Some Upper Bounds for the Spherical Derivative

Abstract

The author answers the question: what can be said on functions with a Nevanlinna deficient value and of positive lower order λ less than 1/2 ? It is given in the following three theorems: Thm. 1. Let f be a meromorphic function of lower order λ, 0 < λ < 1/2, such that δ(∞, f) > 1 - cos πλ. Then

$$ {\lim_{{r \to - \infty }}}\inf \frac{{\log \mu \left( {r,f} \right)}}{{T\left( {r,f} \right)}} \leqslant \frac{{ - \pi \lambda }}{{\sin \pi \lambda }}\left( {\cos \pi \lambda + \delta (\infty, f) - 1} \right) $$

(1)

Thm. 2. Given λ, 0 < λ < 1/2, and d, 1 - cos πλ < d < 1, there exists a meromorphic function f of order λ and of lower order λ such that δ(∞, f) = d and

$$ {\lim_{{r \to - \infty }}}\inf \frac{{\log \mu \left( {r,f} \right)}}{{T\left( {r,f} \right)}} = \frac{{ - \pi \lambda }}{{\sin \pi \lambda }}\left( {\cos \pi \lambda + d - 1} \right) $$

(2)

Thm. 3. Given λ, 0 < λ < 1/2, and d, 0 < d < 1 - cos πλ, there exists a meromorphic function f of order λ and of lower order λ such that δ(∞, f) = d and

$$\mathop{{\lim inf}}\limits_{{{{r}^{{ \to \infty }}}}} \frac{{r\mu (r,f)}}{{T(r,f)}} \geqslant \frac{{\pi {{\lambda }^{2}}}}{2}.$$

(3)