subset \mathbb{R}\to \mathfrak{B}\left(H\right)$ in complex Hilbert spaces. We show that the deviation of the weighted integral mean ${\int }_{\mathcal{X}}\rho {|\mathcal{K}\circ \phi |}^{2}d\mu$ from the pointwise value ${\left|\mathcal{K}\left(\xi \right)\right|}^2$ at any point $\xi \in [\alpha ,\beta ]$ is bounded in terms of the weighted standard deviation ${\sigma }_{\rho }\left(\phi \right)$ of $\phi$, giving the operator-valued analogue of the classical Ostrowski inequality. Taking $\xi ={\displaystyle \frac{\alpha +\beta }{2}}$ yields a midpoint inequality with an explicit constant. We also obtain a distribution-dependent estimate for the Jensen gap that vanishes whenever $\phi$ is constant. New two-point and multi-point Ostrowski inequalities and a superadditivity property of the Jensen gap are also derived.

1. Introduction

Hilbert spaces constitute one of the fundamental structures in modern functional analysis and operator theory. Hilbert spaces play a central role in several branches of mathematics and mathematical physics, including quantum mechanics, harmonic analysis, partial differential equations, approximation theory, and optimization.

The geometric structure of Hilbert spaces provides a suitable framework for studying convex functions and their associated inequalities. Inequalities involving convex functions have been widely studied through various extensions of convexity, such as operator convexity, operator m-convexity, and square modulus convexity. The study of Jensen-type operator inequalities was initiated by Mond and Pečarić [1]. Dragomir [2] used Grüss-type estimates to obtain bounds on the Jensen gap for differentiable convex functions of selfadjoint operators. Alomari [3] extended Popoviciu’s inequality to selfadjoint operators acting on Hilbert spaces, covering both superquadratic and convex functions. Dragomir [4] established several additive refinements of Jensen’s inequality for positive convex functions of selfadjoint operators in Hilbert spaces. Anjidani [5] derived Jensen-type inequalities in the multi-linear algebra setting; Li [6] and Ghazanfari [7] established Hermite–Hadamard-type inequalities for operator convex functions; Rooin et al. [8] investigated operator m-convex functions and obtained related Jensen-type inequalities; and Erdaş, Ünlüyol, and Salaş [9] derived sharp estimates for operator m-convex functions and applied them to synchronous and asynchronous mappings. A comprehensive treatment of Ostrowski and trapezoid inequalities for functions of selfadjoint operators is given in [10].

In addition, one of the most useful inequalities involving convex functions is the Ostrowski inequality [11]. A classical result due to Ostrowski [11] states that if $\mathfrak{g}:[\mathfrak{c},\mathfrak{d}]\to \mathbb{R}$ is differentiable with $|{\mathfrak{g}}^{\prime }\left(\varsigma \right)|\le L$ for all $\varsigma \in [\mathfrak{c},\mathfrak{d}]$, then

$$\left|\mathfrak{g}\left(\varsigma \right)-{\displaystyle \frac{1}{\mathfrak{d}-\mathfrak{c}}}{\int }_{\mathfrak{c}}^{\mathfrak{d}}\mathfrak{g}\left(\upsilon \right)d\upsilon \right|\le L\left[{\displaystyle \frac{\mathfrak{d}-\mathfrak{c}}{4}}+{\displaystyle \frac{{(\varsigma -\frac{\mathfrak{c}+\mathfrak{d}}{2})}^2}{\mathfrak{d}-\mathfrak{c}}}\right]$$

for all $\varsigma \in [\mathfrak{c},\mathfrak{d}]$, and the constant ${\displaystyle \frac{1}{4}}$ is the best possible. This inequality has been extended in many directions, including weighted inequalities, inequalities for convex mappings, and operator versions in Hilbert and Banach space settings [10].

Despite these developments, Ostrowski-type inequalities for square modulus convex operator-valued functions have not yet been investigated.

Following these developments, we consider the class of square modulus convex functions in complex Hilbert spaces. Throughout this paper, $\mathfrak{B}\left(H\right)$ denotes the ${\mathfrak{C}}^{\ast }$-algebra of all bounded linear operators on a complex Hilbert space $(H;⟨·,·⟩)$, and for $S\in \mathfrak{B}\left(H\right)$ the operator modulus is given by $\left|S\right|:={\left(S^{\ast }S\right)}^{1/2}$. We use $(\mathcal{X},\mu )$ for a measure space, $∥·∥$ for the operator norm, and $\mathrm{Re}\left(T\right):=(T+T^{\ast })/2$. The symbols $L_{\rho }^2(\mathcal{X},\mu ,\mathbb{R})$ and ${\mathfrak{L}}_{\rho }(\mathcal{X},\mu ,\mathfrak{B}\left(H\right))$ denote, respectively, the weighted square-integrable scalar functions and the weighted Bochner-integrable operator-valued functions, while ${\mathcal{K}}_{+}^{\prime }\left(\alpha \right)$ and ${\mathcal{K}}_{-}^{\prime }\left(\beta \right)$ denote the strong right and left endpoint derivatives, respectively. In a recent paper, Dragomir [12] introduced the class of square modulus convex functions $\mathcal{K}:[\alpha ,\beta ]\to \mathfrak{B}\left(H\right)$ and proved that for any $\phi :\mathcal{X}\to [\alpha ,\beta ]$ and $\rho \ge 0$ with ${\int }_{\mathcal{X}}\rho d\mu =1$,

$$\begin{array}{cc}\hfill 0& \le {\int }_{\mathcal{X}}\rho \left(\varsigma \right){\left|\mathcal{K}\circ \phi \right|}^{2}d\mu \left(\varsigma \right)-{\left|\mathcal{K}\left({\int }_{\mathcal{X}}\rho \phi d\mu \right)\right|}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}(\beta -\alpha )∥\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)-\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)∥,\hfill \end{array}$$

(1)

where $\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right):=\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)$ and $\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right):=$ $\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)$. We also set $M_{\mathcal{K}}:=\max \{∥\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)∥,$ $∥\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)∥\}$. Setting $\Gamma :=\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)-\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)$, the upper bound in (1) is independent of the weight $\rho$ and the distribution of $\phi$ inside $(\alpha ,\beta )$.

