  In this paper we give an affirmative answer to two conjectures on generalized
$(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised
in [S. Ali and A. Fo\v{s}ner, \textit{On Generalized $(m, n)$-Derivations and
Generalized $(m, n)$-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 torsion
restrictions, that every nonzero generalized $(m,n)$-Jordan derivation (resp.,
a generalized $(m,n)$-Jordan centralizer) is a derivation (resp., a two-sided
centralizer).
