We consider the family of transcendental entire maps given by $f_a\left(z\right)=a(z-(1-a))exp(z+a)$ where a is a complex parameter. Every map has a superattracting fixed point at z = -a and an asymptotic value at z = 0. For a > 1 the Julia set of $f_a$ is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing...

Using the notion of weighted sharing of values which was introduced by Lahiri (2001), we deal with the uniqueness problem for meromorphic functions when two certain types of nonlinear differential monomials namely $h^n{h}^{\left(k\right)}$$(h=f,...$

In the paper, dealing with a question of Lahiri (1999), we study the uniqueness of meromorphic functions in the case when two certain types of nonlinear differential polynomials, which are the derivatives of some typical linear expression, namely $h^n{(h-1)}^m$ ($h=f,g$), share a non-zero polynomial with finite weight. The results obtained in the paper improve, extend, supplement and generalize some recent results due to Sahoo (2013), Li and Gao (2010). In particular, we have shown that under a suitable choice of...

Dealing with a question of Lahiri [6] we study the uniqueness problem of meromorphic functions concerning two nonlinear differential polynomials sharing a small function. Our results will not only improve and supplement the results of Lin-Yi [16], Lahiri Sarkar [12] but also improve and supplement a very recent result of the first author [1].

We consider a transcendental meromorphic function f belonging to the class ℬ (with bounded set of singular values). We show that if the Julia set J(f) is the whole complex plane ℂ, and the closure of the postcritical set P(f) is contained in B(0,R) ∪ {∞} and is disjoint from the set Crit(f) of critical points, then every compact and forward invariant set is hyperbolic, provided that it is disjoint from Crit(f). It is further shown, under general additional hypotheses, that f admits no measurable...

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, $f\left(z\right)=a\iff f^{\left(k\right)}\left(z\right)=b$, and $f^{\left(k\right)}\left(z\right)=d\Rightarrow f\left(z\right)=c$, 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,....

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...

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.

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.

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, $f^{\left(k\right)}\ne 0$; (2) all zeros of $f^{\left(k\right)}-\psi$ have multiplicities at least (k+2)/k; (3) all poles of ψ have multiplicities at most k, then ℱ is normal in D.

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, $f^m+a\left(f^{\left(k\right)}\right)ⁿ-b$ has at most nk zeros in D, ignoring multiplicities, then ℱ is normal in D. The examples show that the result is sharp.