An extremal problem for analytic functions with prescribed zeros and th derivative in

Authors:
A. Horwitz and D. J. Newman

Journal:
Trans. Amer. Math. Soc. **295** (1986), 699-713

MSC:
Primary 30C75; Secondary 30D50

DOI:
https://doi.org/10.1090/S0002-9947-1986-0833704-4

MathSciNet review:
833704

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Abstract: Let be points in the unit disc . Suppose is analytic in , (multiplicities included), and . Then we prove that for all , where and is a Blaschke product of order . We extend this result in a natural way to convex domains with analytic boundary. For not convex we show that there is no extremal function .

**[1]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[2]**S. D. Fisher and Charles A. Micchelli,*The 𝑛-width of sets of analytic functions*, Duke Math. J.**47**(1980), no. 4, 789–801. MR**596114****[3]**A. Horwitz,*Restricted interpolation and optimal recovery of certain classes of functions*, Doctoral Thesis, Temple University, Philadelphia, Pa., 1984.**[4]**Charles A. Micchelli and Theodore J. Rivlin (eds.),*Optimal estimation in approximation theory*, Plenum Press, New York-London, 1977. The IBM Research Symposia Series. MR**0445154****[5]**D. J. Newman,*Polynomials and rational functions*, Approximation theory and applications (Proc. Workshop, Technion—Israel Inst. Tech., Haifa, 1980) Academic Press, New York-London, 1981, pp. 265–282. MR**615416****[6]**H. L. Royden,*The boundary values of analytic and harmonic functions*, Math. Z.**78**(1962), 1–24. MR**0138747**, https://doi.org/10.1007/BF01195147**[7]**J. L. Walsh,*Interpolation and approximation by rational functions in the complex domain*, Third edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR**0218587**

J. L. Walsh,*Interpolation and approximation by rational functions in the complex domain*, Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1965. MR**0218588**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0833704-4

Article copyright:
© Copyright 1986
American Mathematical Society