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In this paper we give an affirmative answer to two conjectures on generalized$(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raisedin [S. Ali and A. Fo\v{s}ner, \textit{On Generalized $(m, n)$-Derivations andGeneralized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21}(2014), 411--420] and [A. Fo\v{s}ner, \textit{A note on generalized$(m,n)$-Jordan centralizers,} Demonstratio Math. \textbf{46} (2013), 254--262].Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsionrestrictions, that every nonzero generalized $(m,n)$-Jordan derivation (resp.,a generalized $(m,n)$-Jordan centralizer) is a derivation (resp., a two-sidedcentralizer).

On Generalized $(m, n)$-Jordan Derivations and Centralizers of Semiprime Rings