The Nevanlinna characteristics and maximum modulus of gap power series

O.B.Skaskiv; I.Е.Chyzhykov

Let $f$ be an~analytic in $\{z:|z|<1\}$ function represented by the lacunary power series $f(z)=\sum_{k=0}^{+\infty} a_k z^{n_k}$. We obtain general conditions on the exponents and the~growth of $f$ which provide the asymptotic equality $$ T(r,f)=(1+o(1))\ln M_f(r)$$ as $r\to 1-0$ outside an~exceptional set, where $T(r,f) $ is the Nevanlinna characteristics of~$f$, $M_f(r)$ is the maximum modulus of $f$ on the circle $\{z:|z|=r\}$.

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3. Hayman W. K., Rossi J. F. Characteristics, maximum modulus and value distribution, Trans. Amer. Math. Soc. 284 (1984), no. 2, 651–664.

4. Paley R. E. A. C. On lacunary power series, Proc. Nat. Acad. Sci. U.S.A. 19 (1933), 271–272.

5. Weiss G., Weiss M. On the Picard property of lacunary power series, Studia Math. 22 (1963), 221–245.

6. Fuchs W. H. J. On the zeros of power series with Hadamard gaps, Nagoya Math. J. 29 (1967), 167–174.

7. Chang I. L. On the zeros of power series with Hadamard gaps-distribution in sectors, Trans. Amer. Math. Soc. 178 (1973), 393–400.

8. Murai T. The value-distribution of lacunary series and a conjecture of Paley, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 1, 135–156.

9. Nicholls P. J., Sons L. R. Minimum modulus and zeros of functions in the unit disc, Proc. London Math. Soc. 31 (1975), no. 1, 99–113.

10. Галь Ю. М. Об аналогах теоремы Вимана для рядов Дирихле, показатели которых имеют положительный шаг, Изв. вузов. Матем. (1986), 194 2, 57–59.

11. Kövari T. A gap theorem for entire functions of infinite order, Michigan Math. J. 12 (1965), 133–140.

12. Гайсин А. М. Оценка ряда Дирихле, показатели которого — нули целой функции с нерегулярным поведением, Мат. cборник 185 (1994), 194 2, 33–56.

	The Nevanlinna characteristics and maximum modulus of gap power series
Author	O.B.Skaskiv, I.Е.Chyzhykov Lviv National University, Faculty of Mechanics and Mathematics
Abstract	Let $f$ be an~analytic in $\{z:\|z\|<1\}$ function represented by the lacunary power series $f(z)=\sum_{k=0}^{+\infty} a_k z^{n_k}$. We obtain general conditions on the exponents and the~growth of $f$ which provide the asymptotic equality $$ T(r,f)=(1+o(1))\ln M_f(r)$$ as $r\to 1-0$ outside an~exceptional set, where $T(r,f) $ is the Nevanlinna characteristics of~$f$, $M_f(r)$ is the maximum modulus of $f$ on the circle $\{z:\|z\|=r\}$.
Keywords	analytic functions in the unit disk, lacunary power series, exponents of power series, growth of analytic functions, asymptotic equality, exceptional sets
DOI	doi:10.30970/ms.13.2.125-133
Reference	1. Гольдберг А. А., Островский И. В. Распределение значений мероморфных функций, М.:Наука, 1970, 592 с. 2. Murai T. The deficiency of entire functions with Fejér gaps, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 39–58. 3. Hayman W. K., Rossi J. F. Characteristics, maximum modulus and value distribution, Trans. Amer. Math. Soc. 284 (1984), no. 2, 651–664. 4. Paley R. E. A. C. On lacunary power series, Proc. Nat. Acad. Sci. U.S.A. 19 (1933), 271–272. 5. Weiss G., Weiss M. On the Picard property of lacunary power series, Studia Math. 22 (1963), 221–245. 6. Fuchs W. H. J. On the zeros of power series with Hadamard gaps, Nagoya Math. J. 29 (1967), 167–174. 7. Chang I. L. On the zeros of power series with Hadamard gaps-distribution in sectors, Trans. Amer. Math. Soc. 178 (1973), 393–400. 8. Murai T. The value-distribution of lacunary series and a conjecture of Paley, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 1, 135–156. 9. Nicholls P. J., Sons L. R. Minimum modulus and zeros of functions in the unit disc, Proc. London Math. Soc. 31 (1975), no. 1, 99–113. 10. Галь Ю. М. Об аналогах теоремы Вимана для рядов Дирихле, показатели которых имеют положительный шаг, Изв. вузов. Матем. (1986), 194 2, 57–59. 11. Kövari T. A gap theorem for entire functions of infinite order, Michigan Math. J. 12 (1965), 133–140. 12. Гайсин А. М. Оценка ряда Дирихле, показатели которого — нули целой функции с нерегулярным поведением, Мат. cборник 185 (1994), 194 2, 33–56.
Pages	125-133
Volume	13
Issue	2
Year	2000
Journal	Matematychni Studii
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