Diffusion-weighted magnetic resonance imaging (dMRI) is widely used to assess the brain white matter. Fiber orientation distribution functions (FODs) are a common way of representing the orientation and density of white matter fibers. However, with standard FOD computation methods, accurate estimation requires a large number of measurements that usually cannot be acquired for newborns and fetuses. We propose to overcome this limitation by using a deep learning method to map as few as six diffusion-weighted measurements to the target FOD. To train the model, we use the FODs computed using multi-shell high angular resolution measurements as target. Extensive quantitative evaluations show that the new deep learning method, using significantly fewer measurements, achieves comparable or superior results than standard methods such as Constrained Spherical Deconvolution and two state-of-the-art deep learning methods. For voxels with one and two fibers, respectively, our method shows an agreement rate in terms of the number of fibers of 77.5% and 22.2%, which is 3% and 5.4% higher than other deep learning methods, and an angular error of 10° and 20°, which is 6° and 5° lower than other deep learning methods. To determine baselines for assessing the performance of our method, we compute agreement metrics using densely sampled newborn data. Moreover, we demonstrate the generalizability of the new deep learning method across scanners, acquisition protocols, and anatomy on two clinical external datasets of newborns and fetuses. We validate fetal FODs, successfully estimated for the first time with deep learning, using post-mortem histological data. Our results show the advantage of deep learning in computing the fiber orientation density for the developing brain from in-vivo dMRI measurements that are often very limited due to constrained acquisition times. Our findings also highlight the intrinsic limitations of dMRI for probing the developing brain microstructure.