Survey on p-adic meromorphic functions sharing five small ones on a work by Ta Thi Hoai An and Nguyen Viet Phuong, with some additional properties

Abstract

Let [Formula: see text] be a complete ultrametric algebraically closed field of characteristic 0, let D be the open disk [Formula: see text] and let [Formula: see text]. Let [Formula: see text] be two meromorphic functions in [Formula: see text] (respectively, two unbounded meromorphic functions in D, respectively, two meromorphic functions in E) having infinitely many zeros or poles in E sharing five small meromorphic functions in the same set (ignoring multiplicity). Then [Formula: see text]. Moreover, if f and g have finitely many poles in [Formula: see text] (respectively, in D, respectively, in E), and share three small functions, (ignoring multiplicity), then [Formula: see text]. We define archi-branched small functions and show that a meromorphic function f (in [Formula: see text], D, or E) cannot have five archi-branched small functions.