Properties of entire solutions of differential equations

Abstract

We investigate the close-to-convexity and l-index boundedness of entire solutions of the differential equations z 2 w + βzw + (γz 2 − β)w = 0 and zw + βw + γzw = 0.

Properties of analytic solutions of a differential equation

“…S. Shah studied the cases when this two-term recurrent formula reduces to a one-term. For example, he proved that if β 1 + γ 2 = 0, −1 ≤ β 0 < 0, β 1 > 0, γ 0 = 0 and −β 1 /2 < γ 1 ≤ 0, then equation 3has an entire solution (2) such that all the derivatives g 3with two-term recurrent formula for the coefficients are continued in papers [6][7][8][9][10], and in [11][12][13][14] this results are generalized for linear differential equation of n-th order with polynomial coefficients of the n-th degree.…”

“…If G ⊂ D R and there exists N ∈ Z + such that inequality (4) holds for all n ∈ Z + and z ∈ G f is said to be function of bounded l-index on (or in) G, and l-index is denoted by N (f, l; G). Boundedness of l-index of an entire solutions of differential equation 3are studied in papers [16][17][18][19][20][21].…”

Властивості Аналітичних Розв'язків Трьох Подібних Диференціальних Рівнянь Другого Порядку

“…If there exists N ∈ Z + such that (5) holds for all n ∈ Z + and for all z ∈ G ⊂ D R , then the function f is said to be of bounded l-index on (or in) G, and the l-index is denoted by N( f ; l, G). The l-index boundedness of entire solutions of the equation (3) for certain conditions on parameters β 0 , β 1 , γ 0 , γ 1 , γ 2 are studied in [19][20][21][22][23][24].…”

“…Some results from [13][14][15][16][17][18][19][20][21][22][23][24] are published also in monograph [10].…”

On the l-index boundedness of some composition of functions

“…For an entire function f the function F (z) = f ( q (1−z) n ), n ∈ N, is of bounded l-index with l(|z|) = β (1−|z|) n+1 , β > 1, if and only if f is of bounded index. Finally, we remark that there is a number of works [15][16][17] devoted to entire solutions of the differential equation (5). Their main results are estimates of the index with additional conditions on the parameters.…”