For our analysis, we make use of the weighted dispersion of $\phi$,

$${\sigma }_{\rho }\left(\phi \right):={\left({\int }_{\mathcal{X}}\rho \left(\varsigma \right){(\phi \left(\varsigma \right)-\overline{\phi })}^{2}d\mu \left(\varsigma \right)\right)}^{1/2},\overline{\phi }:={\int }_{\mathcal{X}}\rho \phi d\mu ,$$

(2)

which satisfies $0\le {\sigma }_{\rho }\left(\phi \right)\le {\displaystyle \frac{\beta -\alpha }{2}}$ and vanishes if and only if $\phi$ is $\mu$-a.e. constant.

Although Dragomir [12] established Jensen-type inequalities for square modulus convex operator-valued functions, the corresponding Ostrowski-type inequalities have not yet been investigated for this class of functions.

Motivated by the work of Dragomir [12], we establish new Ostrowski-type inequalities for square modulus convex operator-valued functions by employing the weighted dispersion ${\sigma }_{\rho }\left(\phi \right)$. In particular, we prove an Ostrowski-type inequality for every evaluation point $\xi \in [\alpha ,\beta ]$:

$$∥{\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\xi \right)\right|}^2∥\le 2M_{\mathcal{K}}{\left[{(\overline{\phi }-\xi )}^2+{\sigma }_{\rho }^2\left(\phi \right)\right]}^{1/2},$$

(3)

together with a distribution-dependent estimate for the Jensen gap obtained by applying the Löwner–Heinz inequality [13] and a midpoint inequality with an explicit constant. New two-point Ostrowski and superadditivity results are also obtained. The results presented in this paper extend some earlier works in the literature on operator inequalities in Hilbert spaces.

2. Preliminaries

This section recalls the main definitions and auxiliary results that will be used throughout this paper.

Definition 1

([12]). A continuous mapping $\mathcal{K}:[{\mu }_0,{\nu }_0]\to \mathfrak{B}\left(H\right)$ is said to be square modulus convex on $[{\mu }_0,{\nu }_0]$ if the operator inequality

$${\left|\mathcal{K}\left(\right(1-\lambda )\zeta +\lambda \theta )\right|}^2\le {(1-\lambda )\left|\mathcal{K}\left(\zeta \right)\right|}^2+\lambda {\left|\mathcal{K}\left(\theta \right)\right|}^2$$

(4)

holds with respect to the partial order of $\mathfrak{B}\left(H\right)$ for every $\zeta ,\theta \in [{\mu }_0,{\nu }_0]$ and $\lambda \in [0,1]$.

As shown in Section 2 of [12], $\mathcal{K}$ is square modulus convex on $[\alpha ,\beta ]$ if and only if for every $\delta \in H\setminus \left\{0\right\}$ the scalar function $\zeta \mapsto {∥\mathcal{K}\left(\zeta \right)\delta ∥}^2$ is convex on $[\alpha ,\beta ]$.

Example 1

([12]). The following are square modulus convex:

• ${\mathcal{K}}_1\left(\xi \right):=\left|(1-\xi )S+\xi R\right|$ for distinct $S,R\in \mathfrak{B}\left(H\right)$, $\xi \in [0,1]$;
• ${\mathcal{K}}_r\left(\xi \right):={((1-\xi )S+\xi R)}^{r/2}$ for $S,R>0$ and $1\le r\le 2$ or $-1\le r\le 0$;
• $\mathcal{K}\left(\xi \right):=e^{\xi S/2}$ for selfadjoint $S\in \mathfrak{B}\left(H\right)$ with spectrum in $[\alpha ,\beta ]$.

Suppose $\mathcal{K}:[\alpha ,\beta ]\to \mathfrak{B}\left(H\right)$ is square modulus convex and strongly differentiable. For $\xi ,\eta \in (\alpha ,\beta )$, the following gradient inequality holds as given in Equation (4) of [12]:

$$2(\xi -\eta )\mathrm{Re}\left({\left(\mathcal{K}\left(\xi \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\xi \right)\right)\ge {\left|\mathcal{K}\left(\xi \right)\right|}^2-{\left|\mathcal{K}\left(\eta \right)\right|}^2\ge 2(\xi -\eta )\mathrm{Re}\left({\left(\mathcal{K}\left(\eta \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\eta \right)\right).$$

(5)

At the endpoints as given in Equations (5) and (6) of [12]:

$$\begin{array}{cc}\hfill {\left|\mathcal{K}\left(\xi \right)\right|}^2-{\left|\mathcal{K}\left(\alpha \right)\right|}^2& \ge 2(\xi -\alpha )\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right),\hfill \\ \hfill 2(\beta -\eta )\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)& \ge {\left|\mathcal{K}\left(\beta \right)\right|}^2-{\left|\mathcal{K}\left(\eta \right)\right|}^2.\hfill \end{array}$$

(6)

The gradient satisfies the monotonicity relation as given in Equation (7) of [12]:

$$\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)\le \mathrm{Re}\left({\left(\mathcal{K}\left(\xi \right)\right)}^{\ast }{\mathcal{K}}_{\pm }^{\prime }\left(\xi \right)\right)\le \mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right),\xi \in [\alpha ,\beta ].$$

(7)

We also make use of the following Cauchy–Bunyakowsky–Schwarz inequality for operator-valued integrals, which was obtained in [12].

Lemma 1.

If $\psi \in L_{\rho }^2(\mathcal{X},\mu ,\mathbb{R})$ and $S\in L_{\rho }^2(\mathcal{X},\mu ,\mathfrak{B}\left(H\right))$ is selfadjoint μ-a.e., then

$$∥{\int }_{\mathcal{X}}\rho \psi Sd\mu ∥\le {\left({\int }_{\mathcal{X}}\rho {\psi }^{2}d\mu \right)}^{1/2}∥{\left({\int }_{\mathcal{X}}\rho S^{2}d\mu \right)}^{1/2}∥.$$

(8)

Lemma 2

(Löwner–Heinz [13]). If $0\le P\le Q$ in $\mathfrak{B}\left(H\right)$, then $∥P^{1/2}∥\le ∥Q^{1/2}∥$.

Throughout, we write $\overline{\phi }:={\int }_{\mathcal{X}}\rho \phi d\mu$ and use ${\sigma }_{\rho }\left(\phi \right)$ as in (2), where ${\sigma }_{\rho }\left(\phi \right)\le {\displaystyle \frac{\beta -\alpha }{2}}$.

