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Coefficient estimates for starlike and convex functions associated with cosine function

Year 2023, Volume: 52 Issue: 3, 596 - 618, 30.05.2023

Marımuthu K , Uma J , Teodor Bulboaca

Abstract

This paper deals with the new classes $\mathcal{S}^{\ast}_{\cos}$ and $\mathcal{S}^{\,\mathbf{c}}_{\cos}$ of starlike and convex functions, respectively, associated with the cosine function. We give initial coefficient bounds for the first seven coefficients of the functions that belong to these classes, and we evaluate the upper bounds for the Hankel determinant of order three and four. We found the upper bound of Zalcman functional for the above mentioned classes for the cases $n=3$ and $n=4$, showing that the Zalcman conjecture holds for these values. Moreover, we determined lower and upper bounds for the difference $\vert a_{4}\vert-\vert a_{3}\vert$ of the coefficients for the functions that belong to these classes.

Keywords

analytic functions subordination cosine function coefficient bounds Hankel determinant Zalcman functional logarithmic coefficients successive coefficients difference

References

  • [1] A. Abubaker and M. Darus, Hankel determinant for a class of analytic functions involving a generalized linear differential operator, Int. J. Pure Appl. Math. 69 (4), 429-435, 2011.
  • [2] A. Alotaibi, M. Arif, M.A. Alghamdi and S. Hussain, Starlikeness associated with cosine hyperbolic function, Mathematics 8 (7), 2020.
  • [3] M. Arif, L. Rani, M. Raza and P. Zaprawa, Fourth Hankel determinant for the set of star-like functions, Math. Probl. Eng. 2021, Art. ID 6674010, 8 pages, 2021.
  • [4] M. Arif, M. Raza, H. Tang, S. Hussain and H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math. 17 (1), 1615-1630, 2019.
  • [5] M. Arif, S. Umar, M. Raza, T. Bulboacă, M. Farooq and H. Khan, On fourth Hankel determinant for functions associated with Bernoulli’s lemniscate, Hacet. J. Math. Stat. 49 (5), 1777-1787, 2020.
  • [6] K.O. Babalola, On $H_{3}(1)$ Hankel determinant for some classes of univalent functions, Inequal. Theory Appl. 6, 1-7, 2007.
  • [7] K. Bano and M. Raza, Starlike functions associated with cosine functions, Bull. Iranian Math. Soc. 47 (5), 1513-1532, 2021.
  • [8] D. Bansal and J. Sokół, Zalcman conjecture for some subclass of analytic functions, J. Fract. Calc. Appl. 8 (1), 1-5, 2017.
  • [9] J.E. Brown and A. Tsao, On the Zalcman conjecture for starlike and typically real functions, Math. Z. 191, 467-474, 1986.
  • [10] C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1), 95-115, 1907.
  • [11] C. Carathéodory, Über den variabilitätsbereich der fourier’schen konstanten von positiven harmonischen funktionen, Rend. Circ. Mat. Palermo 32, 193-217, 1911.
  • [12] N.E. Cho, B. Kowalczyk, O.S. Kwon, A. Lecko and Y.J. Sim, Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha, J. Math. Inequal. 11 (2), 429-439, 2017.
  • [13] N.E. Cho, V. Kumar, S.S. Kumar and V. Ravichandran, Radius problems for starlike functions associated with the sine function, Bull. Iranian Math. Soc. 45 (1), 213-232, 2019.
  • [14] I. Efraimidis, A generalization of Livingston’s coefficient inequalities for functions with positive real part, J. Math. Anal. Appl. 435, 369-379, 2016.
  • [15] P. Goel and S. Kumar, Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc. 43, 957-991, 2020.
  • [16] H.Ö. Güney, G. Murugusundaramoorthy and H.M. Srivastava, The second Hankel determinant for a certain class of bi-close-to-convex functions, Results Math. 74 (3), Article No. 93, 13 pages, 2019.
  • [17] W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23, 159-177, 1970/71.
  • [18] F.R. Keogh and E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20, 8-12, 1969.
