Root and critical point behaviors of certain sums of polynomials

It is known that no two roots of the polynomial equation Pi(n)(j=1)(x - r (j)) +Pi(j=1) (n) (x + r(j)) = 0, where 0 < r(1) = r(2) <= r(2) <= ... <= = r(n), can be equal and the gaps between the roots of (1) in the upper half-plane strictly increase as one proceeds upward, and for 0 < h < r(k), the roots of (x - r(k) - h) Pi (n)(j=1j not equal k) (x - r(j)) + (x + r(k) + h) Pi(n)(j=1j not equal k) (x + r(j)) = 0 and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis. In this paper, we study how the roots and the critical points of (1) and (2) are located.