3. Main Results

The following result estimates the deviation of the weighted integral mean ${\int }_{\mathcal{X}}\rho {\left|\mathcal{K}\circ \phi \right|}^{2}d\mu$ from the pointwise value ${\left|\mathcal{K}\left(\xi \right)\right|}^2$ at any point $\xi \in [\alpha ,\beta ]$, extending the pointwise approach of [14] to the weighted integral mean setting.

Theorem 1.

Let $\mathcal{K}:[\alpha ,\beta ]\to \mathfrak{B}\left(H\right)$ be square modulus convex and strongly differentiable. Let $\rho :\mathcal{X}\to [0,\infty )$ be measurable with ${\int }_{\mathcal{X}}\rho d\mu =1$, and let $\phi :\mathcal{X}\to [\alpha ,\beta ]$ be measurable with $\phi \in L_{\rho }^2(\mathcal{X},\mu ,\mathbb{R})$ and ${\left|\mathcal{K}\circ \phi \right|}^2\in {\mathfrak{L}}_{\rho }(\mathcal{X},\mu ,\mathfrak{B}\left(H\right))$. Set $\overline{\phi }:={\int }_{\mathcal{X}}\rho \phi d\mu$. Then, for every $\xi \in [\alpha ,\beta ]$,

$$\begin{array}{cc}\hfill & ∥{\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\xi \right)\right|}^2∥\hfill \\ \hfill & \le 2M_{\mathcal{K}}{\left[{(\overline{\phi }-\xi )}^2+{\sigma }_{\rho }^2\left(\phi \right)\right]}^{1/2}\hfill \\ \hfill & \le 2M_{\mathcal{K}}{\left[{(\overline{\phi }-\xi )}^2+{\displaystyle \frac{{(\beta -\alpha )}^2}{4}}\right]}^{1/2}.\hfill \end{array}$$

(9)

Proof. 

Set $F\left(t\right):={\left|\mathcal{K}\left(t\right)\right|}^2$. If $\alpha <u<v<\beta$, applying (5) with $\xi =v$ and $\eta =u$, and then using (7), gives

$$\begin{array}{c}\hfill 2(v-u)\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)\le F\left(v\right)-F\left(u\right)\le 2(v-u)\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right).\end{array}$$

The cases $u=\alpha$ or $v=\beta$ follow by continuity, using (6) and passing to the limit in (5). Hence, for every $\alpha \le u<v\le \beta$,

$$\begin{array}{c}\hfill 2(v-u)\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)\le F\left(v\right)-F\left(u\right)\le 2(v-u)\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right).\end{array}$$

(10)

Since $F\left(v\right)-F\left(u\right)$ is selfadjoint, (10) and $∥T∥={\sup }_{∥x∥=1}\left|⟨Tx,x⟩\right|$ for selfadjoint $T\in \mathfrak{B}\left(H\right)$ yield

$$\begin{array}{c}\hfill ∥F\left(v\right)-F\left(u\right)∥\le 2M_{\mathcal{K}}(v-u).\end{array}$$

Consequently, for $\mu$-a.e. $\varsigma \in \mathcal{X}$,

$$\begin{array}{c}\hfill ∥{\left|\mathcal{K}\left(\phi \right(\varsigma \left)\right)\right|}^2-{\left|\mathcal{K}\left(\xi \right)\right|}^2∥\le 2M_{\mathcal{K}}\left|\phi \left(\varsigma \right)-\xi \right|.\end{array}$$

(11)

Multiplying (11) by $\rho \left(\varsigma \right)\ge 0$ and integrating over $\mathcal{X}$,

$$\begin{array}{c}\hfill ∥{\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\xi \right)\right|}^2∥\le 2M_{\mathcal{K}}{\int }_{\mathcal{X}}\rho \left(\varsigma \right)\left|\phi \left(\varsigma \right)-\xi \right|d\mu \left(\varsigma \right).\end{array}$$

(12)

Since

$$\begin{array}{c}\hfill {\int }_{\mathcal{X}}\rho \left(\varsigma \right)\left|\phi \left(\varsigma \right)-\xi \right|d\mu \left(\varsigma \right)\le {\left({\int }_{\mathcal{X}}\rho \left(\varsigma \right){(\phi \left(\varsigma \right)-\xi )}^{2}d\mu \left(\varsigma \right)\right)}^{1/2}\end{array}$$

by the scalar CBS inequality, and since

$$\begin{array}{c}\hfill {\int }_{\mathcal{X}}\rho \left(\varsigma \right){(\phi \left(\varsigma \right)-\xi )}^{2}d\mu \left(\varsigma \right)={(\overline{\phi }-\xi )}^2+{\sigma }_{\rho }^2\left(\phi \right),\end{array}$$

substituting into (12) gives the first bound in (9).

The second bound follows from ${\sigma }_{\rho }^2\left(\phi \right)\le {\displaystyle \frac{{(\beta -\alpha )}^2}{4}}$, giving

$$\begin{array}{c}\hfill {\left[{(\overline{\phi }-\xi )}^2+{\sigma }_{\rho }^2\left(\phi \right)\right]}^{1/2}\le {\left[{(\overline{\phi }-\xi )}^2+{\displaystyle \frac{{(\beta -\alpha )}^2}{4}}\right]}^{1/2},\end{array}$$

completing the proof. □

Remark 1.

Theorem 1 provides the operator-valued analogue of the classical Ostrowski inequality [11] in the setting of square modulus convex functions, extending the pointwise approach of [14] to the weighted integral mean setting.

Corollary 1.

As a particular case of Theorem 1, taking $\xi ={\displaystyle \frac{\alpha +\beta }{2}}$ in (9),

$$\begin{array}{cc}\hfill & ∥{\int }_{\mathcal{X}}\rho {\left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\frac{\alpha +\beta }{2}\right)\right|}^2∥\hfill \\ \hfill & \le 2M_{\mathcal{K}}{\left[{\left(\overline{\phi }-\frac{\alpha +\beta }{2}\right)}^2+{\sigma }_{\rho }^2\left(\phi \right)\right]}^{1/2}\hfill \\ \hfill & \le (\beta -\alpha )M_{\mathcal{K}}.\hfill \end{array}$$

(13)

The constant $(\beta -\alpha )M_{\mathcal{K}}$ is explicit; it arises from the bound ${(\overline{\phi }-\frac{\alpha +\beta }{2})}^2+{\sigma }_{\rho }^2\left(\phi \right)\le {\left(\frac{\beta -\alpha }{2}\right)}^2$, where equality holds whenever $\phi$ takes values in $\{\alpha ,\beta \}$ almost everywhere with respect to $\rho d\mu$, with arbitrary endpoint weights.

