I'm reading William Cherry and Zhuan Ye's book 'Nevanlinna's theory of value distribution, the second main theorem and its error terms'. In Section 1.12, they explains why $N$ and $T$ is used in Nevanlinna theory instead of $n$ and $A$, where $A(f,r)=\int_{D(t)}f^\ast\omega$. Then they gave some results on the comparison of $n(f,a,r)$ and $A(f,r)$. For example,

Gol'dberg in 1978 constructed an entire function $f$ such that for every $a\in \mathbb{C}$ such that $$\limsup_{r\to \infty}\frac{n(f,a,r)}{A(f,r)}=\infty.$$

In another direction, Hayman and Stewart in 1954 formulate a theorem

Let $f$ be a non-constant meromorphic function on $\mathbb{C}$ and set $n(f,r)=\sup_{a\in \mathbb{P}^1}n(f,a,r)$. Then $$1\le\liminf_{r\to \infty}\frac{n(f,r)}{A(f,r)}\le e.$$

**My question is: what can we say about $\frac{N(f,a,r)}{T(f,r)}$?**

Firstly, by FMT we know $\frac{N(f,a,r)}{T(f,r)}\le 1$.

I also checked some elementaty functions. For examples, exponential function $f(z)=e^z$. By a simple calculation, $N(f,\infty,r)=0,N(f,a,r)=\frac{r}{\pi}+O(\log r),T(f,r)=\frac{r}{\pi}$. So
$$\frac{N(f,\infty,r)}{T(f,r)}=0,\frac{N(f,0,r)}{T(f,r)}=1.$$
**When does this ratio nonzero for a general meromorphic function?**

I'm aware that the purpose of Nevanlinna theory is to give the upper bound and lower bound of $N(f,a,r)$ by $T(f,r)$. But I'm still interested in the value of the ratio by taking $r\to \infty$.

Any reply or reference is appreciated.