Korean Journal of Mathematics

  • 제32권2호
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  • Pages.285-295
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  • 2024
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  • 1976-8605(pISSN)
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  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
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FABER POLYNOMIAL COEFFICIENT ESTIMATES FOR ANALYTIC BI-UNIVALENT FUNCTIONS ASSOCIATED WITH GREGORY COEFFICIENTS

  • Serap Bulut (Kocaeli University, Faculty of Aviation and Space Sciences, Arslanbey Campus)
  • 투고 : 2023.08.22
  • 심사 : 2024.04.01
  • 발행 : 2024.06.30
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초록

In this work, we consider the function $${\Psi}(z)=\frac{z}{\ln(1+z)}=1+\sum\limits_{n=1}^{\infty}\,G_nz^n$$ whose coefficients Gn are the Gregory coefficients related to Stirling numbers of the first kind and introduce a new subclass ${\mathcal{G}}^{{\lambda},{\mu}}_{\Sigma}(\Psi)$ of analytic bi-univalent functions subordinate to the function Ψ. For functions belong to this class, we investigate the estimates for the general Taylor-Maclaurin coefficients by using the Faber polynomial expansions. In certain cases, our estimates improve some of those existing coefficient bounds.

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참고문헌

  1. H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math. 130 (3) (2006), 179-222. https://dx.doi.org/10.1016/j.bulsci.2005.10.002 
  2. H. Airault and J. Ren, An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math. 126 (5) (2002), 343-367. https://dx.doi.org/10.1016/S0007-4497(02)01115-6 
  3. G. Akin and S. Sumer Eker, Coefficient estimates for a certain class of analytic and bi-univalent functions defined by fractional derivative, C. R., Math., Acad. Sci. Paris 352 (12) (2014), 1005-1010. https://dx.doi.org/10.1016/j.crma.2014.09.022 
  4. S. Altinkaya and S. Yalcin, Faber polynomial coefficient bounds for a subclass of bi-univalent functions, C. R., Math., Acad. Sci. Paris 353 (12) (2015), 1075-1080. https://dx.doi.org/10.1016/j.crma.2015.09.00 
  5. I.S. Berezin and N.P. Zhidkov, Computing Methods, Pergamon, North Atlantic Treaty Organization and London Mathematical Society, 1965. Zbl 0122.12903 
  6. I.V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π-1, Arxiv. https://arxiv.org/abs/1408.3902v9 
  7. D.A. Brannan and J. Clunie, Aspects of contemporary complex analysis. Academic Pr. (1980). Zbl 0483.00007 
  8. D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math. 31 (2) (1986), 70-77. Zbl 0614.30017 
  9. S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions, C. R., Math., Acad. Sci. Paris 352 (6) (2014), 479-484. https://dx.doi.org/10.1016/j.crma.2014.04.004 
  10. S. Bulut, N. Magesh and V.K. Balaji, Faber polynomial coefficient estimates for certain sub-classes of meromorphic bi-univalent functions, C. R., Math., Acad. Sci. Paris 353 (2) (2015), 113-116. https://dx.doi.org/10.1016/j.crma.2014.10.019 
  11. M. Caglar, H. Orhan and N. Yagmur, Coefficient bounds for new subclasses of bi-univalent functions, Filomat 27 (7) (2013), 1165-1171. https://dx.doi.org/10.2298/FIL1307165C 
  12. P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer, New York, 1983. Zbl 0514.30001 
  13. G. Faber, Uber polynomische Entwicklungen, Math. Ann. 57 (3) (1903) 389-408. https://dx.doi.org/10.1007/BF01444293 
  14. S.G. Hamidi, S.A. Halim and J.M. Jahangiri, Coefficient estimates for a class of meromorphic bi-univalent functions, C. R., Math., Acad. Sci. Paris 351 (9-10) (2013), 349-352. https://dx.doi.org/10.1016/j.crma.2013.05.005 
  15. S.G. Hamidi and J.M. Jahangiri, Faber polynomial coefficient estimates for analytic bi-close-to-convex functions, C. R., Math., Acad. Sci. Paris 352 (1) (2014), 17-20. https://dx.doi.org/10.1016/j.crma.2013.11.005 
  16. S.G. Hamidi and J.M. Jahangiri, Faber polynomial coefficients of bi-subordinate functions, C. R., Math., Acad. Sci. Paris 354 (4) (2016), 365-370. https://dx.doi.org/10.1016/j.crma.2016.01.013 
  17. S.G. Hamidi, T. Janani, G. Murugusundaramoorthy and J.M. Jahangiri, Coefficient estimates for certain classes of meromorphic bi-univalent functions, C. R., Math., Acad. Sci. Paris 352 (4) (2014), 277-282. https://dx.doi.org/10.1016/j.crma.2014.01.010 
  18. T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22 (4) (2012), 15-26. Zbl 1267.30040 
  19. J.M. Jahangiri and S.G. Hamidi, Advances on the coefficients of bi-prestarlike functions, C. R., Math., Acad. Sci. Paris 354 (10) (2016), 980-985. https://dx.doi.org/10.1016/j.crma.2016.08.009 
  20. M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63-68. https://dx.doi.org/10.1090/S0002-9939-1967-0206255-1 
  21. G. Murugusundaramoorthy, K. Vijaya and T. Bulboaca, Initial coefficients bounds for bi-univalent functions related to Gregory coefficients, Mathematics 2023, 11(13), 2857. https://dx.doi.org/10.3390/math11132857 
  22. E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, Arch. Ration. Mech. Anal. 32 (1969), 100-112. https://dx.doi.org/10.1007/BF00247676 
  23. M. Obradovic, A class of univalent functions, Hokkaido Math. J. 27 (2) (1998), 329-335. https://dx.doi.org/10.14492/hokmj/1351001289 
  24. G.M. Phillips, Gregory's method for numerical integration, Amer. Math. Monthly 79 (3) (1972), 270-274. https://dx.doi.org/10.1080/00029890.1972.11993028 
  25. S. Sivasubramanian, R. Sivakumar, T. Bulboaca and T.N. Shanmugam, On the class of bi-univalent functions, C. R., Math., Acad. Sci. Paris 352 (11) (2014), 895-900. https://dx.doi.org/10.1016/j.crma.2014.09.015 
  26. S. Sivasubramanian, R. Sivakumar, S. Kanas and Seong-A Kim, Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions, Ann. Polon. Math. 113 (3) (2015), 295-304. 
  27. H.M. Srivastava, S. Bulut, M. Caglar and N. Yagmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat 27 (5) (2013), 831-842. https://dx.doi.org/10.2298/FIL1305831S 
  28. H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010) 1188-1192. https://dx.doi.org/10.1016/j.aml.2010.05.009 
  29. T.S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981. 
  30. P.G. Todorov, On the Faber polynomials of the univalent functions of class Σ, J. Math. Anal. Appl. 162 (1) (1991), 268-276. https://dx.doi.org/10.1016/0022-247X(91)90193-4 
  31. Q.-H. Xu, Y.-C. Gui and H.M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett. 25 (2012), 990-994. https://dx.doi.org/10.1016/j.aml.2011.11.013 
  32. Q.-H. Xu, H.-G. Xiao and H.M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218 (2012), 11461-11465. https://dx.doi.org/10.1016/j.amc.2012.05.034 
  33. Z.-G. Wang and S. Bulut, A note on the coefficient estimates of bi-close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I 355 (2017). 
  34. P. Zaprawa, Estimates of initial coefficients for bi-univalent functions, Abstr. Appl. Anal. 2014, Art. ID 357480, 6 pp. https://dx.doi.org/10.1155/2014/357480