New function spaces,which generalize the classical Dirichlet space,BMOA or also the recently defined Qpspace,are introduced on Riemann surfaces.Except inclusions between these generalized spaces it is shown that the capacity Bloch space is a maximal space for them.
The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit diskD are generalized to real harmonic mappings of the unit disk,and the results are precise.It is proved that for a harmonic mapping U of D into the open interval I=(1,1),ΛU(z)/cosU(z)π/2≤4/π1/1|z|2 holds for z∈D,whereΛU(z)is the maximum dilation of U at z.The inequality is sharp for any z∈D and any value of U(z),and the equality occurs for some point in D if and only if U(z)=4πRe{arctan(z)},z∈D,with a Mbius transformation of D onto itself.
Let F be a family of functions meromorphic in a domain D,let n≥2 be a positive integer,and let a≠0,b be two finite complex numbers.If,for each f ∈F,all of whose zeros have multiplicity at least k+1,and f+a(f(k))n≠b in D,then F is normal in D.
Let F be a family of functions meromorphic in a domain D, let m, n k , k be three positive integers and b be a finite nonzero complex number. Suppose that, (1) for eachf∈F, all zeros of f have multiplicities at least k ; (2) for each pair of functions f, g ∈F,P(f)H(f) and P(g)H(g) share b, where P(f) and H(f) were defined as (1.1) and (1.2) and nk ≥ max 1≤i≤k-1 {n i }; (3) m ≥ 2 or nk ≥ 2, k ≥ 2, then F is normal in D.
