Hermite–Hadamard-type double-integral inequalities for the operator modulus in Hilbert spaces

Let $ \left(\mathcal{H}; \langle \cdot, \cdot \rangle \right) $ be a complex Hilbert space. Denote by $ \mathcal{L}(\mathcal{H}) $ the Banach $ \mathcal{C}^{\ast}- $algebra of bounded linear operators on $ \mathcal{H} $. For $ \mathcal{A}\in \mathcal{L}(\mathcal{H}), $ we define the modulus of $ \mathcal{A} $ by $ \left\vert \mathcal{A}\right\vert : = \left(\mathcal{A}^{\ast }\mathcal{A} \right) ^{1/2}. $ We say that the continuous function $ S:\left[\mathfrak{a}_{1}, \mathfrak{a}_{2}\right] \rightarrow \mathcal{L}(\mathcal{H}) $ is square modulus convex on $ \left[\mathfrak{a}_{1}, \mathfrak{a}_{2}\right] $ if

$ \begin{align*} &\quad \left\vert S\left( \left( 1-\vartheta \right) \mathfrak{u}+\vartheta \mathfrak{v}\right) \right\vert ^{2}\\&\leq \left( 1-\vartheta \right) \left\vert S\left( \mathfrak{u}\right) \right\vert ^{2}+\vartheta \left\vert S\left( \mathfrak{v}\right) \right\vert ^{2} \end{align*} $

in the operator order of $ \mathcal{L}(\mathcal{H}), $ for all $ \mathfrak{u}, $ $ \mathfrak{v} $ $ \in \left[\mathfrak{a}_{1}, \mathfrak{a}_{2}\right] $ and $ \vartheta $ $ \in \left[0, 1\right]. $ In this paper, we have showed, among other things, that

$ \begin{align*} 0& \leq 2\min \left\{ \vartheta , 1-\vartheta \right\} \times \left[ \frac{1}{\mathfrak{a}_{2}-\mathfrak{a}_{1}}\int_{a}^{ \mathfrak{a}_{2}}\left\vert S\left( s\right) \right\vert ^{2}ds-\frac{1}{ \left( \mathfrak{a}_{2}-\mathfrak{a}_{1}\right) ^{2}}\int_{\mathfrak{a} _{1}}^{\mathfrak{a}_{2}}\int_{\mathfrak{a}_{1}}^{\mathfrak{a}_{2}}\left\vert S\left( \frac{s+\sigma }{2}\right) \right\vert ^{2}dsd\sigma \right] \\ & \leq \frac{1}{\mathfrak{a}_{2}-\mathfrak{a}_{1}}\int_{\mathfrak{a}_{1}}^{ \mathfrak{a}_{2}}\left\vert S\left( s\right) \right\vert ^{2}ds-\frac{1}{ \left( \mathfrak{a}_{2}-\mathfrak{a}_{1}\right) ^{2}}\int_{\mathfrak{a} _{1}}^{\mathfrak{a}_{2}}\int_{\mathfrak{a}_{1}}^{\mathfrak{a}_{2}}\left\vert S\left( \left( 1-\vartheta \right) s+\vartheta \sigma \right) \right\vert ^{2}dsd\sigma \\ & \leq 2\max \left\{ \vartheta , 1-\vartheta \right\} \times \left[ \frac{1}{\mathfrak{a}_{2}-\mathfrak{a}_{1}}\int_{\mathfrak{a} _{1}}^{\mathfrak{a}_{2}}\left\vert S\left( s\right) \right\vert ^{2}ds-\frac{ 1}{\left( \mathfrak{a}_{2}-\mathfrak{a}_{1}\right) ^{2}}\int_{\mathfrak{a} _{1}}^{\mathfrak{a}_{2}}\int_{\mathfrak{a}_{1}}^{\mathfrak{a}_{2}}\left\vert S\left( \frac{s+\sigma }{2}\right) \right\vert ^{2}dsd\sigma \right] \end{align*} $

for all $ \vartheta \in \lbrack 0, 1]. $ Some integral inequalities for symmetric weights were obtained. Applications for exponential functions and modulus quadratic functions were provided as well.