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

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