On left $(\theta,\varphi)$-derivations of prime rings

summary:Let $R$ be a $2$-torsion free prime ring. Suppose that $\theta , \phi $ are automorphisms of $R$. In the present paper it is established that if $R$ admits a nonzero Jordan left $(\theta ,\theta )$-derivation, then $R$ is commutative. Further, as an application of this resul it is shown that every Jordan left $(\theta ,\theta )$-derivation on $R$ is a left $(\theta ,\theta )$-derivation on $R$. Finally, in case of an arbitrary prime ring it is proved that if $R$ admits a left $(\theta ,\phi )$-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of $R$, then $d=0$ on $R$