Inequalities with $\|x-y\|$, $|\langle x,y\rangle|$, and $\sqrt{\|x\|^{2}+\|y\|^{2}}$ in a Hilbert space

Question

Let $H$ be a Hilbert space, and let $\|x\|$ denote the norm of $x\in H$, and $\langle x,y\rangle$ denote the inner product of $x,y\in H$.

For $x,y\in H$ let us denote $\alpha(x,y)=\|x-y\|$, $\beta(x,y)=\sqrt{\|x\|^{2}+\|y\|^{2}}$, $\gamma(x,y)=|\langle x,y\rangle|$.

So, $\alpha$, $\beta$, and $\gamma$ are nonnegative functions.

It is obvious that the following inequalities are true: $$ \alpha\leq\beta\sqrt{2}, \quad \gamma\leq\frac{\beta^{2}}{2}. $$

Are there functions $f,g,\varphi,\psi: [0,\infty)\to [0,\infty)$ such that some of the following inequalities are true: $$ \beta\leq f(\alpha), \quad \beta\leq g(\gamma), \quad \gamma\leq \varphi(\alpha), \quad \alpha\leq \psi(\gamma)? $$

Thank you very much for your help in advance!

Answer 1

• $\beta\leq f(\alpha)$? No such $f$ exists. Consider when $x=y\neq 0$.
• $\beta\leq g(\gamma)$? No such $g$ exists. Consider when $x$ and $y$ are perpendicular and not both zero.
• $\gamma\leq \varphi(\alpha)$? No such $\varphi$ exists. Consider when $x=y\neq 0$.
• $\alpha\leq \psi(\gamma)$? No such $\psi$ exists. Consider when $x$ and $y$ are perpendicular and not both zero.