As a consequence, taking $\xi =\overline{\phi }$ in Theorem 1 and applying the Löwner–Heinz inequality [13], we obtain the following distribution-dependent alternative to inequality (1) of [12].

Theorem 2.

Let $\mathcal{K}:[\alpha ,\beta ]\to \mathfrak{B}\left(H\right)$ be square modulus convex and strongly differentiable, and let $\phi :\mathcal{X}\to [\alpha ,\beta ]$ with $\phi \in L_{\rho }^2(\mathcal{X},\mu ,\mathbb{R})$ and ${\left|\mathcal{K}\circ \phi \right|}^2,\mathrm{Re}\left({(\mathcal{K}\circ \phi )}^{\ast }({\mathcal{K}}^{\prime }\circ \phi )\right),\phi$$\mathrm{Re}\left({(\mathcal{K}\circ \phi )}^{\ast }({\mathcal{K}}^{\prime }\circ \phi )\right)\in {\mathfrak{L}}_{\rho }(\mathcal{X},\mu ,\mathfrak{B}\left(H\right))$, where $\rho \ge 0$ with ${\int }_{\mathcal{X}}\rho d\mu =1$. Define $\Gamma :=\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)-\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)$, assume that $\Gamma \ge 0$, and $\overline{\phi }:={\int }_{\mathcal{X}}\rho \phi d\mu$. Then

$$\begin{array}{cc}\hfill 0& \le {\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\overline{\phi }\right)\right|}^2\hfill \\ \hfill & \le 2{\sigma }_{\rho }\left(\phi \right)∥\Gamma ∥\hfill \\ \hfill & \le (\beta -\alpha )∥\Gamma ∥,\hfill \end{array}$$

(14)

with respect to the partial order of $\mathfrak{B}\left(H\right)$.

Proof. 

Applying (5) with $\eta =\overline{\phi }$, multiplying by $\rho \left(\varsigma \right)\ge 0$, and integrating, we obtain

$$\begin{array}{cc}\hfill & 2{\int }_{\mathcal{X}}\rho \left(\varsigma \right)(\phi \left(\varsigma \right)-\overline{\phi })\mathrm{Re}\left({\left(\mathcal{K}\left(\phi \left(\varsigma \right)\right)\right)}^{\ast }{\mathcal{K}}^{\prime }\left(\phi \left(\varsigma \right)\right)\right)d\mu \left(\varsigma \right)\hfill \\ \hfill & \ge {\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\overline{\phi }\right)\right|}^2\ge 0,\hfill \end{array}$$

(15)

where the lower bound follows by applying (5) with $\xi =\phi \left(\varsigma \right)$ and $\eta =\overline{\phi }$, multiplying by $\rho \left(\varsigma \right)$, and integrating over $\mathcal{X}$. Cases involving $\alpha$ or $\beta$ follow from (6) and continuity.

Since

$$\begin{array}{c}\hfill {\int }_{\mathcal{X}}\rho (\phi -\overline{\phi })\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)d\mu =0,\end{array}$$

the left-hand side of (15) reduces to

$$\begin{array}{c}\hfill 2{\int }_{\mathcal{X}}\rho (\phi -\overline{\phi })(\mathrm{Re}\left({(\mathcal{K}\circ \phi )}^{\ast }({\mathcal{K}}^{\prime }\circ \phi )\right)-\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right))d\mu .\end{array}$$

Set $Q\left(\varsigma \right):=\mathrm{Re}\left({(\mathcal{K}\circ \phi )}^{\ast }({\mathcal{K}}^{\prime }\circ \phi )\right)-\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)$. By (7),

$$\begin{array}{c}\hfill 0\le Q\left(\varsigma \right)\le \Gamma \mu \text{-}\mathrm{a}.\mathrm{e}.\end{array}$$

Thus, Q is essentially bounded and belongs to $L_{\rho }^2(\mathcal{X},\mu ,\mathfrak{B}\left(H\right))$. Applying (8) with $\psi \left(\varsigma \right)=\phi \left(\varsigma \right)-\overline{\phi }$ and $S\left(\varsigma \right)=Q\left(\varsigma \right)$ yields

$$\begin{array}{cc}\hfill & ∥{\int }_{\mathcal{X}}\rho (\phi -\overline{\phi })Qd\mu ∥\hfill \\ & \le {\sigma }_{\rho }\left(\phi \right)∥{\left({\int }_{\mathcal{X}}\rho Q^{2}d\mu \right)}^{1/2}∥.\hfill \end{array}$$

Since $Q\left(\varsigma \right)\ge 0$, functional calculus gives

$$\begin{array}{c}\hfill Q{\left(\varsigma \right)}^2\le ∥Q\left(\varsigma \right)∥Q\left(\varsigma \right)\le {∥\Gamma ∥}^{2}I.\end{array}$$

Consequently,

$$\begin{array}{c}\hfill {\int }_{\mathcal{X}}\rho Q^{2}d\mu \le {∥\Gamma ∥}^{2}I.\end{array}$$

Applying Lemma 2 therefore gives

$$\begin{array}{c}\hfill ∥{\left({\int }_{\mathcal{X}}\rho Q^{2}d\mu \right)}^{1/2}∥\le ∥\Gamma ∥.\end{array}$$

Substituting into (15), we obtain

$$\begin{array}{c}\hfill {\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\overline{\phi }\right)\right|}^2\le 2{\sigma }_{\rho }\left(\phi \right)∥\Gamma ∥.\end{array}$$

(16)

The scalar bound ${\sigma }_{\rho }\left(\phi \right)\le {\displaystyle \frac{\beta -\alpha }{2}}$ applied to (16) gives $(\beta -\alpha )∥\Gamma ∥$, which completes the proof. □

Remark 2.

