If I have the following relation $T(n) \le an\lceil \operatorname{lg} (n) \rceil - an +2bn + n$, is it possible to bound $T(n)$ such that it is in the form $T(n) \le an\operatorname{lg}(n) + bn $ for some constants $a, b \ge0$?
2026-03-27 23:38:26.1774654706
Bounding a logarithmic relation
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Note: $$ an\lceil{\log(n)}\rceil - an + 2bn + n \geq an\log(n)) - an + 2bn + n = an\log(n) + (2b - a + 1)n $$ The RHS is greater than the required expression so long as $2b-a+1 > b$, so it is not necessarily possible.