Meromorphic functions sharing small functions

Let f and g be meromorphic functions sharing four small functions a₁, a₂, a₃, a₄ a_1, a_2, a_3, a_4 ignoring multiplicities. If there is a small function a₅a_5 distinct from a_j, j=1, 2, 3, 4, a_j, j=1, 2, 3, 4, such that [`(N)](r,f=a<sub>5</sub>=g) 1 S(r,f) \overline {N}(r,f=a_5=g)\ne S(r,f) , then f=g f=g , where [`(N)](r,f=a₅=g) \overline{N}(r,f=a_5=g) is the counting function of those common zeros of f(z)-a₅(z) f(z)-a_5(z) and g(z)-a₅(z) g(z)-a_5(z) counted only once ignoring multiplicities.     

Convex functions and subharmonic functions

It is a classical result that a composition of a convex, increasing function and of a subharmonic function is subharmonic. We give related results for a composition of a convex function of several variables and of several subharmonic functions, thus imporving some recent results in this area.     

On functions behaving like additive functions

Let 0 < 1. In the paper we consider the following inequality: |f(x + y) – f(x) – f(y)| min{|f(x + y)|, |f(x) + f(y)|}, wheref: R R. Solutions and continuous solutions of this inequality are investigated. They have similar properties as additive functions, e.g. if the solution is bounded above (below) on a set of positive inner Lebesgue measure then it is continuous. Some sufficient condition for this inequality is also given.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

涉及导函数与分担函数的全纯函数正规定则

针对一类零点个数为有限的全纯函数族,在函数与其导函数分担一个极点均为重级的亚纯函数的条件下,利用Nevanlinna理论及其方法改进了已有文献在分担值条件下得到的一个定理.     

Meromorphic functions sharing four small functions

It is well known that if two nonconstant meromorphic functions f and g on the complex plane ? have the same inverse images counted with multiplicities for four distinct values, then g is a Möbius transformation of f. In this paper, we will show that the above result remains valid if f and g share four distinct small functions counted with multiplicities truncated by 2. This is the best possible truncation level.     

On Chebyshev functions and Klee functions

Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. We show that the function f is 1/λ-hypoconvex if its proximal mapping Pλf is single-valued. When the function f is bounded below, and Pλf is single-valued for every λ>0, the function must be convex. Similarly, we show that the function f is 1/μ-strongly convex if the farthest mapping Qμf is single-valued. When the function is the indicator function of a set, this recovers the well-known Chebyshev problem and Klee problem in Rn. We also give an example illustrating that a continuous proximal mapping (farthest mapping) needs not be locally Lipschitz, which answers one open question by Hare and Poliquin.     

Limiting functions of series of functions

A series with a given set of limit functions is constructed. A theorem proved by Besicovich concerning conditionally convergent series is strengthened.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 25–32, July. 1970.     

Measurable functions and almost continuous functions

I show that if (X, ) is a Radon measure space and Y is a metric space, then a function from X to Y is -measurable iff it is almost continuous (=Lusin measurable). I discuss other cases in which measurable functions are almost continuous.Part of the work of this paper was done during a visit to Japan supported by the United Kingdom Science Research Council and Hokkaido University     

具三个公共小函数的亚纯函数

讨论具三个公共小函数的亚纯函数的唯一性问题,其结果推广并改进了H.Ueda的一个定理。     

Duality between bent functions and affine functions

A Boolean function in an even number of variables is called bent if it is at the maximal possible Hamming distance from the class of all affine Boolean functions. We prove that there is a duality between bent functions and affine functions. Namely, we show that affine function can be defined as a Boolean function that is at the maximal possible distance from the set of all bent functions.     

Derivatives of meromorphic functions and exponential functions

We take up a new method to prove a Picard type theorem. Let f be a meromorphic function in the complex plane, whose zeros are multiple, and let R be a Möbius transformation. If \({\overline {\lim } _{r \to \infty }}\frac{{T\left( {r,f} \right)}}{{{r^2}}} = \infty \) then f′z) = R(e ^z ) has infinitely many solutions in the complex plane.