Picard type theorems concerning certain small functions (Q1682278)

\textit{W. K. Hayman} [Ann. Math. (2) 70, 9--42 (1959; Zbl 0088.28505)] proved the following result: Let \(k\) be a positive integer, \(\alpha\) be a finite nonzero complex number and \(f\) be a transcendental meromorphic function in the complex plane. If \(f\) has no zero, then \(f^{(k)} - \alpha\) has infinitely many zeros in the complex plane. Several extensions and generalizations of this result are available in the literature. In the paper, the following Hayman-type theorem is proved: Let \(f\) be a meromorphic function in the complex plane, whose zeros have multiplicities at least \(k + 1\) (\(k \geq 2\)). If \(\sin z\) is a small function with respect to \(f\), then \(f^{(k)}(z) - P(z)\sin z\) has infinitely many zeros in the complex plane, where \(P\) is a nonzero nonlinear polynomial.