Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On Coefficients of a Certain Subclass of Starlike and Bi-starlike Functions

  • Mahzoon, Hesam (Department of Mathematics, Islamic Azad University) ;
  • Sokol, Janusz (College of Natural Sciences, University of Rzeszow)
  • Received : 2019.06.06
  • Accepted : 2020.07.02
  • Published : 2021.09.30
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Abstract

In this paper we investigate a subclass 𝓜(α) of the class of starlike functions in the unit disk |z| < 1. 𝓜(α), π/2 ≤ α < π, is the set of all analytic functions f in the unit disk |z| < 1 with the normalization f(0) = f'(0) - 1 = 0 that satisfy the condition $$1+\frac{{\alpha}-{\pi}}{2\;sin\;{\alpha}}. The class 𝓜(α) was introduced by Kargar et al. [Complex Anal. Oper. Theory 11: 1639-1649, 2017]. In this paper some basic geometric properties of the class 𝓜(α) are investigated. Among others things, coefficients estimates and bound are given for the Fekete-Szegö functional associated with the k-th root transform [f(zk)]1/k. Also a certain subclass of bi-starlike functions is introduced and the bounds for the initial coefficients are obtained.

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References

  1. D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math., 31(2)(1986), 70-77.
  2. D. A. Brannan, J. Clunie and W. E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math. 22(1970), 476-485. https://doi.org/10.4153/CJM-1970-055-8
  3. M. Dorff, Convolutions of planar harmonic convex mappings, Complex Variables, Theory and Appl., 45(3)(2001), 263-271. https://doi.org/10.1080/17476930108815381
  4. R. Kargar, A. Ebadian and J. Soko l, Radius problems for some subclasses of analytic functions, Complex Anal. Oper. Theory, 11(2017), 1639-1649. https://doi.org/10.1007/s11785-016-0584-x
  5. F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20(1969), 8-12. https://doi.org/10.1090/S0002-9939-1969-0232926-9
  6. M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18(1967), 63-68. https://doi.org/10.1090/S0002-9939-1967-0206255-1
  7. E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, Arch. Rational Mech. Anal., 32(1969), 100-112. https://doi.org/10.1007/BF00247676
  8. Ch. Pommerenke, Univalent Functions, Vandenhoeck and Rupercht, Gottingen, 1975.
  9. W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc., 48(1943), 48-82.
  10. Y. Sun, Z.-G. Wang, A. Rasila and J. Soko l, On a subclass of starlike functions associated with a vertical strip domain, J. Ineq. Appl., (2019) 2019: 35.