Sums of Zeros of Solutions to Second Order Differential Equations with Polynomial Coefficients
نویسندگان
چکیده
We consider the equation u′′ = P (z)u, where P (z) is a polynomial. Let zk(u), k = 1, 2, . . . be the zeros of a solution u(z) to that equation. Inequalities for the sums ∑j k=1 1 |zk(u)| (j = 1, 2, . . .) are derived. They considerably improve the previous result of the author. Some applications of the obtained bounds are also discussed. An illustrative example is presented. It shows that the suggested results are sharp.
منابع مشابه
Entire Functions of Finite Order as Solutions to Certain Complex Linear Differential Equations
When is an entire function of finite order a solution to a complex 2nd order homogeneous linear differential equation with polynomial coefficients? In this paper we will give two (equivalent) answers to this question. The starting point of both answers is the Hadamard product representation of a given entire function of finite order. While the first answer involves certain Stieltjes-like relati...
متن کاملUnique Continuation and Complexity of Solutions to Parabolic Partial Differential Equations with Gevrey Coefficients
In this paper, we provide a quantitative estimate of unique continuation (doubling property) for higher-order parabolic partial differential equations with non-analytic Gevrey coefficients. Also, a new upper bound is given on the number of zeros for the solutions with a polynomial dependence on the coefficients.
متن کاملPolynomial and non-polynomial solutions set for wave equation with using Lie point symmetries
This paper obtains the exact solutions of the wave equation as a second-order partial differential equation (PDE). We are going to calculate polynomial and non-polynomial exact solutions by using Lie point symmetry. We demonstrate the generation of such polynomial through the medium of the group theoretical properties of the equation. A generalized procedure for polynomial solution is pr...
متن کاملSums of Zeros of Solutions to Second Order Ode with Non-polynomial Coefficients
We consider the equation y′′ = F (z)y (z ∈ C) with an entire function F satisfying the condition |F (z)| ≤ A exp ` |z|ρ ρ ́ (ρ ≥ 1, A = const > 0). Let zk(y), k = 1, 2, . . . be the zeros of a solution y(z) to the above equation. Bounds for the sums j X k=1 1 |zk(y)| (j = 1, 2, . . . ) are established. Some applications of these bounds are also considered.
متن کاملBOUNDS FOR THE SUMS OF ZEROS OF SOLUTIONS OF u(m) = P (z)u WHERE P IS A POLYNOMIAL
The main purpose of this paper is to consider the differential equation u = P (z)u (m ≥ 2) where P is a polynomial with complex, in general, coefficients. Let zk(u), k = 1, 2, . . . be the zeros of a nonzero solution u to that equation. We obtain bounds for the sums j
متن کامل