On the \(k\)th derivative of meromorphic functions with zeros of multiplicity at least \(k+1\) (Q947592)

The well known ``Hayman's alternative'' is the starting point of this paper. Namely, it is known that if \(f(z)\) is a transcendental meromorphic function in the plane, then either \(f(z)\) assumes every finite value infinitely often, or every derivative of \(f(z)\) assumes every finite nonzero value infinitely often. I state one of the main results. Let \(f(z)\) be a transcendental meromorphic function on \(\mathbb C\) of finite order \(\rho(f)\), and \(a(z)= P(z)\exp(Q(z))\not\equiv 0\), where \(P\) and \(Q\) are polynomials, let also \(k\geq 2\) be an integer. Suppose that (C1) all zeros of \(f\) have multiplicity at least \(k+1\), except possibly many, and (C2) \(\limsup_{r\to\infty} (\frac{T(r,a)}{T(r,f)}+ \frac{T(r,f)}{T(r,a)})= \infty\). Then the function \(f^{(k)}-a(z)\) has infinitely many zeros. Moreover, in the case that \(\rho(f)\in\mathbb N\), the result holds with condition (C2) only. The main tools in this paper are the Nevanlinna theory, Zalcman's lemma and Phragmen-Lindelöf principle.