Research On Some Questions Of Meromorphic Functions, Entire Curves And Algebroid Functions

In this thesis, we study several important concepts of complex functions: defi-ciencies, deviations, growth order and singular direction. In Chapter2,3, we considerdeficiencies, deviations and growth of two typical complex functions such as mero-morphic functions and entire curves. In Chapter4,5, we consider the growth problemof an algebroid function which could be affected by distribution of the arguments ofits value a points in the complex plane and unit disk respectively. In Chapter6, weconsider the common Borel radii of an algebroid function and its derivative in the unitdisk.In Chapter2, we study the extension problem of controlling the B deviationb(∞, f) with the term p(∞, f) which is introducing by Marchenko, and this generalizesthe case of a constant number φ (r)≡1to the case of a small function φ (r)=o(T (r, f)).We use the tool of Po′lya peaks and general Po′lya peaks of infinite lower order ofmeromorphic functions to gain one’s ends. We use p(∞, f) and B deviation b(∞, f) es-tablishing new spread relation of meromorphic functions, which generalizes the resultof Marchenko. In the last, we consider the second fundamental theorem of an entirefunction respect to polynomials with uniform metric.In Chapter3, we study the extension problem of deficiency, deviation and growthof entire curves, which is case of small entire curves. Firstly, we prove the p-dimensional entire curves with p Nevanlinna small deficiency entire curves has non-zero lower order. Secondly, we consider the relation between the B deviation b(a, G)and Valiron deficiency (a, G) of entire curves. Thirdly, we study the relation of Bdeviation b(a, G) and Nevanlinna deficiency δ (a, G) of entire curves. In the last, wewill obtain the spread relation of the entire curves for the small entire curves and thelower estimate of the logarithmic minimal module.In Chapter4, we take into account how the growth of an algebroid function couldbe affected by distribution of the arguments of its value a points in the complex plane. In this chapter, we first recall the spread relation of algebroid functions in some liter-atures; show that the logarithmic module of algebroid functions can be control by themean valued function and counting function; then using the method of conformal map-ping to map the angular region onto the unit disk to control the logarithmic module ofan algebroid function in an angular region with the number of its value a points; inthe last we use the method of Po′lya peaks of a real function to obtain the results of thischapter.In Chapter5, we take into account how the growth of an algebroid function couldbe affected by distribution of the arguments of its value a points in the unit disk, whichis an affirmative answer of a problem of Zheng Jian-Hua[98].In Chapter6, we study the common (p, q) iteration order Borel radii of an al-gebroid function and its derivative in the unit disk, we prove that they coincide com-pletely.