Zeros and fixed points of the linear combination of shifts of a meromorphic function (Q738482)

Let \(f\) be a meromorphic function in the complex plane. A linear combination of difference shifts of \(f\) is defined as follows: \[ g(z):=\sum\limits_{i=1}^ng_j(z)f(z+c_j),\quad G(z):=\frac{g(z)}{f(z)}\,. \] The authors investigate the existences of zeros and fixed points of linear combinations and quotients of shifts of \(f(z)\), when \(f(z)\) is of order one. The authors also prove a result which extends a result of Bergweiler and Langley. They prove: Let \(f\) be a transcendental meromorphic function in the plane with the order \(\sigma(f )=\sigma=1\). If \(f\) has finitely many poles and infinitely many zeros with the exponent of convergence of the zeros \(\lambda(f )=\lambda\neq1\), and the polynomial coefficients \(\{g_j(z)\}_{j=1}^n\) of \(g(z)\) satisfy \[ \text{deg}\,(g_l(z))>\max\limits_{1\leq j\leq n,\,j\neq l}\{\deg (g_j(z))\}\,, \] then \(g(z)\) has infinitely many zeros and fixed points. If \(f\) is of order less than or equal to one and \(c_l\neq0\), then \(G(z)\) is transcendental provided \(f\) satisfies any one of the following two conditions: (i) \(f\) is transcendental entire and the order of growth of \(f\) satisfies \(\liminf_{r\to+\infty}m(r, f )/r=0\); (ii) \(f\) is transcendental meromorphic and the order of growth of \(f\) satisfies \(\liminf_{r\to+\infty}T(r, f )/r^{1/2}=0\).