On the growth of entire solutions of linear differential equations (Q1605584)

Let \(Q\) be the class of positive continuous functions \(l\) on \([0,+\infty)\) such that \(l(x+O(1/l(x)))=O(l(x))\) as \(x\to+\infty\). For an entire function \(g(z)\) with zeros \(a_k\) denote by \(G_g(r)=\cup_k\{z\colon|z-a_k|\leq r/l(|a_k|)\}, r>0\). If \(l\in Q\), \(g_0,g_1,\dots,g_n,h\) are entire functions of bounded \(l\)-index and for every \(r>0\) there exists \(M=M(r)>0\) such that \(|g_j(z)|\leq Ml^j(|z|)|g_0(z)|\) for all \(z\in{\mathbb C}\backslash G_{g_0}(r)\), then an entire solution \(f\) to the equation \(g_0(z)w^{(n)}+g_1(z)w^{(n-1)}+\cdots+g_n(z)w=h(z)\) is of bounded \(l\)-index [\textit{A. D. Kuzyk} and \textit{M. N. Sheremeta}, Differ. Equations 26, No. 10, 1268-1273 (1990); translation from Differ. Uravn. 26, No. 10, 1716-1722 (1990; Zbl 0732.34006)]. Under some additional conditions on the coefficients of the equation and on the function \(l\), estimates on the growth of \(f\) are derived.