In this paper, linearizable holomorphic solutions for second-order differential equations with state-dependent delay is investigated in the complex field \(\mathbb {C}\) by locally reducing the equation to another nonlinear q-difference-differential equation and by constructing invertible holomorphic solutions of the q-difference-differential equations. We prove the existence of invertible holomorphic solutions of the q-difference-differential equations according to the position of the proportional delay multiplier q on the complex plane. Especially, when q is on the unit circle but not a root of the unity, the proofs involve some techniques to handle small divisor problems. The methods used in this paper mainly include the establishment of the recurrence relations, the construction of the majorant series, and the techniques of dealing with the small divisor problems.
