Hankel determinant problems for certain subclasses of Sakaguchi type functions defined with subordination
Main Article Content
Abstract
The present investigation is concerned with the estimation of initial coefficients, Fekete-Szeg\"{o} inequality, second Hankel determinants, Zalcman functionals and third Hankel determinants for certain subclasses of Sakaguchi type functions defined with subordination in the open unit disc $E=\{z\in\mathbb{C}: |z|<1\}$. The results derived in this paper will pave the way for the further study in this direction.
Article Details
References
[1] R. M. Ali, N. E. Cho, V. Ravichandran and S. Sivaprasad Kumar, Differential subordination for functions associated with the lemniscate of Bernoulli, Taiwanese J. Math. 16 (3) (2012), 1017–1026. Google Scholar
[2] Sahsene Altinkaya and Sibel Yalcin, Third Hankel determinant for Bazilevic functions, Adv. Math., Scientific Journal, 5 (2) (2016), 91–96. Google Scholar
[3] Muhammad Arif, Mohsan Raza, Huo Tang, Shehzad Hussain and Hassan Khan, Hankel determinant of order three for familiar subsets of ananlytic functions related with sine function, Open Math. 17 (2019), 1615–1630. Google Scholar
[4] K. O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, Ineq. Th. Appl. 6 (2010), 1–7. Google Scholar
[5] L. Bieberbach, U ̈ber die koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln, Sitzungsberichte Preussische Akademie der Wissenschaften, 138 (1916), 940–955. Google Scholar
[6] R. N. Das and P. Singh, On subclasses of schlicht mappings, Ind. J. Pure Appl. Math. 8 (1977), 864–872. Google Scholar
[7] L. De-Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152. Google Scholar
[8] M. Fekete and G. Szeg ̈o, Eine Bemer Kung uber ungerade schlichte Functionen, J. Lond. Math. Soc. 8 (1933), 85–89. Google Scholar
[9] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. Google Scholar
[10] I. Iohvidov, Hankel and Toeplitz matrices and forms, Birkh ̈auser, Boston, Mass, 1982. Google Scholar
[11] Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (13) (2007), 619–625. Google Scholar
[12] S. R. Keogh and E.P. Merkes, A coefficienit inequality for certain subclasses of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12. Google Scholar
[13] R. J. Libera and E-J. Zlotkiewiez, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225–230. Google Scholar
[14] R. J. Libera and E-J. Zlotkiewiez, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (1983), 251–257. Google Scholar
[15] W. Ma, Generalized Zalcman conjecture for starlike and typically real functions, J. Math. Anal. Appl. 234 (1999), 328–329. Google Scholar
[16] B. S. Mehrok and Gagandeep Singh, Estimate of second Hankel determinant for certain classes of analytic functions, Scientia Magna, 8 (3) (2012), 85–94. Google Scholar
[17] Sh. Najafzadeh, H. Rahmatan and H. Haji, Application of subordination for estimating the Hankel determinants for subclass of univalent functions, Creat. Math. Inform. 30 (1) (2021), 69–74. Google Scholar
[18] J. W. Noonan and D. K. Thomas, On the second Hankel determinant of a really mean p-valent functions, Trans. Amer. Math. Soc. 223 (2) (1976), 337–346. Google Scholar
[19] Ch. Pommerenke, Univalent functions, Math. Lehrbucher, vandenhoeck and Ruprecht, Gottingen, 1975. Google Scholar
[20] V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, Comptes Rendus mathematique, 353 (2015), 505–510. Google Scholar
[21] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11 (1959),72–80. Google Scholar
[22] G. Shanmugam, B. Adolf Stephen and K. O. Babalola, Third Hankel determinant for α starlike functions, Gulf J. Math. 2 (2) (2014), 107–113. Google Scholar
[23] Gagandeep Singh and Gurcharanjit Singh, On third Hankel determinant for a subclass of analytic functions, Open Sci. J. Math. Appl. 3 (6) (2015), 172–175. Google Scholar
[24] Gurmeet Singh, Gagandeep Singh and Gurcharanjit Singh, Certain subclasses of multivalent functions defined with generalized Salagean operator and related to sigmoid function and lemniscate of Bernoulli, J. Frac. Calc. Appl. 13 (1) (2022), 65–81. Google Scholar
[25] J. Sokol and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19 (1996), 101–105. Google Scholar
[26] J. Sokol and D. K. Thomas, Further results on a class of starlike functions related to the Bernoulli lemniscate, Houston J. Math., 44 (1) (2018), 83–95. Google Scholar
[27] N. Ullah, I. Ali, S. M. Hussain, Jong-Suk Ro, N. Khan and B. Khan, Third Hankel determi- nant for a subclass of univalent functions associated with lemniscate of Bernoulli, Fractal and Fractional, 6 (48) (2022), 1–8. Google Scholar