A well-known result of Lappan states that a meromorphic function f in the unit disc $$\mathbb {D}$$ D is normal if and only if there is a subset $$E\subset \widehat{\mathbb {C}}$$ E ⊂ C ^ consisting of five points such that $$\sup \{(1-|z|^2) f^{\#}(z): z \in f^{-1}(E)\} < \infty ,$$ sup { ( 1 - | z | 2 ) f # ( z ) : z ∈ f - 1 ( E ) } < ∞ , where $$f^{\#}(z)$$ f # ( z ) is the spherical derivative of f at z. An analogous result for normal families is due to Hinkkanen and Lappan: a family $$\mathcal F$$ F of meromorphic functions in a domain $$D\subset \mathbb {C}$$ D ⊂ C is normal if and only if for each compact subset $$K\subset D,$$ K ⊂ D , there are a subset $$E\subset \widehat{\mathbb {C}}$$ E ⊂ C ^ consisting of five points and a positive constant M such that $$\sup \{f^{\#}(z): f\in \mathcal F, z \in f^{-1}(E)\}<M.$$ sup { f # ( z ) : f ∈ F , z ∈ f - 1 ( E ) } < M . In this paper, we extend the above-mentioned results to the case where the set E contains fewer points. In particular, the Pang–Zalcman’s theorem on normality of a family $$\mathcal F$$ F of holomorphic functions f in a domain D, $$f^nf^{(k)}(z)\ne a$$ f n f ( k ) ( z ) ≠ a (for some given constant a), is also extended to the case where the spherical derivative of $$f^nf^{(k)}$$ f n f ( k ) is bounded on the zero set of $$f^nf^{(k)}-a$$ f n f ( k ) - a .