Logarithmic derevative and angular density of zeros for the Blaschke product

Mykola Zabolotskyi, Yuriy Gal, Mariana Mostova


Let z0=1 be the only boundary point of zeros an of the Blaschke product B(z),

Γm=j=1mz: z<1, arg(1-z)=θj=j=1mIθj,
be a finite system of rays, 0<ς<1. We found asymptotics of the logarithmic derivative of B(z) as z=1-re-iφ1-π2<φ<π2, under the condition of existing the angular density of its zeros related to the comparison function 1-r-ρ, 0<ρ<1. We also considered the inverse problem for B(Z), whose zeros lie on Γm.

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P. Kusis, Introduction to the theory of spaces Hp, Mir, Moscow, 1984. (in Russian)

J. B. Garnett, Bounded analytic functions, Mir, Moscow, 1984. (in Russian)

R. S. Haloyan, On the asymptotic properties πp(z,zk), Reports of the Academy of Sciences of the Armenian SSR 59 (1974), no. 2, 65-71.

А. А. Гольдберг, Н. Е. Коренков, Асимптотика логарифмической производной целой функции вполне регулярного роста, Сиб. мат. журн. 21 (1980), no. 3, 63-79. English version: A. A. Goldberg and N. E. Korenkov, Asymptotic behavior of logarithmic derivative of entire function of completely regular growth, Sib. Math. J. 21 (1980), no. 3, 363-375. DOI: 10.1007/BF00968180

J. Miles, A sharp form of the lemma on the logarithmic derivative, J. London Math. Soc. 45 (1992), no. 2, 243-254. DOI: 10.1112/jlms/s2-45.2.243

I. E. Chyzhykov, G. G. Gundersen, and J. Heittokangas, Linear differential equations and logarithmic derivative estimates, Proc. London Math. Soc. 86 (2003), no. 3, 735-754. DOI: 10.1112/S0024611502013965

Sh. I. Strelitz, Asymptotic properties of analytical solutions of differential equations, Mintis, Vilnius, 1972, 468 p. (in Russian)

A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromorphic functions, Nauka, Moscow, 1970 (in Russian).

М. В. Заболоцький, М. Р. Мостова, Логарифмічна похідна і кутова щільність нулів цілої функції нульового порядку, Укр. мат. журн. 66 (2014), no. 4, 473-481. English version: M. V. Zabolotskyj and M. R. Mostova, Sufficient conditions for the existence of the υ-density of zeros for an entire function of order zero, Ukr. Math. J. 68 (2016), no. 4, 570-582. DOI: 10.1007/s11253-016-1242-1

DOI: http://dx.doi.org/10.30970/vmm.2021.91.054-062


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