In spite of many works dedicated to the study of functional differential equations, obtaining the estimates for their solutions still remains actual (e.g., [l]-[7] and references therein). In this paper, we obtain the sharp estimates for strong solutions of difference-differential equations of the neutral type. Note that we essentially use our previous results concerning the traditional initial value problem for a homogeneous equation [8]-[10]. In turn, estimating the solutions of the homogeneous equation, we essentially use the Riesz basisness of the system of exponential solutions of the mentioned equations (see [8]-[14] for details).
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