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# Derivatives of Inner Functions by Javad Mashreghi

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

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12), we did not use the fact that σ is singular with respect to the Lebesgue measure. Hence, these formulas hold for a wider class of functions. 14) also work for any f ∈ H 1 , and we obtain ∞ 1 − |zn |2 f (z) =− − f (z) (1 − z¯n z)(zn − z) n=1 and T 2eit dμ(eit ), (eit − z)2 ∞ f (eiθ ) 1 − |zn |2 = + −iθ iθ e f (e ) n=1 |eiθ − zn |2 T 2dμ(eit ) . 9), we also get the rough estimations ∞ |B (z)| ≤ 1 − |zn |2 , | 1 − z¯n z |2 n=1 and |S (z)| ≤ T dσ(eit ) , |eit − z|2 (z ∈ D), (z ∈ D). 17) From another point of view, by the Schwarz–Pick theorem, we also have |φ (z)| ≤ 1 − |φ(z)|2 , 1 − |z|2 (z ∈ D), for any inner function φ.

R→1 Hence, φ is an inner function. 14, φ is in fact a Blaschke product. e. 1+z exp − = α. 1−z 20 1 Inner Functions By an elementary calculation, the solution of the above equation are zn = log |α| + i(arg α + 2nπ) + 1 , log |α| + i(arg α + 2nπ) − 1 (n ∈ Z). In particular, for α = 1/e, the solutions are zn = inπ , inπ − 1 (n ∈ Z). 15, it is not straightforward that 1+z φ(z) = α − e− 1−z 1+z 1−α ¯ e− 1−z (α ∈ D \ {0}), , is a Blaschke product. 6 The Nevanlinna Class N and Its Subclass N + The Nevanlinna class N is the family of all analytic functions f on the open unit disc which satisfy the growth restriction π sup 0≤r<1 −π log+ |f (reiθ )| dθ < ∞.

Let ε −→ 0 to get μh (E) = 0. Therefore, we have μh,∞ (E) = 0 ⇐⇒ μh (E) = 0. 10) 34 2 The Exceptional Set of an Inner Function The above deﬁnition also implies that μh1 (E) ≤ C μh2 (E) provided that the inequality h1 (t) ≤ C h2 (t) holds at least in a right neighborhood of the origin. This property immediately gives 0 ≤ μα (E) < ∞ =⇒ μβ (E) = 0, (β > α), 0 < μα (E) ≤ ∞ =⇒ μβ (E) = ∞, (β < α). and Hence, for a ﬁxed Borel set E, the graph of α −→ μα (E) is a step function with two possible values ∞ and 0 which breaks at some points in the interval [0, ∞).