This　research　contributes　new　analytical　formulae　for　H　infinity　optimal　tuning　of　non-grounded　tuned　mass-damperinerter　(TMDI)　for　minimizing　free-end　displacement　of　base-excited　cantilevered　structures.　The　derivation　relies　on　the　fixed-point　theory,　making　use　of　single-mode　modelling　of　the　cantilevered　structure　while　accommodating　arbitrary　TMDI　placement　along　the　structural　height.　Optimal　TMDI　tuning　parameters　are　derived　for　given　dominant　mode　shape　of　structure　and　TMDI　inertial　properties　(i.e.,　secondary　mass　and　inertance)　under　two　different　types　of　harmonic　excitations　with　frequency-independent　displacement　and　acceleration　amplitudes.　The　applicability　of　the　derived　TMDI　tuning　formulae　for　response　mitigation　of　lightly　damped　cantilevered　structures　under　stationary　broadband　support　excitation　is　established　through　comparisons　with　numerically　optimal　TMDI　properties　for　a　wide　range　of　TMDI　inertial　properties.　Moreover,　the　analytical　TMDI　tuning　formulae　derived　in　this　study　achieve　enhanced　structural　performance　for　base-excited　structures　compared　to　those　from　the　literature,　derived　under　various　modelling　assumptions　and　optimality　criteria.　Lastly,　the　potential　of　the　proposed　TMDI　tuning　for　structural　response　mitigation　is　numerically　evaluated　by　examining　displacement,　acceleration,　and　energy　dissipation　response　history　data　of　an　experimentally　identified　reinforced　concrete　bridge　pier　model　subjected　to　100　earthquake　ground　motion　(GM)　records.　Overall,　reported　results　demonstrate　that　the　derived　analytical　formulae　can　significantly　extend　the　practical　application　of　TMDI　as　a　bona　fide　dynamic　vibration　absorber　for　base-excited　structures　by　circumventing　the　need　for　computationally　demanding　numerical　TMDI　tuning　optimization.