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In many stability problems, the investigation of pure imaginary roots for a polynomial family is very important. Given a pure imaginary complex number, the set of all images of uncertainty vectors is called the value set corresponding to this pure imaginary complex number. The question whether these sets contain the origin is very important from robust stability point of view of a polynomial family. Cutoff frequency guarantees the noninclusion of the origin to the value set for large frequencies. In this paper, we give a procedure for more strict estimation of cutoff frequency and applications of the obtained result to the constant inertia problem of a polynomial family. [ABSTRACT FROM AUTHOR]