Chebyshev property of complete elliptic integrals and its application to abelian integrals

This paper has two parts. In the first one we study the maximum number of zeros of a function of the form f(k)K(k) + g(k)E(k), where k ∈ (-1, 1), f and g are polynomials, and K(k) = ∫oπ/2 dθ/√1-k2 sin2 θ and E(k) = ∫oπ/2 √1 - k2 sin2 θdθ are the complete normal elliptic integrals of the first and second kinds, respectively. In the second part we apply the first one to obtain an upper bound for the number of limit cycles which appear from a small polynomial perturbation of the planar isochronous differential equation ż = iz + z3, where z = x + iy ∈ ℂ.