When ${\sigma }_{\rho }\left(\phi \right)\le {\displaystyle \frac{\beta -\alpha }{4}}$, the distribution-dependent bound in Theorem 2 is at least as sharp as the uniform bound given in inequality (1) [12].

The following corollary gives the discrete versions of Theorems 1 and 2.

Corollary 2.

Let $\mathcal{K}:[\alpha ,\beta ]\to \mathfrak{B}\left(H\right)$ be square modulus convex and strongly differentiable, let ${\xi }_i\in [\alpha ,\beta ]$, ${\rho }_i\ge 0$ with ${\sum }_{i=1}^n{\rho }_i=1$, and set $\overline{\xi }:={\sum }_{i=1}^n{\rho }_i{\xi }_i$. Then

$$\begin{array}{cc}\hfill 0& \le {\displaystyle \sum _{i=1}^n}{\rho }_i|\mathcal{K}\left({\xi }_i\right){|}^2-{\left|\mathcal{K}\left(\overline{\xi }\right)\right|}^2\hfill \\ \hfill & \le 2{\left({\displaystyle \sum _{i=1}^n}{\rho }_i{({\xi }_i-\overline{\xi })}^2\right)}^{1/2}∥\Gamma ∥\hfill \\ \hfill & \le (\beta -\alpha )∥\Gamma ∥.\hfill \end{array}$$

(17)

In this discrete setting, the variance-dependent bound above is at least as sharp as the corresponding uniform bound whenever${\left({\sum }_{i=1}^n{\rho }_i{({\xi }_i-\overline{\xi })}^2\right)}^{1/2}\le {\displaystyle \frac{\beta -\alpha }{4}}$.

Moreover, for every $\xi \in [\alpha ,\beta ]$,

$$\begin{array}{c}\hfill ∥{\displaystyle \sum _{i=1}^n}{\rho }_i|\mathcal{K}\left({\xi }_i\right){|}^2-{\left|\mathcal{K}\left(\xi \right)\right|}^2∥\le 2M_{\mathcal{K}}{\left[{(\overline{\xi }-\xi )}^2+{\displaystyle \sum _{i=1}^n}{\rho }_i{({\xi }_i-\overline{\xi })}^2\right]}^{1/2}.\end{array}$$

(18)

The following result generalizes Theorem 1 to two evaluation points ${\xi }_1,{\xi }_2\in [\alpha ,\beta ]$.

Theorem 3.

Let $\mathcal{K}:[\alpha ,\beta ]\to \mathfrak{B}\left(H\right)$ be square modulus convex and strongly differentiable, and let $\phi :\mathcal{X}\to [\alpha ,\beta ]$ with ${\phi ,\left|\mathcal{K}\circ \phi \right|}^2,\mathrm{Re}\left({(\mathcal{K}\circ \phi )}^{\ast }({\mathcal{K}}^{\prime }\circ \phi )\right),\phi \mathrm{Re}\left({(\mathcal{K}\circ \phi )}^{\ast }({\mathcal{K}}^{\prime }\circ \phi )\right)\in {\mathfrak{L}}_{\rho }(\mathcal{X},\mu ,\mathfrak{B}\left(H\right))$, where $\rho \ge 0$ with ${\int }_{\mathcal{X}}\rho d\mu =1$. Define $\Gamma :=\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)-\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)$, assume that $\Gamma \ge 0$, and $\overline{\phi }:={\int }_{\mathcal{X}}\rho \phi d\mu$. For every ${\xi }_1,{\xi }_2\in [\alpha ,\beta ]$, with ${\xi }_1\le {\xi }_2$, $\lambda \in [0,1]$, and ${\xi }_{\lambda }:=(1-\lambda ){\xi }_1+\lambda {\xi }_2$,

$$\begin{array}{cc}\hfill & ∥{\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -(1-\lambda )|\mathcal{K}\left({\xi }_1\right){|}^2-\lambda {\left|\mathcal{K}\left({\xi }_2\right)\right|}^2∥\hfill \\ \hfill & \le 2M_{\mathcal{K}}{\left[{(\overline{\phi }-{\xi }_{\lambda })}^2+{\sigma }_{\rho }^2\left(\phi \right)\right]}^{1/2}+2\lambda (1-\lambda )({\xi }_2-{\xi }_1)∥\Gamma ∥.\hfill \end{array}$$

(19)

Proof. 

By the triangle inequality,

$$\begin{array}{cc}\hfill & ∥{\int }_{\mathcal{X}}{\rho |\mathcal{K}\circ \phi |}^{2}d\mu -(1-\lambda )|\mathcal{K}\left({\xi }_1\right){|}^2-\lambda {|\mathcal{K}\left({\xi }_2\right)|}^2∥\le I_1+I_2,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill I_1:=∥{\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left({\xi }_{\lambda }\right)\right|}^2∥,I_2:=∥|\mathcal{K}\left({\xi }_{\lambda }\right){|}^2-(1-\lambda )|\mathcal{K}\left({\xi }_1\right){|}^2-\lambda {|\mathcal{K}\left({\xi }_2\right)|}^2∥.\end{array}$$

Theorem 1 applied at $\xi ={\xi }_{\lambda }$ gives

$$\begin{array}{c}\hfill I_1\le 2M_{\mathcal{K}}{\left[{(\overline{\phi }-{\xi }_{\lambda })}^2+{\sigma }_{\rho }^2\left(\phi \right)\right]}^{1/2}.\end{array}$$

For $I_2$, by the square modulus convexity (4),

$$\begin{array}{c}\hfill (1-\lambda )|\mathcal{K}\left({\xi }_1\right){|}^2+\lambda |\mathcal{K}\left({\xi }_2\right){|}^2-{|\mathcal{K}\left({\xi }_{\lambda }\right)|}^2\ge 0,\end{array}$$

so $I_2=∥(1-\lambda )|\mathcal{K}\left({\xi }_1\right){|}^2+\lambda |\mathcal{K}\left({\xi }_2\right){|}^2-{|\mathcal{K}\left({\xi }_{\lambda }\right)|}^2∥$. Applying (5) at $\xi ={\xi }_2$ and $\eta ={\xi }_{\lambda }$, and using

$$\mathrm{Re}\left({\left(\mathcal{K}\left({\xi }_2\right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left({\xi }_2\right)\right)\le \mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)$$

from (7), as well as ${\xi }_2-{\xi }_{\lambda }=(1-\lambda )({\xi }_2-{\xi }_1)\ge 0$, we obtain

$$|\mathcal{K}\left({\xi }_2\right){|}^2-{|\mathcal{K}\left({\xi }_{\lambda }\right)|}^2\le 2(1-\lambda )({\xi }_2-{\xi }_1)\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right).$$