  • [19] M.G. Khan, B. Ahmad, G. Murugusundaramoorthy, R. Chinram and W.K. Mashwani, Applications of modified sigmoid functions to a class of starlike functions, J. Funct. Spaces 2020, Art. ID 8844814, 8 pages, 2020.
  • [20] M.G. Khan, B. Ahmad, G. Murugusundaramoorthy, W.K. Mashwani, S. Yalçın, T. G. Shaba and Z. Salleh, Third Hankel determinant and Zalcman functional for a class of starlike functions with respect to symmetric points related with sine function, J. Math. Computer Sci. 25, 29-36, 2022.
  • [21] M.G. Khan, B. Ahmad, J. Sokół, Z. Muhammad, W.K. Mashwani, R. Chinram and P. Petchkaew, Coefficient problems in a class of functions with bounded turning associated with Sine function, Eur. J. Pure Appl. Math. 14 (1), 53-64, 2021.
  • [22] N. Khan, M. Shafiq, M. Darus, B. Khan and Q.Z. Ahmad, Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the lemniscate of Bernoulli, J. Math. Inequal. 14 (1), 51-63, 2020.
  • [23] A. Lecko, Y.J. Sim and B. Śmiarowska, The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2, Complex Anal. Oper. Theory 13 (5), 2231-2238, 2019.
  • [24] W.C. Ma, The Zalcman conjecture for close-to-convex functions, Proc. Amer. Math. Soc. 104 (3), 741-744, 1988.
  • [25] W.C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157- 169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994.
  • [26] S. Mahmood, I. Khan, H.M. Srivastava and S.N. Malik, Inclusion relations for certain families of integral operators associated with conic regions, J. Inequal. Appl. 2019, Article No. 59, 11 pages, 2019.
  • [27] S. Mahmood, H.M. Srivastava, N. Khan, Q.Z. Ahmad, B. Khan and I. Ali, Upper bound of the third Hankel determinant for a subclass of q-starlike functions, Symmetry 11, 1-13, 2019.
  • [28] W.K. Mashwani, B. Ahmad, N. Khan, M.G. Khan, S. Arjika, B. Khan and R. Chinram, Fourth Hankel determinant for a subclass of starlike functions based on modified Sigmoid, J. Funct. Spaces 2021, Art. ID 6116172, 10 pages, 2021.
  • [29] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (1), 365-386, 2015.
  • [30] A.K. Mishra and P. Gochhayat, Second Hankel determinant for a class of analytic functions defined by fractional derivative, Int. J. Math. Math. Sci. 2008, Art. ID 153280, 10 pages, 2008.
  • [31] G. Murugusundaramoorthy and N. Magesh, Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant, Bull. Math. Anal. Appl. 1 (3), 85-89, 2009.
  • [32] Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41, 111-122, 1966.
  • [33] Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
  • [34] R.K. Raina and J. Sokół, On coefficient estimates for a certain class of starlike functions, Hacet. J. Math. Stat. 44 (6), 1427-1433, 2015.
  • [35] V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353 (6), 505-510, 2015.
  • [36] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1), 189-196, 1993.
  • [37] K. Sharma, N.K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat. 27 (5-6), 923-939, 2016.
  • [38] L. Shi, M. Ghaffar Khan and B. Ahmad, Some geometric properties of a family of analytic functions involving a generalized q-operator, Symmetry 12, 11 pages, 2020.
  • [39] Y.J. Sim and D.K. Thomas, On the difference of inverse coefficients of univalent functions, Symmetry 12 (12), 2040, 2020.
  • [40] J. Sokół, Radius problems in the class $\mathrm{SL}^{\ast}$, Appl. Math. Comput. 214 (2), 569-573, 2009.
  • [41] H.M. Srivastava, B. Khan, N. Khan, M. Tahir, S. Ahmad, and N. Khan, Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function, Bull. Sci. Math. 167, Article No. 102942, 16 pages, 2021.
  • [42] H. Tang, H.M. Srivastava, S.H. Li, and G.T. Deng, Majorization results for subclasses of starlike functions based on the sine and cosine functions, Bull. Iranian Math. Soc. 46 (2), 381-388, 2020.
  • [43] A. Vasudevarao and A. Pandey, The Zalcman conjecture for certain analytic and univalent functions, J. Math. Anal. Appl. 492, 124466, 2020.

