Research Article
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Year 2019, Volume 48, Issue 6, 1695 - 1705, 08.12.2019
https://doi.org/10.15672/HJMS.2018.632

Abstract

References

  • [1] Y. Abu Muhanna, L. Li, and S. Ponnusamy, Extremal problems on the class of convex functions of order -1/2, Arch. Math. (Basel) 103 (6), 461–471, 2014.
  • [2] K.O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, in: Inequal. Theory and Appl. 6, 1–7, editors: Y. J. Cho, J.K. Kim and S.S. Dragomir, 2010.
  • [3] D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett. 26 (1), 103–107, 2013.
  • [4] D. Bansal, S. Haharana, and J.K. Prajapat, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc. 52, 1139–1148, 2015.
  • [5] S.V. Bharanedhar and S. Ponnusamy, Coefficient conditions for harmonic univelent mappings and hypergeometric mappings, Rocky Mt. J. Math. 44, 753–777, 2014.
  • [6] D. Bshouty, S.S. Joshi, and S.B. Joshi, On close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 58, 1195–1199, 2013.
  • [7] D. Bshouty and A. Lyzzaik, Close-to-convexity criteria for planar harmonic mappings, Complex Appl. Oper. Theory 5, 767–774, 2011.
  • [8] D.G. Cantor, Power series with integral coefficients, Bull. Amer. Math. Soc. 69, 362– 366, 1963.
  • [9] J. Chen, A. Rasila, and X. Wang, Coefficient estimates and radii problems for certain classes of polyharmonic mappings, Complex Var. Elliptic Equ. 60, 354–371, 2015.
  • [10] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 9, 3–25, 1984.
  • [11] P. Dienes, The Taylor Series, Dover, New York, 1957.
  • [12] P.L. Duren, Univalent functions, Springer Verlag, New York Inc., 1983.
  • [13] P.L. Duren, Harmonic mappings in the plane, Cambridge University Press, Cambridge, 2004.
  • [14] A. Edrei, Sur les déterminants récurrents et les singularités d’une fonction donnée par son développement de Taylor, Compos. Math. 7, 20–88, 1940.
  • [15] M. Fekete and G. Szegő, Eine bemerkung über ungerade schlichte functions, J. London Math. Soc. 8, 85–89, 1933.
  • [16] T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal. 4 (52), 2573–2585, 2010.
  • [17] W.K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. Lond. Math. Soc. 18 (3), 77–94, 1968.
  • [18] A. Janteng, S. Halim, and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math. 7 (2), Art. 50, 5 pp, 2006.
  • [19] A. Janteng, S.A. Halim, and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (13), 619–625, 2007.
  • [20] D. Kalaj, S. Ponnusamy, and M. Vuorinen, Radius of close-to-convexity and fully starlikeness of harmonic mappings, Complex Var. Elliptic Equ. 59, 539–552, 2014.
  • [21] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc. 101, 89–95, 1987.
  • [22] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions II, Arch. Math. 49, 420–433, 1987.
  • [23] S.K. Lee, V. Ravichandran, and S. Subramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl. 2013, Art. 281, 17 pp, 2013.
  • [24] R.J. Libera and E.J. Złotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85, 225–230, 1982.
  • [25] R.J. Libera and E.J. Złotkiewicz, Coefficient bounds for the inverse of a function with derivatives in P, Proc. Am. Math. Soc. 87, 251–257, 1983.
  • [26] A.E. Livingston, The coefficients of multivalent close-to-convex functions, Proc. Am. Math. Soc. 21, 545–552, 1969.
  • [27] P.T. Mocanu, Sufficient conditions of univalence for complex functions in the class C1, Rev. Anal. Numér. Théor. Approx. 10, 75–79, 1981.
  • [28] P.T. Mocanu, Injectivity conditions in the complex plane, Complex Appl. Oper. Theory 5, 759–766, 2011.
  • [29] S. Nagpal and V. Ravichandran, A subclass of close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 59, 204–216, 2014.
  • [30] H. Orhan and P. Zaprawa, Third Hankel determinants for starlike and convex functions, Bull. Korean Math. Soc. 55, 165–173, 2018.
  • [31] C. Pommerenke, On the coefficients and Hankel determinant of univalent functions, J. Lond. Math. Soc. 41, 111–122, 1966.
  • [32] C. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14, 108–112, 1967.
  • [33] S. Ponnusamy and A. Rasila, Planar harmonic and quasiregular mappings, in: Topics in modern function theory, Ramanujan Math. Soc. Lect. Notes Ser. 19, 267–333, Ramanujan Math. Soc., Mysore, 2013.
  • [34] S. Ponnusamy and A. Sairam Kaliraj, On harmonic close-to-convex functions, Comput. Methods Funct. Theory 12, 669–685, 2012.
  • [35] S. Ponnusamy and A. Sairam Kaliraj, Constants and characterization for certain classes of univalent harmonic mappings, Mediterr. J. Math. 12, 647–665, 2015.
  • [36] M. Raza and S.N. Malik, Upper bound of the third Hankel determinant for a class of analytic functions related with Lemniscate of Bernoulli, J. Inequal. Appl. 2013, Art. 412, 8 pp., 2013.
  • [37] T.J. Suffridge, Some special classes of conformal mappings, in: Handbook of complex analysis: geometric function theory (Edited by Kühnau), 2, 309–338, Elsevier, Amsterdam, 2005.
  • [38] Y. Sun, Y. Jiang, and A. Rasila, On a subclass of close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 61, 1627–1643, 2016.
  • [39] D. Vamshee Krishna, B. Venkateswarlu, and T. RamReddy, Third Hankel determinant for bounded turning functions of order alpha, J. Nigerian Math. Soc. 34, 121–127, 2015.
  • [40] P. Zaprawa, Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14 (1), Art. 19, 10 pp., 2017.

