On an Area Inequality and Weighted Integrals of Analytic Functions

In this note we investigate the relationship between the following integrals: $$\int_{U}\mid f^{(n)}(z)\mid^{p}\mid f^{(k)}(z)\mid^{q}(1-\mid z\mid )^{np+kq+\alpha}dm\ \ \ {\rm and}\ \ \ \int_{U}\mid f^{\prime}\mid^{p+q}(1-\mid z\mid)^{\alpha}dm$$ where 0 < p,q < ∞, α > ?1, k,n ∈ N ∪{0} and where ? is an arbitrary analytic function on the unit disc U.     

Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk

We completely characterize the boundedness and compactness of composition operators from the space of Cauchy transforms on the unit disk D, into the Bloch-type space B_ν as well as the little Bloch-type space B_ν,0, consisting respectively of all holomorphic functions f on D such that sup_z∈Dν(z)|f^′(z)|<∞, that is, of all holomorphic functions f on D such that lim_|z|→1ν(z)|f^′(z)|=0, for some weight function ν. As a byproduct of our results, norm of the operator is calculated when B_ν is replaced by B_ν/C.     

On the rational (K + 1,K + 1)-type difference equation

In this paper we investigate the boundedness character of the positive solutions of the rational difference equation of the form $$x_{n + 1} = \frac{{a_0 + \sum\nolimits_{j = 1}^k {a_j x_{n - j + 1} } }}{{b_0 + \sum\nolimits_{j = 1}^k {b_j x_{n - j + 1} } }}, n = 0,1,...$$ where k ε N, andaj,bj, j = 0,1,…, k, are nonnegative numbers such thatb ₀+∑ _j=1 ^k b _j x _n-j+1>0 for everyn ∈N ∪{0}. In passing we confirm several conjectures recently posed in the paper: E. Camouzis, G. Ladas and E. P. Quinn, On third order rational difference equations (part 6).     

The global attractivity of a higher order rational difference equation

This paper studies global asymptotic stability for positive solutions to the equation

     

Composition followed by differentiation from H and the Bloch space to nth weighted-type spaces on the unit disk

The boundedness and compactness of the product of the differentiation and composition operator from the space of bounded analytic functions, the Bloch space and the little Bloch space to nth weighted-type spaces on the unit disk are characterized.     

The global attractivity of the rational difference equation

This paper studies the behavior of positive solutions of the recursive equation

with and , where . We prove that if , and , then tends to . This complements several results in the recent literature, including the main result in K. S.  Berenhaut, J. D. Foley and S. Stevic, The global attractivity of the rational difference equation , Proc. Amer. Math. Soc., 135 (2007) 1133-1140.

     

On a nonlinear generalized max-type difference equation

The boundedness character of positive solutions of the following max-type difference equation

     

Zero subspaces of polynomials on

We provide two examples of complex homogeneous quadratic polynomials P on Banach spaces of the form ℓ₁(Γ). The first polynomial P has both separable and nonseparable maximal zero subspaces. The second polynomial P has the property that while the index-set Γ is not countable, all zero subspaces of P are separable.     

On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the open unit ball B of Cⁿ. Let φ=(φ₁,…,φ_n) be a holomorphic self-map of B and g∈H(B) such that g(0)=0. In this paper we study the boundedness and compactness of the following integral-type operator, recently introduced by Xiangling Zhu and the second author