On meromorphic functions with finite logarithmic order

By using a slow growth scale, the logarithmic order, with which to measure the growth of functions, we obtain basic results on the value distribution of a class of meromorphic functions of zero order.

     

On the order of a class of meromorphic functions

This paper proves the following result: Letf(z) be a meromorphic function in thez-plane with a deficient value, and δ(θ _k )(k=1,2, ...,q;0≤θ ₁<θ₂<...<θ _q<θ _q+1=θ ₁+2π) beq rays (1≤q<∞) starting at the origin, and letn≥3 be an integer such that for any given positive numberε,0<ε<π/2, $$\overline {\mathop {\lim }\limits_{r \to \infty } } \frac{{\log ^ + n\left\{ { \cup _{k = 1}^q \Omega \left( {\theta _k + \varepsilon ,\theta _{k + 1} - \varepsilon ,r} \right),f\prime f^n = 1} \right\}}}{{\log r}} \leqslant v< \infty ,$$ whereΝ is a constant independent ofε. IfΜ<∞, then we have $$\lambda \leqslant \frac{\pi }{\omega } + v,$$ whereΜ andλ denote the lower order and order off(z), respectively,Ω=minθ _k+1 ?θ _k ;1≤k≤q, andn(E, f=a) is the number of zeros off(z)?a inE with multiple zeros being counted with their multiplicities.     

On deviations and strong asymptotic functions of meromorphic functions of finite lower order

Let f be a transcendental meromorphic function of finite lower order with N(r,f)=S(r,f), and let qν be distinct rational functions, 1?ν?k. For 0<γ<∞ put

     

On deviations and defects of meromorphic functions of finite lower order

We obtain estimates for the sum of deviations and sum of defects to power 1/2 in terms of the Valiron defect of the derivative at zero. In particular, the Fuchs hypothesis (1958) is verified.     

On sums of roots of infinite order entire functions

A class of infinite order entire functions is considered. Estimates for sums of the roots are derived. These estimates supplement the Hadamard theorem. Moreover, we establish a new estimate for the counting function of the roots, which in appropriate situations can be more useful than the Jensen inequality.     

On the multivalence and order of starlikeness of a class of meromorphic functions

Basically this paper deals with the determination of the radius of starlikeness and radius of univalence of the class of meromorphic functions of the formg(z)=A/z−Φ(z) forA>0, and where Φ(z) is an analytic function defined in the unit disk whose modulus does not exceed unity. We estimate the radius ofp-valence of functions having the fromh(z)=Φ(z)+a/z ^p fora>1,p≧1, and also estimate the radius of starlikeness of certain Blaschke products which is also given as a function of the minimum modulus function. We discuss the question of sharpness of the results and mention some open problems.     

On permutable meromorphic functions

We study the class \({\mathcal{M}}\) of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in \({\mathcal{M}}\), with at least one essential singularity, permutes with a non-constant rational map g, then g is a Möbius map that is not conjugate to an irrational rotation. For a given function \({f \in\mathcal{M}}\) which is not a Möbius map, we show that the set of functions in \({\mathcal{M}}\) that permute with f is countably infinite. Finally, we show that there exist transcendental meromorphic functions \({f : \mathbb{C} \to \mathbb{C}}\) such that, among functions meromorphic in the plane, f permutes only with itself and with the identity map.     

On factorizing meromorphic functions

Summary The paper determines all cases when a meromorphic functionF can be expressed both asf ⊗p andf ⊗q with the same meromorphicf and different polynomialsp andq. In all cases there are constantsk, β, a positive integerm, a root λ of unity of orderS and a polynomialr such thatp=(Lr) ^m+k,q=r ^m+k, whereLz=λz+β. We have eitherm=1,S arbitrary orm=2,S=2, which can occur even ifF andf are entire, or, in the remaining casesS=2, 3, 4 or 6,m dividesS andf(k+t ^m) is a doubly-periodic function.     

On the magnitudes of deviations of entire functions of infinite order from rational functions

The magnitudes of deviations b(a, f) of entire functions of infinite order from rational functions are studied.