A note on partial sharing of values of meromorphic functions with their shifts

We introduce the notion of partial value sharing as follows. Let E‾(a,f) be the set of zeros of f(z)−a(z)f(z)−a(z), where each zero is counted only once and a is a meromorphic function, small with respect to f. A meromorphic function f is said to share a partially with a meromorphic function g   if E‾(a,f)⊆E‾(a,g). We show that partial value sharing of f(z)f(z) and f(z+c)f(z+c) involving 3 or 4 values combined with an appropriate deficiency assumption is enough to guarantee that f(z)≡f(z+c)f(z)≡f(z+c), provided that f(z)f(z) is a meromorphic function of hyper-order strictly less than one and c∈Cc∈C.