In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let $delta$ be a spectrally bounded left Jordan derivation on a Banach algebra A. Then $delta$ maps A into its Jacobson radical. (ii) Let $delta$ be a left Jordan derivation on a unital Banach algebra A with the condition sup{r$(c^{-1}delta(c))$ : c $in$ A invertible} < $infty$. Then $delta$ maps A into its Jacobson radical. Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in [2, p. 260]: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.