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z\\in\\mathbb{D}. \\] If $F_{f^{-1}}(w) = \\log\\!\\left(\\frac{f^{-1}(w)}{w}\\right) = 2\\sum_{n=1}^{\\infty}\\Gamma_n w^n$ denotes the logarithmic expansion corresponding to the inverse function $f^{-1}$, then we establish sharp estimates for the initial inverse logarithmic coefficients and prove that \\[ |\\Gamma_n| \\le \\frac{1}{2n(n+1)}, \\qquad n=1,2,3. \\] We further derive 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