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On two functional equations connected with the characterizations of the distance measures

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Summary

In this paper, the general solutions of the functional equations

$$f_1 (pr,qs) + f_2 (ps,qr) = g(p,q) + h(r,s), p,q,r,s \in ]0, 1]$$

, and

$$f(pr,qs) + f(ps,qr) = g(p,q)h(r,s) + g(r,s)h(p,q), p,q,r,s \in ]0, 1]$$

, are obtained without any regularity assumptions. Heref, f 1, f2, g andh are complex-valued functions defined on the open-closed unit interval [0,1]. The last functional equation is a generalization of

$$f(pr,qs) + f(ps,qr) = g(p,q)f(r,s) + g(r,s)f(p,q), p,q,r,s \in ]0, 1]$$

, which arises in the characterizations of the distance measures.

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Authors and Affiliations

  1. Department of Mathematics, University of Louisville, 40292, Louisville, Kentucky, USA

    Thomas Riedel & Prasanna K. Sahoo

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  1. Thomas Riedel
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  2. Prasanna K. Sahoo
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Riedel, T., Sahoo, P.K. On two functional equations connected with the characterizations of the distance measures. Aequ. Math. 54, 242–263 (1997). https://doi.org/10.1007/BF02755459

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  • DOI: https://doi.org/10.1007/BF02755459

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