Location of zeros Part I: Real polynomials and entire functions

Inequalities for products of zeros of polynomials and entire functions

Estimates for products of the zeros of polynomials and entire functions are derived. By these estimates, new upper bounds for the counting function are suggested. In appropriate situations we improve the Jensen inequality for the counting functions and the Mignotte inequality for products of the zeros of polynomials. Mathematics subject classification (2010): 26C10, 30C15, 30D20.

Conformal Mappings And Entire Functions With Real Zeros

Differential operators and entire functions with simple real zeros

Let φ and f be functions in the Laguerre–Pólya class. Write φ(z) = e−αzφ1(z) and f (z) = e−βzf1(z), where φ1 and f1 have genus 0 or 1 and α,β 0. If αβ < 1/4 and φ has infinitely many zeros, then φ(D)f (z) has only simple real zeros, where D denotes differentiation.  2004 Elsevier Inc. All rights reserved.

Convolution Operators and Zeros of Entire Functions

Let G(z) be a real entire function of order less than 2 with only real zeros. Then we classify certain distributions functions F such that the convolution (G ∗ dF )(z) = ∫∞ −∞G(z − is) dF (s) has only real zeros.

Zeros of differential polynomials in real meromorphic functions

We investigate when differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if g is a real transcendental meromorphic function, c ∈ R \ {0} and n ≥ 3 is an integer, then g′gn − c has infinitely many non-real zeros. If g has only finitely many poles, then this holds for n ≥ 2. Related results for rational functions g are also considered.