Similarly, applying (5) at $\xi ={\xi }_{\lambda }$ and $\eta ={\xi }_1$, and using

$$\mathrm{Re}\left({\left(\mathcal{K}\left({\xi }_1\right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left({\xi }_1\right)\right)\ge \mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)$$

from (7), gives

$$\begin{array}{cc}\hfill |\mathcal{K}\left({\xi }_1\right){|}^2-{|\mathcal{K}\left({\xi }_{\lambda }\right)|}^2& \le -2({\xi }_{\lambda }-{\xi }_1)\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)\hfill \\ \hfill & =-2\lambda ({\xi }_2-{\xi }_1)\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right).\hfill \end{array}$$

Multiplying the estimate for $|\mathcal{K}\left({\xi }_2\right){|}^2-{|\mathcal{K}\left({\xi }_{\lambda }\right)|}^2$ by $\lambda$ and the estimate for $|\mathcal{K}\left({\xi }_1\right){|}^2-{|\mathcal{K}\left({\xi }_{\lambda }\right)|}^2$ by $1-\lambda$, and then adding, we obtain

$$\begin{array}{cc}\hfill & (1-\lambda )|\mathcal{K}\left({\xi }_1\right){|}^2+\lambda |\mathcal{K}\left({\xi }_2\right){|}^2-{|\mathcal{K}\left({\xi }_{\lambda }\right)|}^2\hfill \\ \hfill & \le 2\lambda (1-\lambda )({\xi }_2-{\xi }_1)(\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)-\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right))\hfill \\ \hfill & =2\lambda (1-\lambda )({\xi }_2-{\xi }_1)\Gamma .\hfill \end{array}$$

Hence, $I_2\le 2\lambda (1-\lambda )({\xi }_2-{\xi }_1)∥\Gamma ∥$. Combining with the bound on $I_1$ gives (19). □

Remark 3.

Taking $\lambda =0$ or $\lambda =1$ in Theorem 3 recovers Theorem 1 at $\xi ={\xi }_1$ or $\xi ={\xi }_2$, respectively.

The next result shows that the Jensen gap is superadditive with respect to partitions of the domain $\mathcal{X}$, providing a lower bound that complements the upper bounds of Theorems 1 and 2.

Remark 4.

Let $\mathcal{K}$, φ, ρ be as in Theorem 2, and let $\mathcal{X}={\mathcal{X}}_1\cup {\mathcal{X}}_2$ be a disjoint decomposition with $p_k:={\int }_{{\mathcal{X}}_k}\rho d\mu >0$ and ${\rho }_k:=\rho /p_k$ for $k=1,2$. Setting ${\overline{\phi }}_k:={\int }_{{\mathcal{X}}_k}{\rho }_k\phi d\mu$, the Jensen gap satisfies

$$\begin{array}{c}\hfill {\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\overline{\phi }\right)\right|}^2\ge {\displaystyle \sum _{k=1}^2}{p}_k\left({\int }_{{\mathcal{X}}_k}{\rho }_k{\left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left({\overline{\phi }}_k\right)\right|}^2\right).\end{array}$$

(20)

Indeed, since $\overline{\phi }=p_1{\overline{\phi }}_1+p_2{\overline{\phi }}_2$, the square modulus convexity (4) gives $|\mathcal{K}\left(\overline{\phi }\right){|}^2\le p_1|\mathcal{K}\left({\overline{\phi }}_1\right){|}^2+p_2{|\mathcal{K}\left({\overline{\phi }}_2\right)|}^2$, and (20) follows by subtracting from the decomposed integral. Combined with Theorem 2, this gives the sandwich

$$\begin{array}{c}\hfill {\displaystyle \sum _{k=1}^2}{p}_k\left({\int }_{{\mathcal{X}}_k}{\rho }_k{\left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left({\overline{\phi }}_k\right)\right|}^2\right)\le {\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left(\overline{\phi }\right)\right|}^2\le 2{\sigma }_{\rho }\left(\phi \right)∥\Gamma ∥.\end{array}$$

The following result extends Theorem 3 to n evaluation points ${\xi }_1,\dots ,{\xi }_n\in [\alpha ,\beta ]$.

Theorem 4.

Let $\mathcal{K}:[\alpha ,\beta ]\to \mathfrak{B}\left(H\right)$ be square modulus convex and strongly differentiable, and let $\phi :\mathcal{X}\to [\alpha ,\beta ]$ with ${\phi ,\left|\mathcal{K}\circ \phi \right|}^2,\mathrm{Re}\left({(\mathcal{K}\circ \phi )}^{\ast }({\mathcal{K}}^{\prime }\circ \phi )\right),\phi \mathrm{Re}\left({(\mathcal{K}\circ \phi )}^{\ast }({\mathcal{K}}^{\prime }\circ \phi )\right)\in {\mathfrak{L}}_{\rho }(\mathcal{X},\mu ,\mathfrak{B}\left(H\right))$, where $\rho \ge 0$ with ${\int }_{\mathcal{X}}\rho d\mu =1$. Define $\Gamma :=\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)-\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)$, assume that $\Gamma \ge 0$, and $\overline{\phi }:={\int }_{\mathcal{X}}\rho \phi d\mu$. For ${\xi }_1,\dots ,{\xi }_n\in [\alpha ,\beta ]$ and ${\lambda }_1,\dots ,{\lambda }_n\ge 0$ with ${\sum }_{k=1}^n{\lambda }_k=1$, set ${\xi }_{\lambda }:={\sum }_{k=1}^n{\lambda }_k{\xi }_k$. Then

$$\begin{array}{cc}\hfill & ∥{\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\displaystyle \sum _{k=1}^n}{\lambda }_k{\left|\mathcal{K}\left({\xi }_k\right)\right|}^2∥\hfill \\ \hfill & \le 2M_{\mathcal{K}}{\left[{(\overline{\phi }-{\xi }_{\lambda })}^2+{\sigma }_{\rho }^2\left(\phi \right)\right]}^{1/2}+\begin{array}{c}\hfill \left({\displaystyle \sum _{k=1}^n}{\lambda }_k\left|{\xi }_k-{\xi }_{\lambda }\right|\right)∥\Gamma ∥\end{array}.\hfill \end{array}$$

(21)

Proof. 

