Postnatal growth failure in preterm infants is due to interactions between genetic and environmental factors, which are not fully understood. We assessed dietary supply of nutrients in very-low-birth-weight (VLBW, < 1500 g) infants fed fortified human milk, and examined the association between nutrient intake, medical factors and growth during hospitalisation lasting on average 70 d. We studied 127 VLBW infants during the early neonatal period. Data were obtained from medical records on nutrient intake, growth and growth-related factors. Extra-uterine growth restriction was defined as body weight < 10th percentile of the predicted value at discharge. Using logistic regression, we evaluated nutrient intake and other relevant factors associated with extra-uterine growth restriction in the subgroup of VLBW infants with adequate weight for gestational age at birth. The proportion of growth restriction was 33 % at birth and increased to 58 % at discharge from hospital. Recommended values for energy intake (>500 kJ/kg per d) and intra-uterine growth rate (15 g/kg per d) were not met, neither in the period from birth to 28 weeks post-conceptional age (PCA), nor from 37 weeks PCA to discharge. Factors negatively associated with growth restriction were energy intake (Ptrend = 0·002), non-Caucasian ethnicity (P = 0·04) and weight/predicted birth weight at birth (Ptrend = 0·004). Extra-uterine growth restriction is common in VLBW infants fed primarily fortified human milk. Currently recommended energy and nutrient intake for growing preterm infants was not achieved. Reduced energy supply and non-Caucasian ethnicity were risk factors for growth restriction at discharge from hospital.

For $(\l,a)\in \matbb{C}^*\times \mathbb{C}$, let $f_{\lambda,a}$ be the rational map defined by $f_{\lambda,a}(z) \,{=}\, \lambda z^2 {(az+1)/(z+a)}$. If $\alpha\in \mathbb{R}/\mathbb{Z}$ is a Brjuno number, we let ${\cal D}_\alpha$ be the set of parameters $(\lambda,a)$ such that $f_{\lambda,a}$ has a fixed Herman ring with rotation number $\alpha$ (we consider that $({\it e}^{2i\pi\alpha}{,}0)\,{\in}\, {\cal D}_\alpha$). Results obtained by McMullen and Sullivan imply that, for any $g\in {\cal D}_\alpha$, the connected component of ${\cal D}_\alpha\cap (\mathbb{C}^*\times(\mathbb{C}\setminus \{0,1\}))$ that contains g is isomorphic to a punctured disk.

We show that there is a holomorphic injection $\cal{F}_\alpha\,{:}\,\mathbb{D}\,{\longrightarrow}\, {\cal D}_\alpha$ such that $\cal{F}_\alpha(0) = ({\it e}^{2i\pi \alpha},0)$ and $\cal{F}_\alpha'(0)=(0,r_\alpha),$ where $r_\alpha$ is the conformal radius at 0 of the Siegel disk of the quadratic polynomial $z\longmapsto {\it e}^{2i\pi \alpha}z(1+z)$.

As a consequence, we show that for $a\in (0,1/3)$, if $f_{\l,a}$ has a fixed Herman ring with rotation number $\alpha$ and if $m_a$ is the modulus of the Herman ring, then, as $a\,{\rightarrow}\,0$, we have ${\it e}^{\pi m_a} \,{=} ({r_\alpha}/{a}) + {\cal O}(a).$

We finally explain how to adapt the results to the complex standard family $z\,{\longmapsto} \lambda z {\it e}^{({a}/{2})(z-1/z)}$.