Geodesic
Entire functions with preassigned zero proximate order
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 141-153

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It is known that if the proximate order $\rho(r)$ is such that $\lim\rho(r)=\rho>0$ ($r\to\infty$), then there exists an entire function $f(z)$ of proximate order $\rho(r)$. In the case where $\rho=0$ the question about the existence of such an entire function has remained open until now. This question is investigated in the paper. 
@article{ZNSL_2014_424_a3,
     author = {A. F. Grishin and Nguyen Van Quynh},
     title = {Entire functions with preassigned zero proximate order},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {141--153},
     publisher = {mathdoc},
     volume = {424},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a3/}
}
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A. F. Grishin; Nguyen Van Quynh. Entire functions with preassigned zero proximate order. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 141-153. http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a3/