By the triangle inequality,

$$\begin{array}{cc}\hfill & ∥{\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\displaystyle \sum _{k=1}^n}{\lambda }_k{\left|\mathcal{K}\left({\xi }_k\right)\right|}^2∥\le I_1+I_2,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill I_1:=∥{\int }_{\mathcal{X}}{\rho \left|\mathcal{K}\circ \phi \right|}^{2}d\mu -{\left|\mathcal{K}\left({\xi }_{\lambda }\right)\right|}^2∥,I_2:=∥{\displaystyle \sum _{k=1}^n}{\lambda }_k|\mathcal{K}\left({\xi }_k\right){|}^2-{|\mathcal{K}\left({\xi }_{\lambda }\right)|}^2∥.\end{array}$$

Theorem 1 applied at $\xi ={\xi }_{\lambda }$ gives

$$\begin{array}{c}\hfill I_1\le 2M_{\mathcal{K}}{\left[{(\overline{\phi }-{\xi }_{\lambda })}^2+{\sigma }_{\rho }^2\left(\phi \right)\right]}^{1/2}.\end{array}$$

For $I_2$, set $F\left(t\right):={\left|\mathcal{K}\left(t\right)\right|}^2$, $d_k:={\xi }_k-{\xi }_{\lambda }$, and ${\left(d_k\right)}_{+}:=\max \{d_k,0\}$. Square modulus convexity gives

$$\begin{array}{c}\hfill J:={\displaystyle \sum _{k=1}^n}{\lambda }_{k}F\left({\xi }_k\right)-F\left({\xi }_{\lambda }\right)\ge 0,I_2=∥J∥.\end{array}$$

By (5) and (7),

$$\begin{array}{cc}\hfill F\left({\xi }_k\right)-F\left({\xi }_{\lambda }\right)& \le \left\{\begin{array}{cc}2d_k\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right),\hfill & d_k\ge 0,\hfill \\ 2d_k\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right),\hfill & d_k<0,\hfill \end{array}\right.\hfill \\ \hfill & =2d_k\mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)+2{\left(d_k\right)}_{+}\Gamma ,\hfill \end{array}$$

where the endpoint cases follow from (6) and continuity. Since ${\sum }_{k=1}^n{\lambda }_k{d}_k=0$, summing the preceding inequality yields

$$\begin{array}{cc}\hfill 0\le J& \le 2\left({\displaystyle \sum _{k=1}^n}{\lambda }_k{\left(d_k\right)}_{+}\right)\Gamma \hfill \\ \hfill & =\left({\displaystyle \sum _{k=1}^n}{\lambda }_k\left|d_k\right|\right)\Gamma .\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill I_2\le \left({\displaystyle \sum _{k=1}^n}{\lambda }_k\left|{\xi }_k-{\xi }_{\lambda }\right|\right)∥\Gamma ∥.\end{array}$$

Combining with the bound on $I_1$ gives (21). □

Remark 5.

Taking $n=1$ in Theorem 4 recovers Theorem 1.

4. Examples

This section presents illustrative examples that demonstrate the applicability of the main results developed in this paper.

The following example gives the application of Theorems 2 and 1 to an exponential operator function.

Example 2. 

Let $S\in \mathfrak{B}\left(H\right)$ be selfadjoint with $\sigma \left(S\right)\subset [\alpha ,\beta ]$ and $\mathcal{K}\left(\xi \right):=e^{\xi S/2}$ for $\xi \in [\alpha ,\beta ]$, so ${\left|\mathcal{K}\left(\xi \right)\right|}^2=e^{\xi S}$ and $\mathrm{Re}\left({\left(\mathcal{K}\left(\xi \right)\right)}^{\ast }{\mathcal{K}}^{\prime }\left(\xi \right)\right)={\displaystyle \frac{1}{2}}S{e}^{\xi S}$. Then

$$\begin{array}{c}\hfill \mathrm{Re}\left({\left(\mathcal{K}\left(\alpha \right)\right)}^{\ast }{\mathcal{K}}_{+}^{\prime }\left(\alpha \right)\right)={\displaystyle \frac{1}{2}}S{e}^{\alpha S},\mathrm{Re}\left({\left(\mathcal{K}\left(\beta \right)\right)}^{\ast }{\mathcal{K}}_{-}^{\prime }\left(\beta \right)\right)={\displaystyle \frac{1}{2}}S{e}^{\beta S},∥\Gamma ∥={\displaystyle \frac{1}{2}}∥S\left(e^{\beta S}-e^{\alpha S}\right)∥.\end{array}$$

Moreover, $M_{\mathcal{K}}={\displaystyle \frac{1}{2}}\max \left\{∥S{e}^{\alpha S}∥,∥S{e}^{\beta S}∥\right\}$. Theorem 2 gives

$$\begin{array}{c}\hfill 0\le ∥{\int }_{\mathcal{X}}\rho e^{\phi \left(\varsigma \right)S}d\mu \left(\varsigma \right)-e^{\overline{\phi }S}∥\le {\sigma }_{\rho }\left(\phi \right)∥S\left(e^{\beta S}-e^{\alpha S}\right)∥\le {\displaystyle \frac{\beta -\alpha }{2}}∥S\left(e^{\beta S}-e^{\alpha S}\right)∥.\end{array}$$

For the Ostrowski bound (Theorem 1) at $\xi =\overline{\phi }$,

$$\begin{array}{c}\hfill ∥{\int }_{\mathcal{X}}\rho e^{\phi \left(\varsigma \right)S}d\mu \left(\varsigma \right)-e^{\overline{\phi }S}∥\le 2M_{\mathcal{K}}{\sigma }_{\rho }\left(\phi \right).\end{array}$$

For the following example, we apply the main results to the square modulus convex function generated by two bounded linear operators.

Example 3.

