On critical points of Blaschke products

S.Favorov; L.Golinskii

We obtain an upper bound for the derivative of a Blaschke product, whose zeros lie in a certain Stolz-type region. We show that the derivative belongs to the space of analytic functions in the unit disk, introduced recently in [6]. As an outcome, we obtain a Blaschke-type condition for critical points of such Blaschke products.

1. P. Ahern, The mean modulus of the derivative of an inner function, Indiana Univ. Math. J., 28 (1979), 311--347.

2. P. Ahern, The Poisson integral of a singular measure, Can. J. Math. 35 (1983), 735--749.

3. P. Ahern, D. Clark, On inner functions with $H^p$ derivatives, Michigan Math. J., 21 (1974), 115--127.

4. P. Ahern, D. Clark, On inner functions with $B^p$ derivatives, Michigan Math. J., 23 (1976), 107--118.

5. G. Cargo, Angular and tangential limits of Blaschke products and their successive derivatives, Canad. J. Math., 14 (1962), 334--348.

6. S.~Favorov, L.~Golinskii, A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators, Amer. Math. Soc. Transl., 226 (2009), №2, 37--47.

7. S.~Favorov, L.~Golinskii, Blaschke-type conditions for analytic functions in the unit disk: inverse problems and local analogs, preprint arXive:1007.3020v1 [math CV], 2010.

8. D. Girela, J. Peláez, On the membership in Bergman spaces of the derivative of a Blaschke product with zeros in a Stolz domain, Canad. Math. Bull., 49 (2006), №3, 381--388.

9. D. Girela, J. Peláez, D. Vukotić, Integrability of the derivative of a Blaschke product, Proc. Edinburgh Math. Soc., 50 (2007), 673--687.

10. D. Girela, J. Peláez, D. Vukotić, Interpolating Blaschke products: Stolz and tangential approach regions, Constr. Approx., 27 (2008), 203--216.

11. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, V.199. Springer-Verlag, New York, 2000.

12. J. Mashreghi, M. Shabankhah, Integral means of the logarithmic derivative of Blaschke products, Comp. Meth. Func. Theory, 9 (2009), №2, 421--433.

13. Ch. Pommerenke, On the Green's function of Fuchsian groups, Ann. Acad. Sci. Fenn., 2 (1976), 409--427.

14. D. Protas, Blaschke products with derivative in $H^p$ and $B^p$, Michigan Math. J., 20 (1973), 393--396.

15. D. Protas, Blaschke products with derivatives in function spaces, preprint arXiv:1001.5098v2 [math.CV], 2010.

16. S.A. Vinogradov, Multiplication and division in the space of analytic functions with area integrable derivative, and some related spaces, Zap. Nauchn. Semin. POMI, 222 (1994), 45--77.(Russian)

	On critical points of Blaschke products
Author	S.Favorov, L.Golinskii Karazin Kharkiv National University
Abstract	We obtain an upper bound for the derivative of a Blaschke product, whose zeros lie in a certain Stolz-type region. We show that the derivative belongs to the space of analytic functions in the unit disk, introduced recently in [6]. As an outcome, we obtain a Blaschke-type condition for critical points of such Blaschke products.
Keywords	critical points, Blaschke product, Stolz-type region, unit disk, upper bound of derivative
DOI	doi:10.30970/ms.34.2.168-173
Reference	1. P. Ahern, The mean modulus of the derivative of an inner function, Indiana Univ. Math. J., 28 (1979), 311--347. 2. P. Ahern, The Poisson integral of a singular measure, Can. J. Math. 35 (1983), 735--749. 3. P. Ahern, D. Clark, On inner functions with $H^p$ derivatives, Michigan Math. J., 21 (1974), 115--127. 4. P. Ahern, D. Clark, On inner functions with $B^p$ derivatives, Michigan Math. J., 23 (1976), 107--118. 5. G. Cargo, Angular and tangential limits of Blaschke products and their successive derivatives, Canad. J. Math., 14 (1962), 334--348. 6. S.~Favorov, L.~Golinskii, A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators, Amer. Math. Soc. Transl., 226 (2009), №2, 37--47. 7. S.~Favorov, L.~Golinskii, Blaschke-type conditions for analytic functions in the unit disk: inverse problems and local analogs, preprint arXive:1007.3020v1 [math CV], 2010. 8. D. Girela, J. Peláez, On the membership in Bergman spaces of the derivative of a Blaschke product with zeros in a Stolz domain, Canad. Math. Bull., 49 (2006), №3, 381--388. 9. D. Girela, J. Peláez, D. Vukotić, Integrability of the derivative of a Blaschke product, Proc. Edinburgh Math. Soc., 50 (2007), 673--687. 10. D. Girela, J. Peláez, D. Vukotić, Interpolating Blaschke products: Stolz and tangential approach regions, Constr. Approx., 27 (2008), 203--216. 11. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, V.199. Springer-Verlag, New York, 2000. 12. J. Mashreghi, M. Shabankhah, Integral means of the logarithmic derivative of Blaschke products, Comp. Meth. Func. Theory, 9 (2009), №2, 421--433. 13. Ch. Pommerenke, On the Green's function of Fuchsian groups, Ann. Acad. Sci. Fenn., 2 (1976), 409--427. 14. D. Protas, Blaschke products with derivative in $H^p$ and $B^p$, Michigan Math. J., 20 (1973), 393--396. 15. D. Protas, Blaschke products with derivatives in function spaces, preprint arXiv:1001.5098v2 [math.CV], 2010. 16. S.A. Vinogradov, Multiplication and division in the space of analytic functions with area integrable derivative, and some related spaces, Zap. Nauchn. Semin. POMI, 222 (1994), 45--77.(Russian)
Pages	168-173
Volume	34
Issue	2
Year	2010
Journal	Matematychni Studii
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