Abstract. The estimate of third Hankel determinantof the analytic function f (z) = z + a 2 z 2 + a 3 z 3 + · · · , for which ℜ(1 + zf ′′ (z)/f ′ (z)) > −1/2 are investigated. The corrected version of a known results [2, Theorem 3.1 and Theorem 3.3] are also obtained.

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In the present work, Mittag-Leffler functions with its normalization are considered. Several results are obtained so that these functions have certain geometric properties including starlikeness, convexity, close-to-convexity of order ?, and radius of starlikeness of order ?. Furthermore, we obtain certain condition so that the normalized Mittag-Leffler functions belongs to the Hardy space and to the class of bounded analytic functions. Results obtained are new and their usefulness are depicted by deducing several interesting corollaries and examples.

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