Integrate inequalities for the higher derivatives of Blashke product

Authors

  • Tatsiana S. Mardvilka Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

Keywords:

Blashke product, rational functions, higher derivatives, Lebesgue space

Abstract

Upper and lower inequalities for the higher derivatives of Blashke product in the Lebesgue space Lp are obtained in this work. All p ∈ (0; +∞)\ {1/s}, s ∈ N\1, are considered, where s is order of the derivative. The case p = s/1 was investigated by the author earlier. Let an = {a1,…,an} be a certain set of n complex numbers laying in the unit disc |z| < 1. Let us introduce the Blashke products with zeros at the points a1, a2, …, an. For 0 < p < 1/s and s ∈ N holds the equality infan||bn(s)||Lp = 0. For p > 1 holds the equality infan||bn'||Lp = n. For 1/s < p < ∞ and s ∈ N holds the equality supan||bn(s)||Lp = +∞. In other cases, the obtained estimates are exact in order. The main results of the present paper are stated in theorems 1-5.

Author Biography

  • Tatsiana S. Mardvilka, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

    PhD (physics and mathematics), docent; associate professor at the department of function theory, faculty of mechanics and mathematics

References

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Published

2018-05-05

Issue

No. 1 (2018): Journal of the Belarusian State University. Mathematics and Informatics

Section

Real, Complex and Functional Analysis

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How to Cite

[1]
Mardvilka, T.S. 2018. Integrate inequalities for the higher derivatives of Blashke product. Journal of the Belarusian State University. Mathematics and Informatics. 1 (May 2018), 10–16.