Given \(A(z)\) entire, it is well known that all solutions of \[ y^{(k)}+A(z)y=0 \] are entire functions. The main result in the present paper is the following theorem: Suppose \(\rho(A)<1/2\), \(k\geq2\) and \(y^{(k)}+A(z)y=0\) has a solution \(f\) whose zero-sequence has exponent of convergence \(\lambda(f)<\rho(A)\). Given now \(A_1(z)=A(z)+h(z)\), where \(h\) is a non-vanishing entire function of order \(\rho(h)<\rho(A)\), then the equation \(y^{(k)}+A_1(z)y=0\) has no solution \(g\) such that \(\lambda(g)<\rho(A)\). The main ingredients to prove this theorem are Nevanlinna theory, the \(\cos\pi\rho\)-theorem and the fact that the logarithmic derivative of a meromorphic function of finite order is small in a large set.