  We derive upper bounds for the smallest zero and lower bounds for the largest
zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed
three term recurrence relations satisfied by polynomials corresponding to
different parameter(s) within the same classical family. We prove that
interlacing properties of the zeros impose restrictions on the possible
location of common zeros of the polynomials involved and deduce strict bounds
for the extreme zeros of polynomials belonging to each of these three classical
families. We show numerically that the bounds generated by our method improve
known lower (upper) bounds for the largest (smallest) zeros of polynomials in
these families, notably in the case of Jacobi and Gegenbauer polynomials.
