<p>This paper investigates optimal positivity-preserving exponential integrators for linear non-homogeneous initial-value problems. The central task is formulated as the identification among all polynomials of a given degree with prescribed derivatives at the origin up to a given order, the one with largest interval of absolute monotonicity in the negative axis. Existence and uniqueness of such <i>optimal threshold polynomials</i> is established and their explicit forms for low degrees are derived. Moreover, a computational algorithm for determining these optimal threshold polynomials is provided. Application to optimal positivity-preserving exponential integrators with collocation points the zeros of orthogonal polynomials and related polynomials are investigated. Numerical tests are provided.</p>