On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings

Year 2019, Volume 48, Issue 6, 1695 - 1705, 08.12.2019
https://doi.org/10.15672/HJMS.2018.632

Abstract

In this paper, we obtain the upper bounds to the third Hankel determinants for convex functions of order $\alpha$ and bounded turning functions of order $\alpha$. Furthermore, several relevant results on a new subclass of close-to-convex harmonic mappings are obtained. Connections of the results presented here to those that can be found in the literature are also discussed.

References

  • [1] Y. Abu Muhanna, L. Li, and S. Ponnusamy, Extremal problems on the class of convex functions of order -1/2, Arch. Math. (Basel) 103 (6), 461–471, 2014.
  • [2] K.O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, in: Inequal. Theory and Appl. 6, 1–7, editors: Y. J. Cho, J.K. Kim and S.S. Dragomir, 2010.
  • [3] D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett. 26 (1), 103–107, 2013.
  • [4] D. Bansal, S. Haharana, and J.K. Prajapat, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc. 52, 1139–1148, 2015.
  • [5] S.V. Bharanedhar and S. Ponnusamy, Coefficient conditions for harmonic univelent mappings and hypergeometric mappings, Rocky Mt. J. Math. 44, 753–777, 2014.
  • [6] D. Bshouty, S.S. Joshi, and S.B. Joshi, On close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 58, 1195–1199, 2013.
  • [7] D. Bshouty and A. Lyzzaik, Close-to-convexity criteria for planar harmonic mappings, Complex Appl. Oper. Theory 5, 767–774, 2011.
  • [8] D.G. Cantor, Power series with integral coefficients, Bull. Amer. Math. Soc. 69, 362– 366, 1963.
  • [9] J. Chen, A. Rasila, and X. Wang, Coefficient estimates and radii problems for certain classes of polyharmonic mappings, Complex Var. Elliptic Equ. 60, 354–371, 2015.
  • [10] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 9, 3–25, 1984.
  • [11] P. Dienes, The Taylor Series, Dover, New York, 1957.
  • [12] P.L. Duren, Univalent functions, Springer Verlag, New York Inc., 1983.
  • [13] P.L. Duren, Harmonic mappings in the plane, Cambridge University Press, Cambridge, 2004.
  • [14] A. Edrei, Sur les déterminants récurrents et les singularités d’une fonction donnée par son développement de Taylor, Compos. Math. 7, 20–88, 1940.
  • [15] M. Fekete and G. Szegő, Eine bemerkung über ungerade schlichte functions, J. London Math. Soc. 8, 85–89, 1933.
  • [16] T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal. 4 (52), 2573–2585, 2010.
  • [17] W.K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. Lond. Math. Soc. 18 (3), 77–94, 1968.
  • [18] A. Janteng, S. Halim, and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math. 7 (2), Art. 50, 5 pp, 2006.
  • [19] A. Janteng, S.A. Halim, and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (13), 619–625, 2007.
  • [20] D. Kalaj, S. Ponnusamy, and M. Vuorinen, Radius of close-to-convexity and fully starlikeness of harmonic mappings, Complex Var. Elliptic Equ. 59, 539–552, 2014.
  • [21] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc. 101, 89–95, 1987.
  • [22] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions II, Arch. Math. 49, 420–433, 1987.
  • [23] S.K. Lee, V. Ravichandran, and S. Subramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl. 2013, Art. 281, 17 pp, 2013.
  • [24] R.J. Libera and E.J. Złotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85, 225–230, 1982.
  • [25] R.J. Libera and E.J. Złotkiewicz, Coefficient bounds for the inverse of a function with derivatives in P, Proc. Am. Math. Soc. 87, 251–257, 1983.
  • [26] A.E. Livingston, The coefficients of multivalent close-to-convex functions, Proc. Am. Math. Soc. 21, 545–552, 1969.
  • [27] P.T. Mocanu, Sufficient conditions of univalence for complex functions in the class C1, Rev. Anal. Numér. Théor. Approx. 10, 75–79, 1981.
  • [28] P.T. Mocanu, Injectivity conditions in the complex plane, Complex Appl. Oper. Theory 5, 759–766, 2011.
  • [29] S. Nagpal and V. Ravichandran, A subclass of close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 59, 204–216, 2014.
  • [30] H. Orhan and P. Zaprawa, Third Hankel determinants for starlike and convex functions, Bull. Korean Math. Soc. 55, 165–173, 2018.
  • [31] C. Pommerenke, On the coefficients and Hankel determinant of univalent functions, J. Lond. Math. Soc. 41, 111–122, 1966.
  • [32] C. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14, 108–112, 1967.
  • [33] S. Ponnusamy and A. Rasila, Planar harmonic and quasiregular mappings, in: Topics in modern function theory, Ramanujan Math. Soc. Lect. Notes Ser. 19, 267–333, Ramanujan Math. Soc., Mysore, 2013.
  • [34] S. Ponnusamy and A. Sairam Kaliraj, On harmonic close-to-convex functions, Comput. Methods Funct. Theory 12, 669–685, 2012.
  • [35] S. Ponnusamy and A. Sairam Kaliraj, Constants and characterization for certain classes of univalent harmonic mappings, Mediterr. J. Math. 12, 647–665, 2015.
  • [36] M. Raza and S.N. Malik, Upper bound of the third Hankel determinant for a class of analytic functions related with Lemniscate of Bernoulli, J. Inequal. Appl. 2013, Art. 412, 8 pp., 2013.
  • [37] T.J. Suffridge, Some special classes of conformal mappings, in: Handbook of complex analysis: geometric function theory (Edited by Kühnau), 2, 309–338, Elsevier, Amsterdam, 2005.
  • [38] Y. Sun, Y. Jiang, and A. Rasila, On a subclass of close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 61, 1627–1643, 2016.
  • [39] D. Vamshee Krishna, B. Venkateswarlu, and T. RamReddy, Third Hankel determinant for bounded turning functions of order alpha, J. Nigerian Math. Soc. 34, 121–127, 2015.
  • [40] P. Zaprawa, Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14 (1), Art. 19, 10 pp., 2017.

