STARLIKENESS ASSOCIATED WITH THE SINE HYPERBOLIC FUNCTION

  • Mohsan RAZA ,
  • Hadiqa ZAHID ,
  • Jinlin LIU
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  • 1. Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan;
    2. Department of Information Science, Yangzhou University, Yangzhou 225002, China
E-mail: mohsan976@yahoo.com; hadiqazahid219@gmail.com

Received date: 2022-12-27

  Revised date: 2023-04-11

  Online published: 2024-08-30

Supported by

The first author's work was supported by the Grant No. 20-16367/NRPU/RD/HEC/2021 2021.

Abstract

Let $q_{\lambda }\left( z\right) =1+\lambda \sinh (\zeta ),\ 0<\lambda <1/\sinh \left( 1\right) $ be a non-vanishing analytic function in the open unit disk. We introduce a subclass $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $ of starlike functions which contains the functions $\mathfrak{f}$ such that $z\mathfrak{f}^{\prime }/\mathfrak{f}$ is subordinated by $q_{\lambda }$. We establish inclusion and radii results for the class $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $ for several known classes of starlike functions. Furthermore, we obtain sharp coefficient bounds and sharp Hankel determinants of order two for the class $\mathcal{S}^{\ast }\left( q_{\lambda }\right) $. We also find a sharp bound for the third Hankel determinant for the case $\lambda =1/2$.

Cite this article

Mohsan RAZA , Hadiqa ZAHID , Jinlin LIU . STARLIKENESS ASSOCIATED WITH THE SINE HYPERBOLIC FUNCTION[J]. Acta mathematica scientia, Series B, 2024 , 44(4) : 1244 -1270 . DOI: 10.1007/s10473-024-0404-8

References

[1] Ali R M. Coefficients of the inverse of strongly starlike functions. Bull Malays Math Sci Soc, 2003, 26: 63-71
[2] Ali R M, Jain R N K, Ravichandran V. Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane. Appl Math Comp, 2012, 218: 6557-6565
[3] Banga S, Kumar S S. The sharp bounds of the second and third Hankel determinants for the class $\mathcal{SL}^{\ast }$. Math Slovaca, 2020, 70: 849-862
[4] Bano K, Raza M. Starlike functions associated with cosine functions. Bull Iran Math Soc, 2021, 47: 1513-1532
[5] Cho N E, Kumar V, Kumar S S, Ravichandran V. Radius problems for starlike functions associated with the sine function. Bull Iran Math Soc, 2019, 45: 213-232
[6] Choi J H, Kim Y C, Sugawa T. A general approach to the Fekete-Szegö problem. J Math Soc Japan, 2007, 59: 707-727
[7] Goel P, Kumar S S. Certain class of starlike functions associated with modified sigmoid function. Bull Malays Math Sci Soc, 2020, 43: 957-991
[8] Janowski W. Extremal problems for a family of functions with positive real part and for some related families. Ann Polon Math, 1970, 23: 159-177
[9] Kanas S, Masih V S. On the behaviour of analytic representation of the generalized Pascal snail. Anal Math Phy, 2021, 11: 1-27
[10] Kargar R, Ebadian A, Sokół J. On Booth lemniscate and starlike functions. Anal Math Phys, 2019, 9: 143-154
[11] Kumar S S, Arora K. Starlike functions associated with a petal shaped domain. Bull Korean Math Soc, 2022, 59: 993-1010
[12] Kumar S S, Gangania K. A cardioid domain and starlike functions. Anal Math Phy, 2021, 11: 1-34
[13] Kumar S S, Khan M G, Ahmad B, Mashwani W K.A class of analytic functions associated with sine hyperbolic functions. arXiv: 2011.04875
[14] Kowalczyk B, Lecko A, Sim Y J. The sharp bound of the Hankel determinant of the third kind for convex functions. Bull Austr Math Soc, 2018, 97: 435-445
[15] Kowalczyk B, Lecko A, Lecko M, Sim Y J. The sharp bound of the third Hankel determinant for some classes of analytic functions. Bull Korean Math Soc, 2018, 55: 1859-1868
[16] Kwon O S, Lecko A, Sim Y J. The bound of the Hankel determinant of the third kind for starlike functions. Bull Malays Math Sci Soc, 2019, 42: 767-780
[17] Kwon O S, Lecko A, Sim Y J. On the fourth coefficient of functions in the Carathéodory class. Comput Methods Funct Theory, 2018, 18: 307-314
[18] Lecko A, Sim Y J, Smiarowska B. The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2. Complex Anal Oper Theory, 2019, 13: 2231-2238
[19] Libera R J, Zlotkiewicz E J. Early coefficients of the inverse of a regular convex functions. Proc Amer Math Soc, 1982, 85: 225-230
[20] Ma W, Minda D.A unified treatment of some special classes of univalent functions//Li Z, Ren F, Yang L, Zhang S. Proceeding of the Conference on Complex Analysis. Boston: Int Press, 1994: 157-169
[21] Malik S N, Raza M, Sokół J, Zainab S. Analytic functions associated with cardioid domain. Turk J Math, 2020, 44: 1127-1136
[22] Masih V S, Kanas S. Subclasses of starlike and convex functions associated with the limaçon domain. Symmetry2020, 12: 1-11
[23] Mendiratta R, Nagpal S, Ravichandran V. A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli. Int J Math2014, 25: 1-17
[24] Mendiratta R, Nagpal S, Ravichandran V. On a subclass of strongly starlike functions associated with exponential function. Bull Malays Math Sci Soc, 2015, 38: 365-386
[25] Nehari Z. Conformal Mapping.New York: McGraw-Hill Inc, 1952
[26] Pommerenke C. On the coefficients and Hankel determinants of univalent functions. J London Math Soc, 1966, 14: 111-122
[27] Raina R K, Sokół J. On coefficient for certain class of starlike functions. Hacettepe J Math Stat, 2015, 44: 1427-1433
[28] Ravichandran V, Verma S. Bound for the fifth coefficient of certain starlike functions. C R Math, 2015, 2015, 353: 505-510
[29] Ravichandran V, Ronning F, Shanmugam T N. Radius of convexity and radius of starlikeness for some classes of analytic functions. Compl Var Elli Equ, 1997, 33: 265-280
[30] Riaz A, Raza M, Thomas D K. Hankel determinants for starlike and convex functions associated with sigmoid functions. Forum Math, 2022, 34: 137-156
[31] Shah G M. On the univalence of some analytic functions. Pacific J Math, 1972, 43: 239-250
[32] Sharma K, Jain N K, Ravichandran V. Starlike functions associated with a cardioid. Afr Math, 2016, 27: 923-939
[33] Sokół J. Radius problems in the class $\mathcal{SL}^{\ast }$. Appl Math Comput, 2009, 214: 569-573
[34] Sokół J, Stankiewicz J. Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk Politech Rzeszowskiej Mat, 1996, 19: 101-105
[35] Ullah K, Srivastava H M, Rafiq A, et al. A study of sharp coefficient bounds for a new subfamily of starlike functions. J Inequal Appl, 2021, 2021: Art 194
[36] Wani L A, Swaminathan A. Starlike and convex functions associated with nephroid domain. Bull Malays Math Sci Soc, 2021, 44: 79-104
[37] Yunus Y, Halim S A, Akbarally A B. Subclass of starlike functions associated with a limaçon. AIP Conf Proc, 2018, 1974: 030023
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