Document Type : Research Paper


1 Institute of Mathematics and Applications, Andharua, Bhubaneswar-751029, Odisha, India.

2 School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, Tamilnadu, India.


In this paper, by employing  sine hyperbolic inverse functions,  we  introduced the generalized  subfamily $\mathcal{RK}_{\sinh}(\beta)$ of analytic functions defined on the open unit disk $\Delta:=\{\xi: \xi \in \mathbb{C} \text{ and } |\xi|<1 \}$ associated with the petal-shaped domain. The bounds of the first three Taylor-Maclaurin's coefficients, Fekete-Szeg\"{o} functional and the second Hankel determinants are investigated for $f\in\mathcal{RK}_{\sinh}(\beta)$. We considered Borel distribution as an application to our main results. Consequently, a number of corollaries have been made based on our results, generalizing previous studies in this direction.


Main Subjects

[1] A. Alotaibi, M. Arif, M.A. Alghamdi and S. Hussain, Starlikeness associated with cosine hypergeometric function, Mathematics, 8 , Art. Id: 1118 (2020).
[2] 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 (2019), pp.1615-1630.
[3] K. Bano and M. Raza, Starlike functions asscoiated with cosine function, Bull. Iran. Math. Soc., (2020).
[4] D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett., 26 (2013), pp.103-107.
[5] O. Barukab, M. Arif, M. Abbas and S.K. Khan, Sharp bounds of the coefficient results for the family of bounded turning functions associated with petal-shaped domain, J. Funct. Spaces (2021), Art. Id: 5535629 (2021).
[6] N.E. Cho, V. Kumar, S.S. Kumar and V. Ravichandran, Radius problems for starlike functions associated with sine functions, Bull. Iran. Math. Soc., 45 (2019), pp.213-232.
[7] M. Fekete and G. Szego, Eine Benberkung uber ungerada Schlichte funcktionen, J. London Math. Soc., 8 (1933), pp.85-89.
[8] M. H. Golmohammadi, S. Najafzadeh and M.R. Foroutan, Some Properties of Certain subclass of meromorphic functions associated with $ (p,q) $-derivative, Sahand Commun. Math. Anal., 17 (4) (2020), pp. 71-84.
[9] W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polonic Math, 23 (1971), pp. 159-177.
[10] A. Janteng, S.A. Halim and M. Darus,\emph Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math., 7 (2)(2006), Art. 50, 5 pages.
[11] A. Janteng, S.A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., 1 (13) (2007), pp. 619-625.
[12] W. Koepf, On the Fekete-Szego problem for close-to-convex functions , Proc. Amer. Math. Soc., 101 (1987), pp. 89-95.
[13] S.S. Kumar and K. Arora, Starlike function associated with a petal shaped domain, arXiv Preprint 2010.10072.
[14] S.K. Lee, V. Ravichandran and S. Subramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl., 281 (2013), pp.1-17.
[15] R.J. Libera and E.J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in $P$, Proc. Amer. Math. Soc., 87 (2) (1983), pp. 251-257.
[16] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential fuction, Bull. Malays. Math. Sci. Soc., 38 (2015), pp. 365-386.
[17] S.S. Miller and P.T. Mocanu, Differential Subordination: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, 225, CRC Press, 2000.
[18] A.K. Mishra and T. Panigrahi, The Fekete-Szego problem for a class defined by Hohlov operator, Acta Univ. Apul., 29 (2012), pp. 241-254.
[19] R.N. Mohapatra and T. Panigrahi, Second Hankel determinant for a class of analytic functions defined by Komatu integral operator, Rend. Mat. Appl., 41 (1) (2020), pp. 51-58.
[20] G. Murugusundaramoorthy and T. Bulboaca, Hankel determinants for new subclasses of analytic functions related to shell shaped region, Mathematics, 8 (2020), 1041.
[21] G. Murugusundaramoorthy and K. Vijaya,Certain subclasses of analytic functions associated with generalized telephone numbers, Symmetry, 2022, 14, 1053.
[22] K.I. Noor and S.A. Shah, On certain generalized Bazilevic type functions associated with conic regions, Sahand Commun. Math. Anal., 17 (4) (2020), pp. 13-23. 
[23] C. Pommerenke, Univalent Functions, Studia Mathematica/ Mathematische Lehrbucher, 25, Vandenhoeck and Ruprecht, Gottingen, Germany, (1975).
[24] C. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc., 41 (1966), pp. 111-122.
[25] C. Pommerenke, On the Hankel determinant of univalent function ,Mathematika, 14 (1967), pp. 108-112.
[26] S.H. Sayedain Boroujeni and S.Najafzadeh, Error function and certain subclasses of analytic univalent functions, Sahand Commun. Math. Anal., 20 (1) (2023), pp. 107-117.
[27] K. Sharma, N.K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat., 27 (2016), pp. 923-939.
[28] L. Shi, I. Ali, M. Arif, N.E. Cho, S. Hussain and H. Khan, A study of third Hankel determinant problem for certain subfamilies of anlaytic functions involving cardioid domain, Mathematics, 7 (2019), 418.
[29] L. Shi, H.M. Srivastava, M. Arif, S.Hussain and H. Khan, An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function, Symmetry, 11 (2019), 598.
[30] J. Sokol and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike function, Zeszyty Naukowei oficyna Wydawnicza al. Powstancow Warszawy, 19 (1996), pp.101-105.
[31] H.M. Srivastava, Operators of basic (or $q$-)calculus and fractional $q$-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Tran. Sci., 44 (2020), pp. 327-344.
[32] A.K. Wanas and J.A. Khuttar, Applications of Borel distribution series on analytic functions, Earthline J. Math. Sci., 4 (1) (2020), pp. 71-82.