We study the sharp \(\textrm{L}^\infty \) estimates for fully non-linear elliptic equations on compact complex manifolds. For the case of Kähler manifolds, we prove that the oscillation of any admissible solution to a degenerate fully non-linear elliptic equation satisfying several structural conditions can be controlled by the \(\textrm{L}^1(\log \textrm{L})^n(\log \log \textrm{L})^r(r>n)\) norm of the right-hand function (in a regularized form). This result improves that of Guo-Phong-Tong. In addition to their method of comparison with auxiliary complex Monge-Ampère equations, our proof relies on an inequality of Hölder-Young type and an iteration lemma of De Giorgi type. For the case of Hermitian manifolds with non-degenerate background metrics, we prove a similar \(\textrm{L}^\infty \) estimate which improves that of Guo-Phong. An explicit example is constucted to show that the \(\textrm{L}^\infty \) estimates given here may fail when \(r\leqslant n-1\) . The construction relies on a gluing lemma of smooth, radial, strictly plurisubharmonic functions.
