We characterize the boundedness and compactness of weighted composition operators between weighted Banach spaces of analytic functions and . we estimate the essential norm of a weighted composition operator and compute it for those Banach spaces which are isomorphic to c0. We also show that, when such an operator is not compact, it is an isomorphism on a subspace isomorphic to c0 or l∞. Finally, we apply these results to study composition operators between Bloch type spaces and little Bloch type spaces.

We investigate the asymptotics and zero distribution of solutions of ω″ + Aω = 0 where A is an entire function of very slow growth. The results parallel the classical case when A is assumed to be a polynomial.

Chen and Gu [1] have given some results relating to normal families, and, in this paper, we give versions of these results valid for normal functions. In the process, we improve some of our previous results involving products of certain spherical derivatives as they relate to normal functions. Some examples are given to show the sharpness of our results.

Normal functions and Bloch functions are respectively functions of bounded spherical expansion and bounded Euclidean expansion. In this paper we discuss the behaviour of normal functions and of Bloch functions in terms of the maximal ideal space of H∞, the Bergman projection and the Ahlfors-Shimizu characteristic.

Pommerenke initiated the study of linearly invariant families of locally schlicht holomorphic functions defined on the unit disk The concept of linear invariance has proved fruitful in geometric function theory. One aspect of Pommerenke's work is the extension of certain results from classical univalent function theory to linearly invariant functions. We propose a definition of a related concept that we call hyperbolic linear invariance for locally schlicht holomorphic functions that map the unit disk into itself. We obtain results for hyperbolic linearly invariant functions which generalize parts of the theory of bounded univalent functions. There are many similarities between linearly invariant functions and hyperbolic linearly invariant functions, but some new phenomena also arise in the study of hyperbolic linearly invariant functions.

In this article we localize the zeros of some polynomials and the derivatives of some entire functions of finite genus. If we put m = 1 in the condition of Theorem 1 we obtain the famous Obreshkoff Theorem which can be regarded as a ‘complex version’ of a well-known theorem due to Laguerre. The nonreal zeros of the derivative of the real entire funtion of Theorem 3 must belong to circles Vk which are similar to the Jen circles for polynomials.

We examine what can be said about a polynomial p and an entire function f given that p o f is an even, or an odd, function.

Let f be a transcendental entire function and denote the n-th iterate fn. For n ≥ 2, we give an explict estimate of the number of periodic points of f with period n, that is, fix-points of fn which are not fix-points of fk for 1 ≤ k <n.

In this paper we prove a number of results on Cauchy transforms of generalized type given by Borel measures supported on the class of analytic functions mapping the unit disc into the unit disk. We also give a BMOA characterization using these families.

In this paper we treat transcendental meromorphic solutions of some algebraic differential equations. We consider the number of distinct transcendental meromorphic solutions. Algebraic relations between meromorphic solutions and comparisons of the growth of transcendental meromorphic solutions are also discussed.

In this paper, we obtain some normality criteria for families of meromorphic functions that concern the exceptional functions of derivatives, which improve and generalize related results of Gu, Yang, Schwick, Wang-Fang, and Pang-Zalcman. Some examples are given to show the sharpness of our results.

We introduce a sequence of polynomials which are extensions of the classic Bernoulli polynomials. This generalization is obtained by using the Bessel functions of the first kind. We use these polynomials to evaluate explicitly a general class of series containing an entire function of exponential type.

A point of departure for this paper is the famous theorem of Hermite and Biehler: If f (z) is a polynomial with complex coefficients ak and its zeros zk satisfy Im Zk < 0, then the polynomials with coefficients Re ak, and Im ak have only real zeros.

We generalize this theorem for some entire functions. The entire functions in Theorem 2 and Theorem 3 are of first and second genus respectively.

In this paper we give the definition of a meromorphic function which is geometrically finite and investigate some properties of geometrically finite meromorphic functions and the Lebesgue measure of their Julia sets.

We generalise the classical Bernstein's inequality: . Moreover we obtain a new representation formula for entire functions of exponential type.

In this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

Suppose that f is meromorphic in the plane, and that there is a sequence Zn → ∞ and a sequence of positive numbers ∈n → 0, such that ∈n|zn|f#(zn)/log|zn| → ∞. It is shown that if f is analytic and non-zero in the closed discs Δn = {z: |z – zn| ≦∈n|zn|}, n = 1, 2, 3 …, then, given any positive integer K, there are arbitrarily large values of n and there is a point z in Δn such that │f (z)| 〉 |Z│k. Examples are given to show that the hypotheses cannot be relaxed.

Let R(z, w) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation possesses an admissible solution, then , where αj, are distinct complex constants. In particular, when R(z, w) is independent of z, it is shown that if (*) possesses an admissible solution w(z), then by some Möbius transformation u = (aw + b) / (cw + d) (ad – bc ≠ 0), the equation can be reduced to one of the following forms: where τj (j = 1, … 4) are distinct constants, and σj (j = 1, … 4) are constants, not necessarily distinct.

Let k be a positive integer and b a nonzero constant. Suppose that F is a family of meromorphic functions in a domain D. If each function f ∈ F has only zeros of multiplicity at least k + 2 and for any two functions f, g ∈ F, f and g share 0 in D and f(k) and g(k) share b in D, then F is normal in D. The case f ≠ 0, f(k) ≠ b is a celebrated result of Gu.

In this paper we continue our previous studies and derive all possible expressions for a meromorphic function and its differential polunomials when they share two finite distinct values a1, a2, CM (counting multiplicities) in majority.