Turán Inequalities and Subtraction-free Expressions (original) (raw)

Journal of Inequalities and Applications, 2006

We prove Turán-type inequalities for some special functions by using a generalization of the Schwarz inequality.

Turan type inequalities for rational functions with prescribed poles

International Journal of Nonlinear Analysis and Applications, 2022

In this paper, we establish some inequalities for rational functions with prescribed poles having t-fold zeros at the origin. The estimates obtained generalise as well as refine some known results for rational functions and in turn, produce extensions of some polynomial inequalities earlier proved by Turan, Jain etc.

ITERATED LAGUERRE AND TURÁN INEQUALITIES

Journal of Inequalities in Pure and Applied Mathematics

New inequalities are investigated for real entire functions in the Laguerre-Pólya class. These are generalizations of the classical Turán and Laguerre inequalities. They provide necessary conditions for certain real entire functions to have only real zeros.

On an inequality of Paul Turan concerning polynomials

Lobachevskii Journal of Mathematics, 2016

Let P (z) be a polynomial of degree n and P s (z) be its sth derivative. In this paper, we shall prove some inequalities for the sth derivative of a polynomial having zeros inside a circle, which as a special case give generalizations and refinements of some results of Turan, Govil, Malik and others.

Higher order Turán inequalities for the partition function

Transactions of the American Mathematical Society, 2018

The Turán inequalities and the higher order Turán inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. A real sequence {a n } is said to satisfy the Turán inequalities if for n ≥ 1, a 2 n − a n−1 a n+1 ≥ 0. It is said to satisfy the higher order Turán inequalities if for n ≥ 1, 4(a 2 n − a n−1 a n+1)(a 2 n+1 − a n a n+2) − (a n a n+1 − a n−1 a n+2) 2 ≥ 0. A sequence satisfying the Turán inequalities is also called log-concave. For the partition function p(n), DeSalvo and Pak showed that for n > 25, the sequence {p(n)} n>25 is log-concave, that is, p(n) 2 − p(n − 1)p(n + 1) > 0 for n > 25. It was conjectured by Chen that p(n) satisfies the higher order Turán inequalities for n ≥ 95. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for p(n + 1)p(n − 1)/p(n) 2. Consequently, for n ≥ 95, the Jensen polynomials g 3,n−1 (x) = p(n − 1) + 3p(n)x + 3p(n + 1)x 2 + p(n + 2)x 3 have only real zeros. We conjecture that for any positive integer m ≥ 4 there exists an integer N (m) such that for n ≥ N (m), the polynomials m k=0 m k p(n + k)x k have only real zeros. This conjecture was independently posed by Ono.

Polynomial inequalities and universal Taylor series

Mathematische Zeitschrift, 2016

We derive several new properties concerning both universal Taylor series and Fekete universal series from classical polynomial inequalities. In particular, we study some density properties of their approximating subsequences. Moreover we exhibit summability methods which preserve or imply the universality of Taylor series in the complex plane. Likewise we show that the partial sums of the Taylor expansion around zero of a C ∞ function is universal if and only if the sequence of its Cesàro means satisfies the same universal approximation property.

The Riemann Hypothesis and the Turan Inequalities

Transactions of the American Mathematical Society, 1986

A solution is given to a fifty-eight year-old open problem of G Pólya, involving the Taylor coefficients of the Riemann ¿-function. On setting z = -x2 in (1.4), the function F(z), defined by 00 h zm (1.6) F(z):= £ 0(2m)!'

On inequalities for zeros of entire functions

Proceedings of the American Mathematical Society, 2004

We derive inequalities for zeros of an entire function of finite order in terms of the coefficients of its Taylor series. Our results are new even for polynomials.