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 is an element of (-1,1), f and g are polynomials, and K (k) = integral(0)(pi/ 2) dtheta/root1-k(2) sin(2) theta and E(k) = integral(0)(pi/2) root1-k(2) sin(2) thetadtheta 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 (z)over dot = iz + z(3), where z = x + iy is an element of C.