Sharp bounds associated with the Zalcman conjecture for the initial coefficients and second Hankel determinants for certain subclass of analytic functions. (English) Zbl 1535.30046
Summary: In this paper, we obtain sharp bounds in the Zalcman conjecture for the initial coefficients, the second Hankel determinant \(H_{2,2}(f ) = a_2 a_4 - a^2_3\) and an upper bound for the second Hankel determinant \(H_{2,3}(f ) = a_3 a_5 - a^2_4\) for the functions belonging to a certain subclass of analytic functions. The practical tools applied in the derivation of our main results are the coefficient inequalities of the Carathéodory class \(\mathcal{P}\).
MSC:
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 30C50 | Coefficient problems for univalent and multivalent functions of one complex variable |
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