Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at the origin. Then we define the weighted length of curves in the family as $$L(\gamma):= \int_{\gamma} \frac{1}{1+|z|^2}d|z|,$$where $d|z|$ is the classical length element.

My question is that, is it true that $\inf_{\gamma \in \mathcal{A}}L(\gamma) =\pi$?

I have done some computations for some curves with explicit formulas. For example, if $\gamma$ is a unit circle centered at $z=1$, then $\gamma \in \mathcal{A}$ and $L(\gamma)=4\pi/\sqrt{5}$. Also, it seems that when $\gamma$ is more and more closed to the line segment $[0,1]$ with multiplicity 2, the weighted length $L(\gamma)$ is getting smaller and approaching to $\pi$.

Any ideas or comments are really appreciated. Thank you very much for your time.

Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at the origin. Then we define the weighted length of curves in the family as $$L(\gamma):= \int_{\gamma} \frac{2}{1+|z|^2}d|z|,$$where $d|z|$ is the classical length element.

My question is that, is it true that $\inf_{\gamma \in \mathcal{A}}L(\gamma) =\pi$?

I have done some computations for some curves with explicit formulas. For example, if $\gamma$ is a unit circle centered at $z=1$, then $\gamma \in \mathcal{A}$ and $L(\gamma)=4\pi/\sqrt{5}$. Also, it seems that when $\gamma$ is more and more closed to the line segment $[0,1]$ with multiplicity 2, the weighted length $L(\gamma)$ is getting smaller and approaching to $\pi$.

Any ideas or comments are really appreciated. Thank you very much for your time.