Shared Values and Normal Families of Meromorphic Functions

AbstractIn this paper, we study the normality of a family of meromorphic functions concerning shared values and prove the following theorem: Let F be a family of meromorphic functions in a domain D, let k≥2 be a positive integer, and let a, b, c be complex numbers such that a≠b. If, for each f∈F, f and f(k) share a and b in D, and the zeros of f(z)−c are of multiplicity ≥k+1, then F is normal in D

Lacunary series and Q

In this paper, we establish a necessary condition for a kind of lacunary series on the unit ball to be in QK. As a consequence, we prove a necessary and sufficient condition for that QK coincides with the Bloch space. In the case of Qp spaces we show that the condition, which is similar to that obtained by Hu, is also sufficient. This is a generalization of the result of Aulaskari, Xiao and Zhao for Qp spaces on the unit disk

NORMAL FUNCTIONS: L p ESTIMATES

ABSTRACT. For a meromorphic (or harmonic) function f, let us call the dilation of f at z the ratio of the (spherical) metric at f (z) and the (hyperbolic) metric at z. Inequalities are known which estimate the sup norm of the dilation in terms of its Lp norm, for p Ù 2, while capitalizing on the symmetries of f. In the present paper we weaken the hypothesis by showing that such estimates persist even if the Lp norms are taken only over the set of z on which f takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that p ≥ 2

Normal Families, Orders Of Zeros, And Omitted Values

. For k a positive integer,\Omega a region in the complex plane, and ff a complex number, let M k(\Omega ; ff) denote the collection of all functions f meromorphic in\Omega such that each zero of f \Gamma ff has multiplicity at least k . Let D denote the unit disk in the complex plane. We give three conditions, each of which is sufficient for a subset F of M k (D; ff) to be a normal family. These conditions are: (1) for each compact subset K of D and for some fi ? 0 there exists a constant MK (fi) (depending on both K and fi ) such that, for each f 2 F , fz 2 K : jf(z)j ! fig ae fz 2 K : jf (k) (z)j MK (fi)g; (2) for ? 2=k and for each compact subset K of D there exists a constant C K; (depending on both K and ) such that ZZ fz2K:jf(z)j!1g jf (k) (z)j dx dy ! C K; for each f 2 F ; (3) for each compact subset K of D there is a constant M k (K) such that the product of the spherical derivatives of two or three consecutive derivatives of f , up to the derivative of order k \..