Counting Real Zeros of Analytic Functions Satisfying Linear Ordinary Differential Equations

We suggest an explicit procedure to establish upper bounds for the number of real zeros of analytic functions satisfying linear ordinary differential equations with meromorphic coefficients. If the equation[formula]has no singular points in a small neighborhoodUof a real segmentK, all the coefficientsaj(t) have absolute value AonUanda0(t)≡1, then any solution of this equation may have no more thanβ(A+ν) zeros onK, whereβ=β(U, K) is a geometric constant depending only onKandU. If the principal coefficienta0(t) is nonconstant, but its modulus is at leasta>0 somewhere onK, then the number of real zeros onKof any solution analytic inU, does not exceed (A/a+ν)μwith someμ=μ(U, K).