Rouché's Theorem

Given two functions $f$ and $g$ Analytic in $A$ with $\gamma$ a simple loop Homotopic to a point in $A$, if $\vert g(z)\vert < \vert f(z)\vert$ for all $z$ on $\gamma$, then $f$ and $f+g$ have the same number of Roots inside $\gamma$.

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 22, 1975.