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Let f be a transcendental entire function of finite lower order, and let $q_{\nu }$ be rational functions. For 0 < γ < ∞ let B(γ):= πγ/sinπγ if γ ≤ 0.5, B(γ):= πγ if γ > 0.5. We estimate the upper and lower logarithmic density of the set $r:{\sum }_{1\le \nu \le k}log⁺ma{x}_{\left|\right|z\left|\right|=r}{\left|f\left(z\right)-q_{\nu }\left(z\right)\right|}^{-1}<B\left(\gamma \right)T(r,f)$.

The notion of a strong asymptotic tract for subharmonic functions is defined. Eremenko's value b(∞,u) for subharmonic functions is introduced and it is used to provide an exact upper estimate of the number of strong tracts of subharmonic functions of infinite lower order. It is also shown that b(∞,u) ≤ π for subharmonic functions of infinite lower order.