Year 2023, Volume: 52 Issue: 3, 596 - 618, 30.05.2023

Marımuthu K , Uma J , Teodor Bulboaca

Abstract

References

  • [1] A. Abubaker and M. Darus, Hankel determinant for a class of analytic functions involving a generalized linear differential operator, Int. J. Pure Appl. Math. 69 (4), 429-435, 2011.
  • [2] A. Alotaibi, M. Arif, M.A. Alghamdi and S. Hussain, Starlikeness associated with cosine hyperbolic function, Mathematics 8 (7), 2020.
  • [3] M. Arif, L. Rani, M. Raza and P. Zaprawa, Fourth Hankel determinant for the set of star-like functions, Math. Probl. Eng. 2021, Art. ID 6674010, 8 pages, 2021.
  • [4] M. Arif, M. Raza, H. Tang, S. Hussain and H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math. 17 (1), 1615-1630, 2019.
  • [5] M. Arif, S. Umar, M. Raza, T. Bulboacă, M. Farooq and H. Khan, On fourth Hankel determinant for functions associated with Bernoulli’s lemniscate, Hacet. J. Math. Stat. 49 (5), 1777-1787, 2020.
  • [6] K.O. Babalola, On $H_{3}(1)$ Hankel determinant for some classes of univalent functions, Inequal. Theory Appl. 6, 1-7, 2007.
  • [7] K. Bano and M. Raza, Starlike functions associated with cosine functions, Bull. Iranian Math. Soc. 47 (5), 1513-1532, 2021.
  • [8] D. Bansal and J. Sokół, Zalcman conjecture for some subclass of analytic functions, J. Fract. Calc. Appl. 8 (1), 1-5, 2017.
  • [9] J.E. Brown and A. Tsao, On the Zalcman conjecture for starlike and typically real functions, Math. Z. 191, 467-474, 1986.
  • [10] C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1), 95-115, 1907.
  • [11] C. Carathéodory, Über den variabilitätsbereich der fourier’schen konstanten von positiven harmonischen funktionen, Rend. Circ. Mat. Palermo 32, 193-217, 1911.
  • [12] N.E. Cho, B. Kowalczyk, O.S. Kwon, A. Lecko and Y.J. Sim, Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha, J. Math. Inequal. 11 (2), 429-439, 2017.
  • [13] N.E. Cho, V. Kumar, S.S. Kumar and V. Ravichandran, Radius problems for starlike functions associated with the sine function, Bull. Iranian Math. Soc. 45 (1), 213-232, 2019.
  • [14] I. Efraimidis, A generalization of Livingston’s coefficient inequalities for functions with positive real part, J. Math. Anal. Appl. 435, 369-379, 2016.
  • [15] P. Goel and S. Kumar, Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc. 43, 957-991, 2020.
  • [16] H.Ö. Güney, G. Murugusundaramoorthy and H.M. Srivastava, The second Hankel determinant for a certain class of bi-close-to-convex functions, Results Math. 74 (3), Article No. 93, 13 pages, 2019.
  • [17] W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23, 159-177, 1970/71.
  • [18] F.R. Keogh and E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20, 8-12, 1969.
  • [19] M.G. Khan, B. Ahmad, G. Murugusundaramoorthy, R. Chinram and W.K. Mashwani, Applications of modified sigmoid functions to a class of starlike functions, J. Funct. Spaces 2020, Art. ID 8844814, 8 pages, 2020.
  • [20] M.G. Khan, B. Ahmad, G. Murugusundaramoorthy, W.K. Mashwani, S. Yalçın, T. G. Shaba and Z. Salleh, Third Hankel determinant and Zalcman functional for a class of starlike functions with respect to symmetric points related with sine function, J. Math. Computer Sci. 25, 29-36, 2022.
  • [21] M.G. Khan, B. Ahmad, J. Sokół, Z. Muhammad, W.K. Mashwani, R. Chinram and P. Petchkaew, Coefficient problems in a class of functions with bounded turning associated with Sine function, Eur. J. Pure Appl. Math. 14 (1), 53-64, 2021.
  • [22] N. Khan, M. Shafiq, M. Darus, B. Khan and Q.Z. Ahmad, Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the lemniscate of Bernoulli, J. Math. Inequal. 14 (1), 51-63, 2020.
  • [23] A. Lecko, Y.J. Sim and B. Śmiarowska, The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2, Complex Anal. Oper. Theory 13 (5), 2231-2238, 2019.
  • [24] W.C. Ma, The Zalcman conjecture for close-to-convex functions, Proc. Amer. Math. Soc. 104 (3), 741-744, 1988.
  • [25] W.C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157- 169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994.
  • [26] S. Mahmood, I. Khan, H.M. Srivastava and S.N. Malik, Inclusion relations for certain families of integral operators associated with conic regions, J. Inequal. Appl. 2019, Article No. 59, 11 pages, 2019.
  • [27] S. Mahmood, H.M. Srivastava, N. Khan, Q.Z. Ahmad, B. Khan and I. Ali, Upper bound of the third Hankel determinant for a subclass of q-starlike functions, Symmetry 11, 1-13, 2019.
  • [28] W.K. Mashwani, B. Ahmad, N. Khan, M.G. Khan, S. Arjika, B. Khan and R. Chinram, Fourth Hankel determinant for a subclass of starlike functions based on modified Sigmoid, J. Funct. Spaces 2021, Art. ID 6116172, 10 pages, 2021.
  • [29] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (1), 365-386, 2015.
  • [30] A.K. Mishra and P. Gochhayat, Second Hankel determinant for a class of analytic functions defined by fractional derivative, Int. J. Math. Math. Sci. 2008, Art. ID 153280, 10 pages, 2008.
  • [31] G. Murugusundaramoorthy and N. Magesh, Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant, Bull. Math. Anal. Appl. 1 (3), 85-89, 2009.
  • [32] Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41, 111-122, 1966.
  • [33] Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
  • [34] R.K. Raina and J. Sokół, On coefficient estimates for a certain class of starlike functions, Hacet. J. Math. Stat. 44 (6), 1427-1433, 2015.
  • [35] V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353 (6), 505-510, 2015.
  • [36] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1), 189-196, 1993.
  • [37] K. Sharma, N.K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat. 27 (5-6), 923-939, 2016.
  • [38] L. Shi, M. Ghaffar Khan and B. Ahmad, Some geometric properties of a family of analytic functions involving a generalized q-operator, Symmetry 12, 11 pages, 2020.
  • [39] Y.J. Sim and D.K. Thomas, On the difference of inverse coefficients of univalent functions, Symmetry 12 (12), 2040, 2020.
  • [40] J. Sokół, Radius problems in the class $\mathrm{SL}^{\ast}$, Appl. Math. Comput. 214 (2), 569-573, 2009.
  • [41] H.M. Srivastava, B. Khan, N. Khan, M. Tahir, S. Ahmad, and N. Khan, Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function, Bull. Sci. Math. 167, Article No. 102942, 16 pages, 2021.
  • [42] H. Tang, H.M. Srivastava, S.H. Li, and G.T. Deng, Majorization results for subclasses of starlike functions based on the sine and cosine functions, Bull. Iranian Math. Soc. 46 (2), 381-388, 2020.
  • [43] A. Vasudevarao and A. Pandey, The Zalcman conjecture for certain analytic and univalent functions, J. Math. Anal. Appl. 492, 124466, 2020.
There are 43 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Marımuthu K 0000-0002-4076-2494

