Geodesic
On an estimate for a function represented by a Dirichlet series
Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 525-532 
Z. Sh. Karimov. On an estimate for a function represented by a Dirichlet series. Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 525-532. http://geodesic.mathdoc.fr/item/SM_1975_25_4_a3/
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     title = {On~an estimate for a~function represented by {a~Dirichlet} series},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The article considers the series $$ f(z)=\sum_{k=1}^\infty a_ke^{\lambda_kz},\qquad0<\lambda_k\uparrow\infty,\quad\sum_{k=1}^\infty\lambda_k^{-1}<\infty, $$ convergent in the whole plane. Theorem 1. {\it Let $|f(x)| $-\infty where $0. For given $\varepsilon>0$ and $h>0$ there exists a constant $A$, not depending on $f(z)$ and $H(x)$ such that $|f(z)| $x=\operatorname{Re}z,$ $|y|.} Theorem 2. {\it If in addition $$ \delta=\varlimsup_{k\to\infty}\frac1{\lambda_k}\ln\biggl|\frac1{L'(\lambda_k)}\biggr|<\infty,\qquad L(\lambda)=\prod_{k=1}^\infty\biggl(1-\frac\lambda{\lambda_k}\biggr), $$ then for arbitrary $z$ we have $|f(z)| $x=\operatorname{Re}z$.} The quantity $\delta$ cannot be replaced by a smaller one. These results strengthen corresponding results due to Gaier (RZhMat., 1967, 10B155) and Anderson and Binmore (RZhMat., 1972, 7B1115). Bibliography: 7 titles.

[1] D. Gaier, “On the coefficients and the growth of gap power series”, SIAM J. Numer. Anal., 3:2 (1966), 248–265 | DOI | MR | Zbl

[2] J. M. Anderson, K. G. Binmore, “Glosure theorems with applications to entire functions with gaps”, Trans. Amer. Math. Soc., 161 (1971), 381–400 | DOI | MR | Zbl

[3] A. F. Leontev, “Rasprostranenie svoistv tselykh funktsii poryadka menshe poloviny na nekotorye drugie funktsii”, Trudy Matem. in-ta im. V. A. Steklova, LXIV (1961), 126–146 | MR

[4] A. F. Leontev, “O polnote sistemy $\{e^{\lambda_k^z}\}$ v zamknutoi polose”, DAN SSSR, 152:2 (1963), 266–268 | MR

[5] R. P. Boas, Entire Functions, Academic Press, New York, 1954 | MR | Zbl

[6] A. F. Leontev, “Ob odnom dopolnenii k teoreme Adamara”, DAN SSSR, 206:5 (1972), 1049–1051 | MR

[7] A. F. Leontev, “O summirovanii ryada Dirikhle s kompleksnymi pokazatelyami i ego primenenii”, Trudy Matem. in-ta im. V. A. Steklova, LXII (1971), 300–326 | MR