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Let S be a self mapping on a complex normed space X . In this paper, we study the class of mappings satisfying the following condition for all x, y ∈ X, where m is a positive integer. We prove some of the properties of these classes of mappings. Mathematics Subject Classification. 54E40, 62H86. Key words and phrases. (m, p)-isometries; m-isometric operators; expansive and contractive operator.doi:10.28924/2291-8639-18-2020-194 fatcat:5rg3a3wiwzc47av54e5nfpvc54