Let R be an associative ring. An additive mapping d : R ! R is called a Jordan derivation
if dðx2Þ ¼ dðxÞx þ xdðxÞ holds for all x 2 R. The objective of the present paper is to characterize
a prime ring R which admits Jordan derivations d and g such that ½dðxmÞ; gðynÞ ¼ 0 for all
x; y 2 R or dðxmÞ  gðynÞ ¼ 0 for all x; y 2 R, where m P 1 and n P 1 are some fixed integers. This
partially extended Herstein's result in [6, Theorem 2], to the case of (semi)prime ring involving pair
of Jordan derivations. Finally, we apply these purely algebraic results to obtain a range inclusion
result of continuous linear Jordan derivations on Banach algebras.