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Coefficient Problems for Subclasses of Close-to-Star Functions

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Abstract

In this paper, two subclasses of close-to-star functions and two subclasses of close-to-convex functions are studied. By using the coefficient estimates of the known Carathéodory function classes, the upper bounds of the Zalcman functional \({J}_{3,4}\), Hankel determinant \({{H}_{2}}(3)\) and Toeplitz determinant of given function classes are estimated. Some sharp results are obtained.

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References

  • Ali MF, Thomas DK, Vasudevarao A (2018) Toeplitz determinants whose elements are the coefficients of analytic and univalent functions. Bull Aust Math Soc 97(2):253–264

    Article  MathSciNet  Google Scholar 

  • Ali MF, Vasudevarao A (2018) On logarithmic coefficients of some close-to-convex functions. Proc Am Math Soc 146(3):1131–1142

    Article  MathSciNet  Google Scholar 

  • Causey WM, Merkes EP (1970) Radii of starlikeness of certain classes of analytic functions. J Math Anal Appl 31:579–586

    Article  MathSciNet  Google Scholar 

  • Cho NE, Kowalczyk B, Kwon OS et al (2018) The bounds of some determinants for starlike functions of order alpha. Bull Malays Math Sci Soc 41(1):523–535

    Article  MathSciNet  Google Scholar 

  • Cho NE, Kowalczyk B, Kwon OS et al (2017) Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha. J Math Ineq 11(2):429–439

    Article  MathSciNet  Google Scholar 

  • Duren PL (1983) Univalent functions. Springer Verlag, New York

    MATH  Google Scholar 

  • Efraimidis I (2016) A generalization of Livingston’s coefficient inequalities for functions with positive real part. J Math Anal Appl 435(1):369–379

    Article  MathSciNet  Google Scholar 

  • Goodman AW (1983) Univalent functions. Mariner, Tampa

    MATH  Google Scholar 

  • Janteng A, Halim SA, Darus M (2007) Hankel determinant for starlike and convex functions. Int J Math Anal 1(13–16):619–625

    MathSciNet  MATH  Google Scholar 

  • Kaplan W (1952) Close to convex schlicht functions. Mich Math J 1:169–185

    Article  MathSciNet  Google Scholar 

  • Kowalczyk B, Lecko A, Sim YJ (2018) The sharp bound of the Hankel determinant of the third kind for convex functions. Bull Aust Math Soc 97(3):435–445

    Article  MathSciNet  Google Scholar 

  • Lecko A, Sim YJ (2019) Coefficient problems in the subclasses of close-to-star functions. Results Math 74(3): Art. 104, 14 pp

  • Lee SK, Ravichandran V, Supramaniam S (2013) Bound for the second Hankel determinant of certain univalent functions. J Ineq Appl 281:1–7

    MathSciNet  MATH  Google Scholar 

  • Ma W (1999) Generalized Zalcman conjecture for starlike and typically real functions. J Math Anal Appl 234(1):328–339

    Article  MathSciNet  Google Scholar 

  • MacGregor TH (1963) The radius of univalence of certain analytic functions. Proc Am Math Soc 14:514–520

    Article  MathSciNet  Google Scholar 

  • Pommerenke C (1967) On the Hankel determinants of univalent functions. Mathematika 14:108–112

    Article  MathSciNet  Google Scholar 

  • Pommerenke C (1966) On the coefficients and Hankel determinants of univalent functions. J Lond Math Soc 41(1):111–122

    Article  MathSciNet  Google Scholar 

  • Radhika V, Sivasubramanian S, Murugusundaramoorthy G et al (2016) Toeplitz matrices whose elements are the coefficients of functions with bounded boundary rotation. J Complex Anal 2016:1–4

    Article  MathSciNet  Google Scholar 

  • Ravichandran V, Verma S (2017) Generalized Zalcman conjecture for some classes of analytic functions. J Math Anal Appl 450(1):592–605

    Article  MathSciNet  Google Scholar 

  • Reade MO (1955) On close-to-convex univalent functions. Mich Math J 3:59–62

    Article  MathSciNet  Google Scholar 

  • Robertson MS (1936) Analytic functions starlike in one direction. Am J Math 58(3):465–472

    Article  Google Scholar 

  • Sakaguchi K (1964) The radius of convexity for a certain class of regular functions. J Nara Gakugei Univ Nat Sci 12:5–8

    MathSciNet  Google Scholar 

  • Sun Y, Wang Z, Rasila A (2019) On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings. Hacet J Math Stat 48(6):1695–1705

    MathSciNet  Google Scholar 

  • Thomas DK, Halim SA (2017) Toeplitz matrices whose elements are the coefficients of starlike and close-to-convex functions. Bull Malays Math Sci Soc 40(4):1781–1790

    Article  MathSciNet  Google Scholar 

  • Ye K, Lim LH (2016) Every matrix is a product of Toeplitz matrices. Found Comput Math 16(3):577–598

    Article  MathSciNet  Google Scholar 

  • Zaprawa P (2018) On Hankel determinant \({{H}_{2}}(3)\) for univalent functions. Results Math 73(3): Art. 89, 12 pp

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Funding

This study was funded by Natural Science Foundation of Anhui Province (1908085MA18) and Foundation of Anhui Educational Committee (KJ2020A0002), China.

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, Anhui, China

    Dong-rong Wang, Hua-ying Huang & Bo-yong Long

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  1. Dong-rong Wang
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  2. Hua-ying Huang
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  3. Bo-yong Long
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Correspondence to Bo-yong Long.

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Wang, Dr., Huang, Hy. & Long, By. Coefficient Problems for Subclasses of Close-to-Star Functions. Iran J Sci Technol Trans Sci 45, 1071–1077 (2021). https://doi.org/10.1007/s40995-021-01115-2

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  • DOI: https://doi.org/10.1007/s40995-021-01115-2

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