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The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

Department of Mathematics, Kyungsung University, Busan 48434, Korea
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Mathematics 2019, 7(8), 721; https://doi.org/10.3390/math7080721
Received: 15 July 2019 / Revised: 28 July 2019 / Accepted: 3 August 2019 / Published: 8 August 2019
(This article belongs to the Special Issue Complex Analysis and Its Applications 2019)
Let SR * be the class of starlike functions with real coefficients, i.e., the class of analytic functions f which satisfy the condition f ( 0 ) = 0 = f ( 0 ) 1 , Re { z f ( z ) / f ( z ) } > 0 , for z D : = { z C : | z | < 1 } and a n : = f ( n ) ( 0 ) / n ! is real for all n N . In the present paper, it is obtained that the sharp inequalities 4 / 9 H 3 , 1 ( f ) 3 / 9 hold for f SR * , where H 3 , 1 ( f ) is the third Hankel determinant of order 3 defined by H 3 , 1 ( f ) = a 3 ( a 2 a 4 a 3 2 ) a 4 ( a 4 a 2 a 3 ) + a 5 ( a 3 a 2 2 ) . View Full-Text
Keywords: starlike functions; hankel determinant; carathéodory functions; schwarz functions starlike functions; hankel determinant; carathéodory functions; schwarz functions
MDPI and ACS Style

Kwon, O.S.; Sim, Y.J. The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients. Mathematics 2019, 7, 721.

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