Rigidity and Structural Asymmetry of Bounded Solutions

We introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter. We identify a "Rotation number hypothesis" on the non-homogeneous term, which a structural asymmetry between the solutions corresponding to the parameters $s$ and $1-{s}$. More precisely, if both solutions with initial value $1$ are bounded on $[1,+\infty)$, then necessarily $\Re(s)=\tfrac12$.