Normal families and shared values of meromorphic functions.
Normal families and shared values of meromorphic functions
Let ℱ be a family of meromorphic functions on a plane domain D, all of whose zeros are of multiplicity at least k ≥ 2. Let a, b, c, and d be complex numbers such that d ≠ b,0 and c ≠ a. If, for each f ∈ ℱ, , and , then ℱ is a normal family on D. The same result holds for k=1 so long as b≠(m+1)d, m=1,2,....
Normal families of bicomplex meromorphic functions
We introduce the extended bicomplex plane 𝕋̅, its geometric model: the bicomplex Riemann sphere, and the bicomplex chordal metric that enables us to talk about convergence of sequences of bicomplex meromorphic functions. Hence the concept of normality of a family of bicomplex meromorphic functions on bicomplex domains emerges. Besides obtaining a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the fundamental normality tests for families...
Normal families, orders of zeros, and omitted values.
Normal functions and normal families.
Normality and shared sets of meromorphic functions
The purpose of this paper is to investigate the normal families and shared sets of meromorphic functions. The results obtained complement the related results due to Fang, Liu and Pang.
Normality and value sharing with a linear differential polynomial
We prove some normality criteria for a family of meromorphic functions and as an application we prove a value distribution theorem for a differential polynomial.
Normality criteria and multiple values II
Let ℱ be a family of meromorphic functions defined in a domain D, let ψ (≢ 0, ∞) be a meromorphic function in D, and k be a positive integer. If, for every f ∈ ℱ and z ∈ D, (1) f≠ 0, ; (2) all zeros of have multiplicities at least (k+2)/k; (3) all poles of ψ have multiplicities at most k, then ℱ is normal in D.
Normality criteria for families of zero-free meromorphic functions
Let ℱ be a family of zero-free meromorphic functions in a domain D, let n, k and m be positive integers with n ≥ m+1, and let a ≠ 0 and b be finite complex numbers. If for each f ∈ ℱ, has at most nk zeros in D, ignoring multiplicities, then ℱ is normal in D. The examples show that the result is sharp.
Normality criteria of Lahiri's type and their applications.
On Bloch functions and gap series.
On Bloch functions and gap series.
Kennedy obtained sharp estimates of the growth of the Nevanlinna characteristic of the derivative of a function f analytic and with bounded characteristic in the unit disc. Actually, Kennedy's results are sharp even for VMOA functions. It is well known that any BMOA function is a Bloch function and any VMOA function belongs to the little Bloch space. In this paper we study the possibility of extending Kennedy's results to certain classes of Bloch functions. Also, we prove some more general results...
On Bloch functions and normal functions.
On Entire Functions with Infinite Domains of Normality.
On Extreme Bloch Functions with Prescribed Critical Points.
On groups and normal polymorphic functions.
On Normal Meromorphic Functions.
On spaces and pseudoanalytic extension.
On radial behaviour and balanced Bloch functions.
A Bloch function g is a function analytic in the unit disk such that (1 - |z|2) |g' (z)| is bounded. First we generalize the theorem of Rohde that, for every bad Bloch function, g(rζ) (r → 1) follows any prescribed curve at a bounded distance for ζ in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that |g'(z)| does not vary much on each circle {|z| = r} except for small exceptional arcs. We show e.g. that∫01 |g'(rζ)|dr <...
On rotation-automorphic functions.