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Dedicated with great pleasure to Eduard Tsekanovskii on the occasion of his 80th birthday.Abstract. In 1961, Birman proved a sequence of inequalities {In}, for n ∈ N, valid for functions in C n 0 ((0, ∞)) ⊂ L 2 ((0, ∞)). In particular, I 1 is the classical (integral) Hardy inequality and I 2 is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite b > 0, these inequalities hold on the standard Sobolev space H n 0 ((0, b)). Furthermore, in all cases, the Birman constants [(2n − 1)!!] 2 /2 2n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in L 2 ((0, ∞)) (resp., L 2 ((0, b))). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.

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The home food environment (HFE), the availability and accessibility of foods and food products within the home, has a strong influence on healthy eating behaviors. Studies assessing the HFE commonly utilize a home food inventory (HFI) for data collection. However, this approach tends to be burdensome for participants. This study validated a low-burden digital photo method for assessing the home food environment and confirmed that this method is preferred by participants. Study participants completed an HFI, submitted photos of household foods, then identified preference for the HFI or photo reporting method. Researchers completed an HFI based on each participant’s photo submissions. Researcher-to-participant and researcher-to-researcher comparisons were made through ANOVA and randomized block analyses to determine concurrent validity and inter-rater reliability. Method preference was assessed using the Z-test. The participant group (N = 53) was predominantly female (68%), young adult (90.5%), and Hispanic or Latino (71.7%). Concurrent validity was initially moderate (ĸ = 0.54); adjustments yielded substantial agreement (ĸ = 0.61). The inter-rater reliability (p = 0.98) demonstrated significant consistency among reviewers. The photo-documentation method was found to be valid and preferred (p = 0.01) for reporting on the HFE. The photo method can be used advantageously to collect quality data.