Uma J 0000-0002-1547-823X

Teodor Bulboaca 0000-0001-8026-218X

Publication Date May 30, 2023
Published in Issue Year 2023 Volume: 52 Issue: 3

Cite

APA K, M., J, U., & Bulboaca, T. (2023). Coefficient estimates for starlike and convex functions associated with cosine function. Hacettepe Journal of Mathematics and Statistics, 52(3), 596-618.
AMA K M, J U, Bulboaca T. Coefficient estimates for starlike and convex functions associated with cosine function. Hacettepe Journal of Mathematics and Statistics. May 2023;52(3):596-618.
Chicago K, Marımuthu, Uma J, and Teodor Bulboaca. “Coefficient Estimates for Starlike and Convex Functions Associated With Cosine Function”. Hacettepe Journal of Mathematics and Statistics 52, no. 3 (May 2023): 596-618.
EndNote K M, J U, Bulboaca T (May 1, 2023) Coefficient estimates for starlike and convex functions associated with cosine function. Hacettepe Journal of Mathematics and Statistics 52 3 596–618.
IEEE M. K, U. J, and T. Bulboaca, “Coefficient estimates for starlike and convex functions associated with cosine function”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 3, pp. 596–618, 2023.
ISNAD K, Marımuthu et al. “Coefficient Estimates for Starlike and Convex Functions Associated With Cosine Function”. Hacettepe Journal of Mathematics and Statistics 52/3 (May 2023), 596-618.
JAMA K M, J U, Bulboaca T. Coefficient estimates for starlike and convex functions associated with cosine function. Hacettepe Journal of Mathematics and Statistics. 2023;52:596–618.
MLA K, Marımuthu et al. “Coefficient Estimates for Starlike and Convex Functions Associated With Cosine Function”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 3, 2023, pp. 596-18.
Vancouver K M, J U, Bulboaca T. Coefficient estimates for starlike and convex functions associated with cosine function. Hacettepe Journal of Mathematics and Statistics. 2023;52(3):596-618.

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