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For $\\alpha \\geq 0$, we consider the following class $$\\mathcal{W}^0_{\\mathcal{H}}(\\alpha):= \\{f = h + \\overline{g}\\in\\mathcal{H}: {\\rm Re\\,}(h'(z) + \\alpha z h''(z)) >|g'(z) + \\alpha z g''(z)|, \\quad z\\in \\mathbb{D}\\}. $$ In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for functions in the class $\\mathcal{W}^0_{\\mathcal{H}}(\\alpha)$. 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