Locally univalence of Dirichlet series satisfying a linear differential equation of second order with exponential coefficients |
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| Author |
m.m.sheremeta@gmail.com
Ivan Franko National University of Lviv,
Lviv, Ukraine
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| Abstract |
Let $h>0$, $\gamma_0\not=0$, $\gamma_1\not=0$, $\gamma_2\le -1$
and $|\gamma_1|/4+|\gamma_0|/6\le h/5$. It is well-proven that the differential equation $w''+(\gamma_0 e^{2hs}+\gamma_1 e^{hs}+\gamma_2) w=0$ has an entire solution $F(s)=\exp\{s\lambda_1\}+\sum\nolimits_{k=2}^{\infty}f_k\exp\{s\lambda_k\}$ locally univalent in the half-plane $\{s\colon \,\text{Re}\,s\le 0\}$
and such that $\ln\,M(\sigma,F)\sim(\sqrt{|\gamma_0|}/h)e^{h\sigma}$ as $\sigma\to+\infty$, where $M(\sigma,F)=\sup\{|F(\sigma+it)|\colon t\in\mathbb{R}\}$.
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| Keywords |
Dirichlet series; locally univalence; linear differencial equation of second order
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| DOI |
doi:10.15330/ms.51.2.152-158
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| Reference |
1. G.M. Golusin, Geometrical theory of functions of complex variable, M.: Nauka, 1966, 628 p. (in Russian),
Engl. transl.: AMS: Translations of Mathematical monograph, 26 (1969), 676 p.
2. W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), ¹2, 169–185. 3. S.M. Shah, Univalence of a function f and its successive derivatives when f satisfies a differential equation, II. J. Math. anal. and appl., 142 (1989), 422–430. 4. P. Montel, Sur les fonctions localement univalentes ou multivalentes, Ann. Sci. Ecole Norm. Sup. (3), 54 (1937), 39–54. 5. M.S. Robertson, Schlicht Dirichlet series, Canad. J. Math., 10 (1958), 161–176. 6. M.N. Sheremeta, On the derivative of an entire Dirichlet series, Mat. sb., 137 (1988), ¹1, 128–139 (in Russian); Engl. transl.: Math. USSR Sbornik, 65 (1990), ¹1, 133–145. 7. A.F. Leont’ev, Series of exponents, M.: Nauka., 1976, 536 p. (in Russian) 8. M.N. Sheremeta, Full equivalence of the logarithms of the maximum modulus and the maximal term of an entire Dirichlet series, Mat. Zametki, 47 (1990), ¹6, 119–123 (in Russian); Engl. transl.: Math. Notes, 47 (1990), ¹6, 119–123. 9. M.N. Sheremeta, On the correlations between the maximal term and the maximum modulus of an entire Dirichlet series, Mat. Zametki, 51 (1992), ¹5, 141–148 (in Russian); Engl. transl.: Math. Notes, 51 (1992), ¹5, 141–148. |
| Pages |
152-158
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| Volume |
51
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| Issue |
2
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| Year |
2019
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| Journal |
Matematychni Studii
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| Full text of paper | |
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