MATH502_HW4 - Serge Ballif MATH 502 Homework 4(1 Consider a...

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Serge Ballif MATH 502 Homework 4 February 15, 2008 (1) Consider a continuous family { f t , t ( - 1 , 1) } of functions that are continuous on the closed unit disc U and holomorphic on the open disc U . (In other words, t 7→ f t is a continuous map from ( - 1 , 1) to the Banach space C ( U ) .) Suppose that f 0 has a single, simple zero in U , say z 0 . Show that for | t | sufficiently small the function f t also has a single, simple zero in U , say z t and that the mapping t 7→ z t is continuous. What can you say in the case of a multiple zero? Define ε = min {| f 0 ( z ) | : | z | = 1 } (the minimum exists on a compact set). Since the map t 7→ f t is continuous, we know that for each u U there exists δ > 0 such that | t - s | < δ implies | f t ( u ) - f s ( u ) | < ε . We are concerned with the case where s = 0, and u = z 0 . For | t | < δ we have | f t ( z 0 ) - f 0 ( z 0 ) | < ε = min {| f 0 ( z ) | : | z | = 1 } . Hence, we can invoke Rouch´ es theorem to conclude that f 0 and f t ( | t | < δ ) have the same number of zeros inside the unit disc; namely, they both have a single simple zero. To see that the mapping t 7→ z t is continuous, we need only note that the
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