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 Approximate Lie $*$-derivations on $\rho$-complete convex modular algebras Hwan-Yong Shin Keywords:modular $*$-algebra, convex modular, $\Delta_\mu$-condition, $(m,n)$-Cauchy-Jensen mapping, Lie $*$-derivation Abstract: In this paper, we obtain generalized Hyers--Ulam stability results of a $(m,n)$-Cauchy-Jensen functional equation associated with approximate Lie $*$-derivations on $\rho$-complete convex modular $*$-algebras $\chi_\rho$ with $\Delta_\mu$-condition on the convex modular $\rho.$