For distinct $S,R\in \mathfrak{B}\left(H\right)$, define $\mathcal{K}\left(\xi \right):=(1-\xi )S+\xi R$ for $\xi \in [0,1]$. Then ${\left|\mathcal{K}\left(\xi \right)\right|}^2={\left|(1-\xi )S+\xi R\right|}^2$ is square modulus convex, and a direct calculation gives $\Gamma ={\left|S-R\right|}^2$, so $∥\Gamma ∥={∥S-R∥}^2$. Equation (27) of [12] states the bound as ${∥S-R∥}^2$, while the correct value from Corollary 3 of [12] is ${\displaystyle \frac{1}{2}}{∥S-R∥}^2$. Theorem 2 gives the following chain

$$\begin{array}{c}\hfill 0\le {\int }_{\mathcal{X}}{\rho \left|(1-\phi )S+\phi R\right|}^{2}d\mu -{\left|(1-\overline{\phi })S+\overline{\phi }R\right|}^2\le 2{\sigma }_{\rho }\left(\phi \right){∥S-R∥}^2\le {∥S-R∥}^2.\end{array}$$

Example 4.

Let $n\ge 2$ and $S\in M_n\left(\mathbb{C}\right)$ be positive definite with eigenvalues in $[\alpha ,\beta ]\subset (0,\infty )$. Define $\mathcal{K}\left(\xi \right):=e^{\xi /2}{S}^{1/2}$ for $\xi \in [\alpha ,\beta ]$, so ${\left|\mathcal{K}\left(\xi \right)\right|}^2=e^{\xi }S$ and $\mathrm{Re}\left({\left(\mathcal{K}\left(\xi \right)\right)}^{\ast }{\mathcal{K}}^{\prime }\left(\xi \right)\right)={\displaystyle \frac{1}{2}}{e}^{\xi }S$. Then $∥\Gamma ∥={\displaystyle \frac{1}{2}}(e^{\beta }-e^{\alpha })∥S∥$. Theorem 2 with the uniform weight $\rho \equiv {\displaystyle \frac{1}{\beta -\alpha }}$ on $\mathcal{X}=[\alpha ,\beta ]$ and $\phi =\mathrm{id}$ gives $\overline{\phi }={\displaystyle \frac{\alpha +\beta }{2}}$ and ${\sigma }_{\rho }\left(\phi \right)={\displaystyle \frac{\beta -\alpha }{2\sqrt{3}}}$, so

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\beta -\alpha }}{\int }_{\alpha }^{\beta }{e}^{\xi }Sd\xi -e^{(\alpha +\beta )/2}S\le {\displaystyle \frac{(\beta -\alpha )(e^{\beta }-e^{\alpha })}{2\sqrt{3}}}∥S∥I_n,\end{array}$$

with respect to the partial order of $M_n\left(\mathbb{C}\right)$, where $S\le ∥S∥I_n$ since S is positive definite. Since ${\displaystyle \frac{1}{\beta -\alpha }}{\int }_{\alpha }^{\beta }{e}^{\xi }d\xi ={\displaystyle \frac{e^{\beta }-e^{\alpha }}{\beta -\alpha }}$, we obtain

$$\begin{array}{c}\hfill \left({\displaystyle \frac{e^{\beta }-e^{\alpha }}{\beta -\alpha }}-e^{(\alpha +\beta )/2}\right)S\le {\displaystyle \frac{(\beta -\alpha )(e^{\beta }-e^{\alpha })}{2\sqrt{3}}}∥S∥I_n.\end{array}$$

For a concrete numerical comparison, let $n=2$, $S=\left(\begin{array}{cc}1& 0\\ 0& 2\end{array}\right)$, $[\alpha ,\beta ]=[1,2]$, and take a two-point probability space with ${\rho }_1={\rho }_2={\displaystyle \frac{1}{2}}$, ${\phi }_1={\displaystyle \frac{7}{5}}$, and ${\phi }_2={\displaystyle \frac{8}{5}}$. Then $\overline{\phi }={\displaystyle \frac{3}{2}}$, ${\sigma }_{\rho }\left(\phi \right)={\displaystyle \frac{1}{10}}$, and $∥\Gamma ∥=e^2-e$. Hence, Theorem 2 gives

$$\begin{array}{c}\hfill 2{\sigma }_{\rho }\left(\phi \right)∥\Gamma ∥={\displaystyle \frac{e^2-e}{5}}\approx 0.9342,\end{array}$$

whereas the uniform bound in (1) is

$$\begin{array}{c}\hfill {\displaystyle \frac{\beta -\alpha }{2}}∥\Gamma ∥={\displaystyle \frac{e^2-e}{2}}\approx 2.3354.\end{array}$$

The actual deviation is

$$\begin{array}{c}\hfill ∥{\displaystyle \frac{e^{7/5}+e^{8/5}}{2}}S-e^{3/2}S∥=2\left({\displaystyle \frac{e^{7/5}+e^{8/5}}{2}}-e^{3/2}\right)\approx 0.0449.\end{array}$$

Thus, in this example, the distribution-dependent bound is $40\%$ of the uniform bound.

5. Conclusions

In this paper, we have established several new operator inequalities for square modulus convex functions $\mathcal{K}:[\alpha ,\beta ]\to \mathfrak{B}\left(H\right)$ in complex Hilbert spaces.

We proved an Ostrowski-type inequality that bounds the deviation of the weighted integral mean ${\int }_{\mathcal{X}}\rho {\left|\mathcal{K}\circ \phi \right|}^{2}d\mu$ from the pointwise value ${\left|\mathcal{K}\left(\xi \right)\right|}^2$ at any point $\xi \in [\alpha ,\beta ]$. Taking $\xi ={\displaystyle \frac{\alpha +\beta }{2}}$ yields a midpoint inequality with an explicit constant.

We also established a distribution-dependent estimate for the Jensen gap with the bound $2{\sigma }_{\rho }\left(\phi \right)∥\Gamma ∥$, where ${\sigma }_{\rho }\left(\phi \right)$ is the weighted standard deviation of $\phi$ under $\rho$. This bound vanishes whenever $\phi$ is constant and depends on the spread of $\phi$.

Further results include a two-point Ostrowski inequality, a multi-point Ostrowski inequality for n evaluation points, and a superadditivity property of the Jensen gap. Discrete versions of all results are provided, and the bounds are illustrated by three concrete examples.

In future work, our results can be extended through different classes of operator convexity, such as operator m-convex functions and operator h-convex functions, to obtain new inequalities in Hilbert spaces.