Structural Asymmetry and Rigidity of Bounded Solutions for a Parametrized Complex Linear Differential Equation

We introduce a family of parametrized nonhomogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter lying in the critical strip. We establish a phenomenon of structural asymmetry and rigidity: When $\Re(s)\neq \tfrac{1}{2}$, the two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\bar{s}$, cannot both be bounded on $[1,+\infty)$.