On the second Hankel determinant for the \(k\)th-root transform of analytic functions. (English) Zbl 1488.30023
Summary: Let \(f\) be a normalized analytic function in the open unit disk of the complex plane satisfying \(z f'(z)/f(z)\) is subordinate to a given analytic function \(\varphi\). A sharp bound is obtained for the second Hankel determinant of the \(k\)th-root transform \(z[f(z^k)/z^k]^{\frac{1}{k}}\). Best bounds for the Hankel determinant are also derived for the \(k\)th-root transform of several other classes, which include the class of \(\alpha\)-convex functions and \(\alpha\)-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function \(\varphi\), and thus connect with earlier known results for particular choices of \(\varphi\).
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 |
| 30C80 | Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination |
Keywords:
second Hankel determinant; subordination; \(k\)th-root transformation; Ma-Minda starlike functions; \(\alpha\)-convex functionsReferences:
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