대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제34권3호
  • /
  • Pages.841-854
  • /
  • 2019
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

A SUBCLASS OF HARMONIC UNIVALENT MAPPINGS WITH A RESTRICTED ANALYTIC PART

  • 투고 : 2018.06.22
  • 심사 : 2018.09.17
  • 발행 : 2019.07.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this article, a subclass of univalent harmonic mapping is introduced by restricting its analytic part to lie in the class $S^{\delta}[{\alpha}]$, $0{\leq}{\alpha}<1$, $-{\infty}<{\delta}<{\infty}$ which has been introduced and studied by Kumar [17] (see also [20], [21], [22], [23]). Coefficient estimations, growth and distortion properties, area theorem and covering estimates of functions in the newly defined class have been established. Furthermore, we also found bound for the Bloch's constant for all functions in that family.

키워드

참고문헌

  1. A. G. Akritas, A. W. Strzebonski, and P. S. Vigklas, Improving the performance of the continued fractions method using new bounds of positive roots, Nonlinear Anal. Model. Control 13 (2008), no. 3, 265-279. https://doi.org/10.15388/NA.2008.13.3.14557
  2. A. Alesina and M. Galuzzi, Vincent's theorem from a modern point of view, Rend. Circ. Mat. Palermo (2) Suppl. No. 64 (2000), 179-191.
  3. H. Chen, P. M. Gauthier, and W. Hengartner, Bloch constants for planar harmonic mappings, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3231-3240. https://doi.org/10.1090/S0002-9939-00-05590-8
  4. Sh. Chen, S. Ponnusamy, and X.Wang, Bloch constant and Landau's theorem for planar p-harmonic mappings, J. Math. Anal. Appl. 373 (2011), no. 1, 102-110. https://doi.org/10.1016/j.jmaa.2010.06.025
  5. J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3-25. https://doi.org/10.5186/aasfm.1984.0905
  6. F. Colonna, The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J. 38 (1989), no. 4, 829-840. https://doi.org/10.1512/iumj.1989.38.38039
  7. P. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.
  8. P. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156, Cambridge University Press, Cambridge, 2004. https://doi.org/10.1017/CBO9780511546600
  9. M. O. Gonzalez, Complex Analysis: Selected Topics, Marcel Dekker, INC, New York, 1992.
  10. A. W. Goodman, Univalent Functions. Vol. I and Vol. II, Mariner Publishing Co., Inc., Tampa, FL, 1983.
  11. I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Monographs and Textbooks in Pure and Applied Mathematics, 255, Marcel Dekker, Inc., New York, 2003. https://doi.org/10.1201/9780203911624
  12. I. Hotta and A. Michalski, Locally one-to-one harmonic functions with starlike analytic part, Bull. Soc. Sci. Lett. Lodz Ser. Rech. Deform. 64 (2014), no. 2, 19-27.
  13. S. Kanas and D. Klimek-Smet, Harmonic mappings related to functions with bounded boundary rotation and norm of the pre-Schwarzian derivative, Bull. Korean Math. Soc. 51 (2014), no. 3, 803-812. https://doi.org/10.4134/BKMS.2014.51.3.803
  14. S. Kanas and D. Klimek-Smet, Coefficient estimates and Bloch's constant in some classes of harmonic mappings, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 2, 741-750. https://doi.org/10.1007/s40840-015-0138-9
  15. D. Klimek-Smet and A. Michalski, Univalent anti-analytic perturbations of the identity in the unit disc, Sci. Bull. Che lm 1 (2006), 67-76.
  16. D. Klimek-Smet and A. Michalski, Univalent anti-analytic perturbations of convex analytic mappings in the unit disc, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 61 (2007), 39-49.
  17. V. Kumar, Quasi-Hadamard product of certain univalent functions, J. Math. Anal. Appl. 126 (1987), no. 1, 70-77. https://doi.org/10.1016/0022-247X(87)90074-6
  18. H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), no. 10, 689-692. https://doi.org/10.1090/S0002-9904-1936-06397-4
  19. M. Liu, Estimates on Bloch constants for planar harmonic mappings, Sci. China Ser. A 52 (2009), no. 1, 87-93. https://doi.org/10.1007/s11425-008-0090-3
  20. A. K. Mishra, Quasi-Hadamard product of analytic functions related to univalent functions, Math. Student 64 (1995), no. 1-4, 221-225.
  21. A. K. Mishra and M. Choudhury, A class of multivalent functions with negative Taylor coecients, Demonstratio Math. 28 (1995), no. 1, 223-234. https://doi.org/10.1515/dema-1995-0126
  22. A. K. Mishra and M. K. Das, Fractional integral operators and distortion theorems for a class of multivalent functions with negative coefficients, J. Anal. 4 (1996), 185-199.
  23. A. K. Mishra and P. Gochhayat, Coefficients of inverse functions in a nested class of starlike functions of positive order, J. Ineq. Pure and Appl. Math. 7 (2006), no. 3, Art. 94, 1-15.
  24. M. I. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), no. 2, 374-408. https://doi.org/10.2307/1968451
  25. E. C. Titchmarsh, The Theory of Functions, second edition, Oxford University Press, Oxford, 1939.
  26. A. J. H. Vincent, Sur la resolution des equations numeriques, J. Math. Pures Appl. 1 (1836), 341-372.
  27. M. Zhu and X. Huang, The distortion theorems for harmonic mappings with analytic parts convex or starlike functions of order $\beta$, J. Math. (2015), Art. ID 460191, 1-6. https://doi.org/10.1155/2015/460191