Geodesic
Sufficient sets in a certain space of entire functions
Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 389-400 
R. S. Yulmukhametov. Sufficient sets in a certain space of entire functions. Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 389-400. http://geodesic.mathdoc.fr/item/SM_1983_44_3_a8/
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     author = {R. S. Yulmukhametov},
     title = {Sufficient sets in a~certain space of entire functions},
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Voir la notice de l'article provenant de la source Math-Net.Ru

For any trigonometrically convex function $h(\varphi)$ an entire function $L(z)$ is constructed, satisfying the relation $$ \ln|L(re^{i\varphi})|=h(\varphi)r+O(r^{1/2}\ln r),\qquad re^{i\varphi}\notin\Omega(a_n), $$ where the $a_n$ are the zeros of $L(z)$ and $\Omega(a_n)=\{z:|z-a_n|\leqslant1\}$. The set of zeros of such a function is sufficient in the space of entire functions $F(z)$ satisfying $$ \sup_{r,\varphi}\frac{\ln|F(re^{i\varphi})|}{h(\varphi)r-r^{q+\varepsilon}}<\infty $$ for some $\varepsilon>0$, where $q\in(1/2,1)$ is a parameter of the space. Bibliography: 5 titles.

[1] Napalkov V. V., “O diskretnykh dostatochnykh mnozhestvakh v nekotorykh prostranstvakh tselykh funktsii”, DAN SSSR, 250:4 (1980), 809–812 | MR | Zbl

[2] Levin B. Ya., Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956

[3] Yulmukhametov R. S., “Prostranstva analiticheskikh funktsii, imeyuschikh zadannyi rost vblizi granitsy”, Matem. zametki, 32:1 (1982), 41–57 | MR | Zbl

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

[5] Melnik Yu. I., “O predstavlenii regulyarnykh funktsii ryadami Dirikhle v zamknutom kruge”, Matem. sb., 97 (139) (1975), 493–501 | MR