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On an arithemtical inequality

Here we extend an arithmetical inequality about multiplicative functions obtained by K. Alladi, P. Erdős and J. D. Vaaler, to include also the case of submultiplicative functions. Also an alternative proof of an extension of a result used for this purpose is given. Let Uk, for integral k, denote the set {1,2,…, k}, and Vk denote the collection of all subsets of Uk. In the following, all unspecified sets like A,…, are assumed to be subsets of Uk. Let σ = {Si} and τ = {Tj} be two given collections of subsets of Uk. Set and Let ′ denote complementation in Uk (but for in the proof of (3) where it denotes complementation in C). For any collection p of subsets of Uk, let p′ denote the collection of the complements of members of p.