On belonging of entire Dirichlet series to a modified generalized convergence class |
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| Author |
info@nuft.edu.ua
Kyiv National University of Food Technologies, Kyiv, Ukraine
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| Abstract |
For entire Dirichlet series $F(s)=\sum\nolimits_{n=0}^{+\infty}a_n e^{s\lambda_n}$ we found conditions on $a_n$, $\lambda_n$
and on positive functions $\alpha$ and $\beta$ continuous increasing to $+\infty$ on $[0,\,+\infty)$ are found,
under which the condition $\int\nolimits_{\sigma_0}^{+\infty}\frac{1}{\beta(\sigma)}
\alpha\left(\frac1{\sigma}{\ln M(\sigma, F)}\right)d\sigma\le+\infty$ is equivalent to
the condition
$$\sum\limits_{n=1}^{+\infty}(\alpha(\lambda_{n})-\alpha(\lambda_{n-1}))
\beta_1\left(\frac{1}{\lambda_n}\ln\frac{1}{|a_n|}\right) \le+\infty$$,
where $\beta_1(x)=\int\nolimits_{x}^{+\infty}\frac{dt}{\beta(t)}$,
and $M(\sigma, F)=\sup\{|F(\sigma+it)|\colon t\in {\mathbb R}\}$.
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| Keywords |
entire Dirichlet series; convergence class
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| DOI |
doi:10.15330/ms.50.2.135-142
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| Reference |
1. G. Valiron, General theory of integral functions, Toulouse, 1923, 382 p.
2. P.K. Kamthan, A theorem of step functions. II, Instambul univ. fen. fac. mecm. A. 28 (1963), 65–69. 3. O.M. Mulyava, On convergence classes of Dirichlet series, Ukr. Math. J., 51 (1999), ¹1, 1485–1494. (in Ukrainian) 4. O.M. Mulyava, Convergence classes in theory of Dirichlet series, Dopov. Nac. acad. Ukr., (1999), ¹3, 35–39. (in Ukrainian) 5. M.N. Sheremeta, Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion, Izv. Vyssh. Uchebn. Zaved. Mat., (1967), ¹2, 100–108. (in Russian) 6. M.M. Sheremeta, Entire Dirichlet series, K.: ISDÎ, (1993), 168 p. (in Ukrainian) 7. M.M. Sheremeta, On two classes of positive functions and the belonging of main characteristics of entire functions then, Mat. Stud., 19 (2003), ¹1, 75–82. 8. E. Seneta, Regularly varying functions, Lecture Notes in Mathematics, V.508, Springer-Verlag, Berlin, 1976, 112 p. 9. A.F. Leont’ev, Series of exponents, M.: Nauka, 1956, 536 p. (in Russian) 10. O.M. Mulyava, M.M. Sheremeta, On the belonging of entire Dirichlet series to logarithmic convergence class, Mat. Stud., 3 (2010), ¹1, 17–21. (in Ukrainian) |
| Pages |
135-142
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| Volume |
50
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| Issue |
2
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| Year |
2018
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| Journal |
Matematychni Studii
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| Full text of paper | |
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