Geodesic
Application of expansions of entire functions in series of exponentials
Sbornik. Mathematics, Tome 54 (1986) no. 1, pp. 135-159 
A. F. Leont'ev. Application of expansions of entire functions in series of exponentials. Sbornik. Mathematics, Tome 54 (1986) no. 1, pp. 135-159. http://geodesic.mathdoc.fr/item/SM_1986_54_1_a7/
@article{SM_1986_54_1_a7,
     author = {A. F. Leont'ev},
     title = {Application of expansions of entire functions in series of exponentials},
     journal = {Sbornik. Mathematics},
     pages = {135--159},
     year = {1986},
     volume = {54},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1986_54_1_a7/}
}
TY  - JOUR
AU  - A. F. Leont'ev
TI  - Application of expansions of entire functions in series of exponentials
JO  - Sbornik. Mathematics
PY  - 1986
SP  - 135
EP  - 159
VL  - 54
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1986_54_1_a7/
LA  - en
ID  - SM_1986_54_1_a7
ER  - 
%0 Journal Article
%A A. F. Leont'ev
%T Application of expansions of entire functions in series of exponentials
%J Sbornik. Mathematics
%D 1986
%P 135-159
%V 54
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1986_54_1_a7/
%G en
%F SM_1986_54_1_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

The equality \begin{equation} \varlimsup_{r\to\infty}\frac{\ln|F(re^{i\varphi})|}{r^\rho}=\varlimsup_{r\to\infty}\frac{\ln\Phi(re^{i\varphi})}{r^\rho} \end{equation} is established for those values of $\varphi$ for which the left-hand side is nonnegative. Here $F(z)=\sum_1^\infty a_ke^{\lambda_kz}$, $\Phi(z)=\sum_1^\infty |a_ke^{\lambda_kz}|$, $\rho>1$. It is assumed that the $\lambda_k$ ($k\geqslant1$) are the zeros of an entire function $L(\lambda)\in[\rho_1,0]$ ($1/\rho+1/\rho_1=1$), that $$ \lim_{k\to\infty}\frac1{|\lambda_k|^{\rho_1}}\ln\biggl|\frac1{L'(\lambda_k)}\biggr|=0 $$ and that the right-hand side of (1) is finite. It follows from this result that the indicator $h_F(\varphi)$ of $F(z)$ is determined by the moduli of the coefficients $a_k$. The equation \begin{equation} \sum_0^\infty c_k F^{(k)}(z)=f(z)\qquad\biggl(\sum_0^\infty c_k\lambda^k=L(\lambda)\biggr) \end{equation} is also considered. Let $0 and let $H(\varphi)r^\rho$ be a convex function of $z=re^{i\varphi}$. If $h_f(\varphi)\leqslant H(\varphi)$ ($h_f(\varphi)$ is the indicator of $f(z)$ for order $\rho$) then equation (2) has a solution with $h_F(\varphi)\leqslant H(\varphi)$. It is shown by using the results stated above that there are not always solutions of (2) satisfying the condition $h_F(\varphi)\leqslant h_f(\varphi)$. Bibliography: 8 titles.

[1] Leontev A. F., “Predstavlenie tselykh funktsii ryadami eksponent”, Tr. MIAN, 157 (1981), 68–89 | MR | Zbl

[2] Leontev A. F., “Predstavlenie tselykh funktsii ryadami po funktsiyam Mittag-Lefflera”, DAN SSSR, 1982, no. 264, 1313–1315 | MR | Zbl

[3] Leontev A. F., “Predstavlenie tselykh funktsii ryadami po funktsiyam Mittag-Lefflera”, Analysis Mathematica, 9 (1983), 177–205 | DOI | MR | Zbl

[4] Rakhimkulov N. I., Dostatochnye mnozhestva v nekotorykh prostranstvakh tselykh funktsii i ikh primenenie, Rukopis dep. v VINITI, No 3371-81 Dep.

[5] Leontev A. F., Obobscheniya ryadov eksponent, Nauka, M., 1981 | MR

[6] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966

[7] Leontev A. F., Ryady eksponent, Nauka, M., 1976 | MR

[8] Leontev A. F., Posledovatelnosti polinomov iz eksponent, Nauka, M., 1980 | MR