106 Thu Hien Nguyen

Thu Hien Nguyen (Leipzig University, Germany & V. N. Karazin Kharkiv National University, Ukraine)// Some results on entire functions from the Laguerre-Pólya class: proof ideas and techniques // Abstract: The Laguerre-Pólya class is a class of entire functions that are locally the uniform limit of a sequence of real polynomials that have only real zeros. We present some simple necessary and sufficient conditions for entire functions to belong to the Laguerre-Pólya class in terms of their Taylor coefficients. For an entire function f(z) = \sum_{k=0}^{\infty} a_k z^k, we define the second quotients of Taylor coefficients as q_n(f) := \frac{a_{n-1}^2}{a_{n-2} a_{n}}, n\geq 2, and find conditions on q_n(f) for f to belong to the Laguerre-Pólya class, or to have only real zeros. In this talk, we will focus on the entire functions with increasing second quotients of Taylor coefficients, and discuss proof ideas and techniques we used. // This is joint work with Anna Vishnyakova.