On fourth Hankel determinant for functions associated with Bernoulli’s lemniscate

The aim of this paper is to find an upper bound of the fourth Hankel determinant H4(1) for a subclass of analytic functions associated with the right half of the Bernoulli’s lemniscate of the form (x2 + y2)2−2 (x2 − y2) = 0. The problem is also discussed for 2-fold and 3-fold symmetric functions. The key tools in the proof of our main results are the coefficient inequalities for class P of functions with positive real part.

PDF

___

[1] R.M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. 26 (1), 63–71, 2003.
[2] R.M. Ali, N.E. Cho, V. Ravichandran and S.S. Kumar, First order differential subor-dination for functions associated with the lemniscate of Bernoulli, Taiwanese J. Math. 16 (3), 1017–1026, 2012.
[3] M. Arif, L. Rani, M. Raza and P. Zaprawa, Fourth Hankel determinant for a family of functions with bounded turning, Bull. Korean Math. Soc. 55 (6), 1703-1711, 2018.
[4] M. Arif, L. Rani, M. Raza and P. Zaprawa, Fourth Hankel determinant for a set of starlike function, submitted.
[5] K.O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, Inequal. Theory Appl. 6, 1–7, 2007.
[6] C. Carathéodory, Über den variabilitätsbereich der koeffizienten von potenzreihen die gegebene werte nicht annehmen, Math. Ann. 64, 95–115, 1907.
[7] C. Carathéodory, Über den variabilitätsbereich der fourier’schen konstanten von pos-itiven harmonischen funktionen, Rend. Circ. Mat. Palermo, 32, 193-127, 1911.
[8] E. Deniz and L. Budak, Second Hankel determinat for certain analytic functions satisfying subordinate condition, Math. Slovaca, 68 (2), 463–471, 2018.
[9] M. Fekete and G. Szegö, Eine bemerkung uber ungerade schlichte funktionen, J. Lon-don Math. Soc. 8, 85–89, 1933.
[10] A.W. Goodman, Univalent Functions, Mariner Publications, Tampa, FLorida, 1983.
[11] S.A. Halim and R. Omar, Applications of certain functions associated with lemniscate Bernoulli, J. Indones. Math. Soc. 18 (2), 93–99, 2012.
[12] A. Janteng, S.A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (13), 619–625, 2007.
[13] R.J. Libera and E.J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85, 225-230, 1982.
[14] J.W. Noonan and D.K. Thomas, On second Hankel determinant of a really mean p-valent functions, Trans. Amer. Math. Soc. 223 (2), 337–346, 1976.
[15] 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.
[16] M. Raza and S.N. Malik, Upper bound of the third Hankel determinant for a class of analytic functions related with with the lemniscate of Bernoulli, J. Inequal. Appl. (2013), art. 412, 2013.
[17] J. Sokól, On application of certain sufficient condition for starlikeness, J. Math. Appl. 30, 40–53, 2008.
[18] J. Sokól, Radius problem in the class SL, Appl. Math. Comput. 214, 569–573, 2009.
[19] J. Sokól, Coefficient estimates in a class of strongly starlike functions, Kyungpook Math. J. 49, 349–353, 2009.
[20] J. Sokól and D.K. Thomas, Further results on a class of starlike functions related to the Bernoulli lemniscate, Houston J. Math. 44, 83–95, 2018.
[21] J. Sokól and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Folia Scient. Univ. Tech. Resoviensis, 147, 101–105, 1996.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

Arşiv