Uniform exponential-power estimate for the solution to a family of the Cauchy problems for linear differential equations

We consider a solution to a parametric family of the Cauchy problems for \(m\)th-order linear differential equations with constant coefficients. Parameters of the family are the coefficients of the differential equation and the initial values of the solution and its derivatives up to the \((m-1)\)th-order (by a solution to a family of problems we mean a function of the parameters of the given family that maps each tuple of parameters to a solution to the problem with these parameters). We obtain an exponential-power estimate for the functions of this parametric family that is uniform (with respect to parameters) on any bounded set. We also prove that the maximal element of the set of real parts of monic polynomial roots is a continuous function (of the coefficients of the polynomial). The continuity of this element is used for obtaining the estimate mentioned above (since to each tuple of coefficients of the differential equation there corresponds its characteristic polynomial with these coefficients, the set of the roots of the characteristic polynomial and the maximal element of this set are also functions of the coefficients of the differential equation).