In this article, we prove inequalities for the Taylor coefficients of functions G holomorphic in the unit disc satisfying the condition |G(z)|(1-|z|2)α≤1,|z|<1,\documentclass[12pt]{minimal}
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\begin{document}$$|G(z)|(1-|z|^2)^{\alpha }\le 1, |z|<1,$$\end{document} for fixed α≥0.\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \ge 0.$$\end{document} The upper bound for the modulus of the k-th Taylor coefficient is dependent on the moduli of some initial coefficients. As corollaries we get similar estimates for α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}-Bloch functions and an estimate for an area type functional on α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}-Bloch functions.
