Let $H := \{z \in \mathbb{C}: Re(z) > 0\}$ be the open right half-plane. I would like to construct a holomorphic function $f:H\rightarrow \mathbb{C}$ such that $|f|$ is bounded and bounded away from zero in $H$, and such that there exist $x_1, x_2 \in \mathbb{R}$ such that $$\sup\limits_{z\in H} |f(z)| = \lim\limits_{t\rightarrow+\infty} |f(x_1+it)|$$ $$\inf\limits_{z\in H} |f(z)|= \lim\limits_{t\rightarrow+\infty} |f(x_2-it)|.$$ That is, the supremum and the infimum of $|f|$ is attached by taking limits in two vertical lines, the first one with increasing imaginary part and the second one with decreasing imaginary part.
Any help would be appreciated, thank you very much in advance!!