Thecomplex harmonic spline(C.H.S.) is defined by the formulawhereSM(s) is any complex spline function on the unit circler= ∂U, withMknots and degreen. In this paper, it is proved thatP(z) =f(ø(z)), where ø is a homeomorphism mapping function andfis an analytic function, the number of zeros ofP(z) onUis denoted byN, and we obtain the following result:N= 1/2π ΔrargP(z) <nM