Normal families of meromorphic functions concerning shared values

Let a(≠ 0), b ∈ ℂ and k ∈ ℤ+;. Let ℱ be a family of meromorphic functions in a domain D such that each f in ℱ has only zeros of multiplicity at least k + 2. For each f in ℱ, if f + a(f(k))k+1-b has at most 1 zero in D, then ℱ is normal in D. We also give some examples to illustrate that some restricted conditions of our main results are necessary.