Some Hermite–Hadamard type inequalities for the square norm in Hilbert spaces

Abstract

Let \(\left( H;\left\langle \cdot ,\cdot \right\rangle \right)\) be a complex Hilbert space and \(f:[0,\infty )\rightarrow \mathbb{R}\) be convex (concave) on \([0,\infty )\). If \(x, y\in H\) with \(Re \left\langle x,y\right\rangle \geq 0\), then
\begin{align*}
f\left( \frac{\left\Vert x\right\Vert ^{2}+Re \left\langle x,y\right\rangle +\left\Vert y\right\Vert ^{2}}{3}\right) & \leq \left( \geq
\right) \int_{0}^{1}f\left( \left\Vert \left( 1-t\right) x+ty\right\Vert
^{2}\right) dt \\
& \leq \left( \geq \right) \frac{1}{3}\left[ f\left( \left\Vert x\right\Vert
^{2}\right) +f\left[ Re \left\langle x,y\right\rangle \right] +f\left(
\left\Vert y\right\Vert ^{2}\right) \right] .
\end{align*}
Some examples for power functions and exponential are also provided.