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

Author

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  • Oh Sang Kwon

    (Department of Mathematics, Kyungsung University, Busan 48434, Korea)

  • Young Jae Sim

    (Department of Mathematics, Kyungsung University, Busan 48434, Korea)

Abstract

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 ) .

Suggested Citation

  • Oh Sang Kwon & Young Jae Sim, 2019. "The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients," Mathematics, MDPI, vol. 7(8), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:721-:d:255880
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    Citations

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    Cited by:

    1. Dong Guo & Huo Tang & Zongtao Li & Qingbing Xu & En Ao, 2023. "Coefficient Problems for a Class of Univalent Functions," Mathematics, MDPI, vol. 11(8), pages 1-14, April.
    2. Isra Al-Shbeil & Muhammad Imran Faisal & Muhammad Arif & Muhammad Abbas & Reem K. Alhefthi, 2023. "Investigation of the Hankel Determinant Sharp Bounds for a Specific Analytic Function Linked to a Cardioid-Shaped Domain," Mathematics, MDPI, vol. 11(17), pages 1-22, August.
    3. Lei Shi & Hari M. Srivastava & Ayesha Rafiq & Muhammad Arif & Muhammad Ihsan, 2022. "Results on Hankel Determinants for the Inverse of Certain Analytic Functions Subordinated to the Exponential Function," Mathematics, MDPI, vol. 10(19), pages 1-15, September.
    4. Paweł Zaprawa & Anna Futa & Magdalena Jastrzębska, 2020. "On Coefficient Functionals for Functions with Coefficients Bounded by 1," Mathematics, MDPI, vol. 8(4), pages 1-14, April.
    5. Lei Shi & Muhammad Arif & Ayesha Rafiq & Muhammad Abbas & Javed Iqbal, 2022. "Sharp Bounds of Hankel Determinant on Logarithmic Coefficients for Functions of Bounded Turning Associated with Petal-Shaped Domain," Mathematics, MDPI, vol. 10(11), pages 1-19, June.
    6. Mohsan Raza & Hari Mohan Srivastava & Qin Xin & Fairouz Tchier & Sarfraz Nawaz Malik & Muhammad Arif, 2023. "Starlikeness Associated with the Van Der Pol Numbers," Mathematics, MDPI, vol. 11(10), pages 1-22, May.
    7. Yue-Juan Sun & Muhammad Arif & Lei Shi & Muhammad Imran Faisal, 2023. "Some Further Coefficient Bounds on a New Subclass of Analytic Functions," Mathematics, MDPI, vol. 11(12), pages 1-14, June.

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