The growth of solutions of higher order differential equations (Q1885179)

Considering linear differential equations of type \[ f''+e^{-z}f'+Q(z)f=0,\tag{1} \] where \(Q(z)\) is an entire function of finite order, it is well known that all solutions of (1) are entire functions. Moreover, if \(f_1,f_2\) are linearly independent, then \(\max(\rho(f_1),\rho(f_2))=\infty\). The question about conditions on \(Q(z)\) to ensure that every nontrivial solution of (1) is of infinite order has been in a number of papers recently, see the reference list of the present paper. In particular, see [\textit{Z. X. Chen}, Science in China (Ser.~A) 31, 775--784 (2001)], given distinct complex numbers \(a,b\), if \(Q(z)=h(z)e^{bz}\), \(h(z)\) a nonvanishing polynomial, then every nontrivial solution of \(f''+e^{az}f'+Q(z)f=0\) has hyperorder \(\rho_2(f)=1\), hence being of infinite order. In this paper, the same conclusion will be proved for nontrivial solutions of \[ f^{(k)}+h_{k-1}(z)e^{a_{k-1}z}f^{(k-1)}+\dots+h_1(z)e^{a_1z}f'+h_0(z)e^{a_0z}f=0 \] with \(h_0\not\equiv0\), \(h_1,\dots,h_{k-1}\) entire of order \(\rho(h_j)<1\) and \(a_0\neq0\), \(a_1,\dots,a_{k-1}\) complex constants such that either \(a_j=0\), or \(\arg a_j\neq\arg a_0\), or \(\arg a_j=\arg a_0\), if \(a_j=c_ja_0\), \(0<c_j<1\), \(j=1,\dots,k-1\). This assertion, as well as a related further result, will be proved by using a number of well known standard lemmas.