Normality Criteria for Families of Meromorphic Function Concerning Shared Values

Let k be a positive integer and let ₰ be a family of meromorphic functions in the plane domain D all of whose zeros with multiplicity at least k. Let P = apz^p+... +a2z^2+z be a polynomial, ap;a2 ≠ 0 and p = deg(P) ≥ k+2. If, for each f , g ϵ ₰ , P( f )G( f ) and P(g)G(g) share a non-zero constant b in D, where G( f ) = f^ (k) +H( f ) be a differential polynomial of f satisfying w/deg │H ≤ k/ l+1 +1 or w(H)-deg(H) k, then ₰ is normal in D.