Geodesic
On the Zeros of Laplace Transforms
Matematičeskie zametki, Tome 76 (2004) no. 6, pp. 883-892 
A. M. Sedletskii. On the Zeros of Laplace Transforms. Matematičeskie zametki, Tome 76 (2004) no. 6, pp. 883-892. http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a8/
@article{MZM_2004_76_6_a8,
     author = {A. M. Sedletskii},
     title = {On the {Zeros} of {Laplace} {Transforms}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {883--892},
     year = {2004},
     volume = {76},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a8/}
}
TY  - JOUR
AU  - A. M. Sedletskii
TI  - On the Zeros of Laplace Transforms
JO  - Matematičeskie zametki
PY  - 2004
SP  - 883
EP  - 892
VL  - 76
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a8/
LA  - ru
ID  - MZM_2004_76_6_a8
ER  - 
%0 Journal Article
%A A. M. Sedletskii
%T On the Zeros of Laplace Transforms
%J Matematičeskie zametki
%D 2004
%P 883-892
%V 76
%N 6
%U http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a8/
%G ru
%F MZM_2004_76_6_a8

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

Suppose that $f$ is a positive, nondecreasing, and integrable function in the interval $(0,1)$. Then, by Pólya's theorem, all the zeros of the Laplace transform $$ F(z)=\int_0^1e^{zt}f(t)\,dt $$ lie in the left-hand half-plane $\operatorname{Re} z\le0$. In this paper, we assume that the additional condition of logarithmic convexity of $f$ in a left-hand neighborhood of the point $1$ is satisfied. We obtain the form of the left curvilinear half-plane and also, under the condition $f(+0)>0$, the form of the curvilinear strip containing all the zeros of $F(z)$.

[1] Hardy G. H, “On the zeroes of certain classes of integral Taylor series, II”, Proc. London Math. Soc. Ser. 2, 2 (1905), 401–431 | DOI

[2] Polya G., “Über die Nullstellen gewisser ganzer Funktionen”, Math. Z., 2 (1918), 352–383 | DOI | MR

[3] Titchmarsh E. C., “The zeros of certain integral functions”, Proc. London Math. Soc. Ser. 2, 25 (1926), 283–302 | DOI | MR

[4] Cartwright M., “The zeros of certain integral functions”, Quart. J. Math., 1 (1930), 38–59 | DOI

[5] Cartwright M., “The zeros of certain integral functions, II”, Quart. J. Math., 2 (1931), 113–129 | DOI | Zbl

[6] Khardi G. G., Littlvud D. E., Polia G., Neravenstva, IL, M., 1948

[7] Khardi G. G., Rogozinskii V. V., Ryady Fure, Fizmatgiz, M., 1959

[8] Levin B. Ya., Tselye funktsii, Kurs lektsii, MGU, M., 1971

[9] Sedletskii A. M., “On zeros of Laplace transforms of finite measure”, Integral Transforms and Special Functions, 1 (1993), 51–59 | DOI | MR

[10] Seneta E., Pravilno menyayuschiesya funktsii, Nauka, M., 1985 | Zbl

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

[12] Sedletskii A. M., “O nulyakh funktsii tipa Mittag-Lefflera”, Matem. zametki, 68:5 (2000), 710–724 | MR