Hybrid moments of the Riemann zeta-function

The “hybrid” moments $${\int }_T^{2T}\mathrm{\mid }\zeta (1\mathrm{/}2+it){\mathrm{\mid }}^k{({\int }_{t-G}^{t+G}\mathrm{\mid }\zeta (1\mathrm{/}2+ix){\mathrm{\mid }}^{\mathrm{\ell }}dx)}^{m}dt$$ of the Riemann zeta-function $\zeta (s)$ on the critical line $\mathrm{\Re }s=1\mathrm{/}2$ are studied. The expected upper bound for the above expression is $O_{ϵ}(T^{1+ϵ}{G}^m)$. This is shown to be true for certain specific values of the natural numbers $k,\mathrm{\ell },m$, and the explicitly determined range of $G=G(T;k,\mathrm{\ell },m)$. The application to a mean square bound for the Mellin transform function of $\mathrm{\mid }\zeta (1\mathrm{/}2+ix){\mathrm{\mid }}^4$ is given.