The Laguerre inequality and the distribution of zeros of entire functions
Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 2, p. 331-345

The Laguerre inequality and the distribution of zeros of real entire functions are investigated with the aid of certain infinite-order differential operators. The paper includes new proofs, problems, conjectures and many illustrative examples and counterexamples.

@article{AMBP_2005__12_2_331_0,
     author = {Csordas, George and Escassut, Alain},
     title = {The Laguerre inequality and the distribution of zeros of entire functions},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {12},
     number = {2},
     year = {2005},
     pages = {331-345},
     doi = {10.5802/ambp.210},
     zbl = {1096.30023},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2005__12_2_331_0}
}
Csordas, George; Escassut, Alain. The Laguerre inequality and the distribution of zeros of entire functions. Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 2, pp. 331-345. doi : 10.5802/ambp.210. http://www.numdam.org/item/AMBP_2005__12_2_331_0/

[1] Boas, R.P. Entire Functions, Academic Press, New York (1954) | MR 68627 | Zbl 0058.30201

[2] Cardon, D. A.; De Gaston, S. A. Differential operators and entire functions with simple real zeros, J. Math. Anal. Appl., Tome 301 (2005), pp. 286-393 | Article | MR 2105679 | Zbl 1066.30031

[3] Craven, T.; Csordas, G. Differential operators of infinite-order and the distribution of zeros of entire functions, J. Math. Anal. Appl., Tome 186 (1994), pp. 799-820 | Article | MR 1293856 | Zbl 0810.30006

[4] Csordas, G.; Varga, R. S. Necessary and sufficient conditions and the Riemann Hypothesis, Adv. in Appl. Math., Tome 11 (1990), pp. 328-357 | Article | MR 1061423 | Zbl 0707.11062

[5] De Bruijn, N.G. The roots of Trigonometric integrals, Duke Math, Tome 7 (1950), pp. 197-226 | Article | MR 37351 | Zbl 0038.23302

[6] G. Csordas, W. Smith; Varga, R. S. Level sets of real entire functions and the Laguerre inequalities, Analysis, Tome 12 (1992), pp. 377-402 | MR 1182637 | Zbl 0759.30016

[7] Jensen, J. L. W. V. Recherches sur la théorie des équations, Acta Math., Tome 36 (1913), pp. 181-195 | Article | JFM 43.0158.01 | MR 1555086

[8] Kuipers, L. Note on the location of zeros of polynomials, Simon Stevin, Tome 31 (1957), pp. 61-72 | MR 83970 | Zbl 0082.02203

[9] Levin, B. Ja. Distribution of Zeros of Entire Functions, Transl. Math. Mono. Vol 5. Amer. Math. Soc, Providence, R.I. (1980) | MR 589888

[10] Obreschkoff, N. Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin (1963) | MR 164003 | Zbl 0156.28202

[11] Pólya, G. Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Klg. Danske Vid. Sel. Math.-Fys. Medd, Tome 7 (1927), pp. 3-33

[12] Pólya, G. Location of Zeros, Collected papers, Vol.2, R.P. Boas, MIT Press, Cambridge, MA (1974) | MR 505093

[13] Sikkema, P. C. Differential Operators and Differential Equations of Infinite Order with Constant Coefficients, P. Noordhoff N.V., Groningen (1953) | MR 60082 | Zbl 0051.34402