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We consider the equation y′′ +P(z)y′ +Q(z)y = 0 (z ∈ C) , where P(z) = nP ∑ k=0 pkz k−1 and Q(z) = nQ ∑ k=0 qkz k−2 with real coefficients pk,qj (k = 0, ...,nP ; j = 0, ...,nQ ;nP ,nQ < ∞) . Let zk(y),k = 1,2, ... be the nontrivial zeros of a solution y(z) to that equation. Estimates for the sums ∑ j k=1 1 |zk(y)| ( j = 1,2, ...) are derived. Applications of the obtained estimates to the counting function of the zeros of solutions are also discussed.

Inequalities for zeros of solutions to second order ODE with one singular point