On the logarithmic derivative of a meromorphic function

We derive the following estimate for the quantity m(r,f′/tf) of the Nevanlinna theory of the distribution of values characterizing the growth of the logarithmic derivative of a meromorphic functionf(z),f(0) = 1, 0 < r < R < ∞: m(r,f′/f) < 1n+ [T(R,f)/r (R/R?r)²] + 6.0084. This estimate is more accurate than that obtained earlier by Vu Ngoyan and I. V. Ostrovskii.     

The logarithmic derivative of a meromorphic function

A well-known lemma on the logarithmic derivative for a function f(z), f(0) = 1 (0 < r="> m( r,\fracf￠f ) < ln+ { \fracT(r,f)r\fracrr- r } + 5.8501.m\left( {r,\frac{{f'}}{f}} \right)< \ln + \left\{ {\frac{{T(\rho ,f)}}{r}\frac{\rho }{{\rho - r}}} \right\} + 5.8501.     

Nevanlinna's lemma on the logarithmic derivative of a meromorphic function

In proving the second fundamental theorem for functions meromorphic in the half-plane, Nevanlinna has asserted (in 1925) that they satisfy a lemma similar to the well-known lemma on the logarithmic derivative, but his proof was based on additional assumptions. These assumptions were later relaxed by Dufresnoy (in 1939) and Ostrovskii (in 1961). Here we shall show that in the general case, functions which are meromorphic in the half-plane do not satisfy a lemma similar to the lemma on the logarithmic derivative.     

On the derivative of meromorphic functions with multiple zeros

Let f be a transcendental meromorphic function and let R be a rational function, R?0. We show that if all zeros and poles of f are multiple, except possibly finitely many, then f′−R has infinitely many zeros. If f has finite order and R is a polynomial, then the conclusion holds without the hypothesis that poles be multiple.     

Classes of meromorphic functions characterizable by growth of the spherical derivative

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 1, pp. 96–104, January–February, 1994.     

On the growth of meromorphic functions with a radially distributed value

The purpose of this paper is to investigate the growth of meromorphic functions with a radially distributed value. The paper is closely related to a more recent result due to Fang and Zalcman [M.L. Fang and L. Zalcman, On the value distribution of f + a(f′) ⁿ , Sci. China Ser. A-Math. 38(2008), 279–285].     

Common Borel directions of a meromorphic function with zero order and its derivative

There is a meromorphic function of zero order for which the function and its derivative have no common Borel direction.

     

On deficient values of meromorphic functions satisfying a certain functional equation

Aequationes mathematicae -     

A meromorphic function of zero order with nonasymptotic defect value

We construct an example of a meromorphic function of zero order with nonasymptotic defect value.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 195–206, February, 1973.In conclusion, the author would like to express his deep gratitude to N. U. Arakelyan for valuable discussions.     

On the second derivative of the spherical contact distribution function of smooth grain models

We consider a stationary grain model Ξ in ℝ ^d with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula for the(d− 1)-st quermass density. Received: 22 November 1999 / Revised version: 2 November 2000 /?Published online: 14 June 2001     

On entire functions with zero as a deficient value

We consider entire functions of finite order for which zero is a Nevanlinna deficient value. Estimates are determined for the ratio of the integrated moduli of the logarithmic derivative of the function on a circle of radius r to the Nevanlinna characteristic of the function at r.     

关于亚纯函数的充满圆

考虑亚纯函数的零点或极点时，每个值点只取一次，而不计其重复度，本文证明了拟充满圆存在定理。