A mean value result involving the fourth moment of $\mathrm{\mid }\zeta (1\mathrm{/}2+it)\mathrm{\mid }$

If $(k,\mathrm{\ell })$ is an exponent pair such that $k+\mathrm{\ell }<1$, then we have $${\int }_1^T\mathrm{\mid }\zeta (1\mathrm{/}2+it){\mathrm{\mid }}^4\mathrm{\mid }\zeta (\sigma +it){\mathrm{\mid }}^{2}dt{\ll }_{ϵ}{T}^{1+ϵ}(\sigma >\min (\frac{5}{6},\max (\mathrm{\ell }-k,\frac{5k+\mathrm{\ell }}{4k+1})),$$ while if $(k,\mathrm{\ell })$ is an exponent pair such that $3k+\mathrm{\ell }<1$, then we have $${\int }_1^T\mathrm{\mid }\zeta (1\mathrm{/}2+it){\mathrm{\mid }}^4\mathrm{\mid }\zeta (\sigma +it){\mathrm{\mid }}^{4}dt{\ll }_{ϵ}{T}^{1+ϵ}(\sigma >\frac{11k+\mathrm{\ell }+1}{8k+2})\mathrm{.}$$