 .-Preface.-1. internal Functions.-2. the phenomenal Set of an internal Function.-3. The by-product of Finite Blaschke Products.-4. Angular Derivative.-5. Hp-Means of S'.-6. Bp-Means of S'.-7. The spinoff of a Blaschke Product.-8. Hp-Means of B'.-9. Bp-Means of B'.-10. the expansion of necessary technique of B'.-References.-Index

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Extra info for Derivatives of Inner Functions

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Zm }. Hence, B has at most n + m − 2 zeros in C. These are the zeros of B, repeated appropriately, and the zeros of P . In this case, P might have zeros at the origin. 2 applies. Therefore, P can have zeros at the origin and the rest are of the form w1 , 1/w ¯1 , w2 , 1/w ¯2 , . . , w , 1/w ¯, where w1 , w2 , . . , w ∈ D \ {0, z1 , . . , zm }. Certainly 2m − 2. Hence, ≤ m − 1. 3 A Formula for |B | Let B(z) = z0 − z . 1 − z¯0 z Then it is easy to verify that |B(z)|2 = 1 − This identity can be rewritten as (1 − |z|2 ) (1 − |z0 |2 ) .

Hence, ≤ m − 1. 3 A Formula for |B | Let B(z) = z0 − z . 1 − z¯0 z Then it is easy to verify that |B(z)|2 = 1 − This identity can be rewritten as (1 − |z|2 ) (1 − |z0 |2 ) . 3 A Formula for |B | 43 1 − |B(z)|2 1 − |z0 |2 = , 1 − |z|2 |1 − z¯0 z|2 and we can generalize it in the following manner. 4 Let n B(z) = j=1 zj − z . 1 − z¯j z Put B1 = 1 and k−1 Bk (z) = j=1 zj − z , 1 − z¯j z (2 ≤ k ≤ n). Then, for each z ∈ C \ T, 1 − |B(z)|2 = 1 − |z|2 n 1 − |zk |2 . |1 − z¯k z|2 |Bk (z)|2 k=1 Proof. The proof is by induction on the order of B.

But, it might have fewer zeros in De . The reason is that, in De , the zeros of Bε may cluster at the poles of B or at ∞. We now study the zeros of B in De . First assume that j0 = 0. Then, by direct veriﬁcation, we see that 42 3 The Derivative of Finite Blaschke Products B (z) = z j0 −1 × m jk −1 k=1 (z − zk ) m zk )jk +1 k=1 (z − 1/¯ × P (z), where P is a polynomial of degree 2m and it has no zeros among {0, z1, . . , zm }. Hence, B has n + m − 1 zeros in C. These are the zeros of B, repeated appropriately, and the zeros of P .