Details

Primary Language English
Subjects Mathematics
Journal Section Mathematics
Authors

Yong SUN This is me (Primary Author)
Hunan Institute of Engineering
0000-0001-8020-4684
China


Zhi-gang WANG This is me
Hunan First Normal University
0000-0001-6118-7196
China


Antti RASİLA
Aalto University
0000-0003-3797-942X
Finland

Publication Date December 8, 2019
Published in Issue Year 2019, Volume 48, Issue 6

Cite

Bibtex @research article { hujms479171, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2019}, pages = {1695 - 1705}, doi = {10.15672/HJMS.2018.632}, title = {On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings}, key = {cite}, author = {Sun, Yong and Wang, Zhi-gang and Rasila, Antti} }
APA Sun, Y. , Wang, Z. & Rasila, A. (2019). On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings . Hacettepe Journal of Mathematics and Statistics , 48 (6) , 1695-1705 . DOI: 10.15672/HJMS.2018.632
MLA Sun, Y. , Wang, Z. , Rasila, A. "On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings" . Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1695-1705 <https://dergipark.org.tr/en/pub/hujms/article/479171>
Chicago Sun, Y. , Wang, Z. , Rasila, A. "On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1695-1705
RIS TY - JOUR T1 - On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings AU - Yong Sun , Zhi-gang Wang , Antti Rasila Y1 - 2019 PY - 2019 N1 - doi: 10.15672/HJMS.2018.632 DO - 10.15672/HJMS.2018.632 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1695 EP - 1705 VL - 48 IS - 6 SN - 2651-477X-2651-477X M3 - doi: 10.15672/HJMS.2018.632 UR - https://doi.org/10.15672/HJMS.2018.632 Y2 - 2018 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings %A Yong Sun , Zhi-gang Wang , Antti Rasila %T On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 6 %R doi: 10.15672/HJMS.2018.632 %U 10.15672/HJMS.2018.632
ISNAD Sun, Yong , Wang, Zhi-gang , Rasila, Antti . "On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings". Hacettepe Journal of Mathematics and Statistics 48 / 6 (December 2019): 1695-1705 . https://doi.org/10.15672/HJMS.2018.632
AMA Sun Y. , Wang Z. , Rasila A. On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1695-1705.
Vancouver Sun Y. , Wang Z. , Rasila A. On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1695-1705.
IEEE Y. Sun , Z. Wang and A. Rasila , "On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings", Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 6, pp. 1695-1705, Dec. 2019, doi:10.15672/HJMS.2018.632