The key tools in the proof of our main results are the coefficient inequalities for class $\mathcal{P}$ of functions with positive real part.***************************************************************************************************************************
📌 3 Key Takeaways
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Objective & Hypothesis:The aim of this paper is to find an upper bound of the fourth Hankel determinant $H_{4}(1)$ for a subclass of analytic functions associated with the right half of the Bernoulli's lemniscate of the form $\left(x^{2}+y^{2}\right) ^{2}-2\left( x^{2}-y^{2}\right) =0$.
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Key Findings & Data:The problem is also discussed for 2-fold and 3-fold symmetric functions.
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Conclusion & Impact:The key tools in the proof of our main results are the coefficient inequalities for class $\mathcal{P}$ of functions with positive real part.***************************************************************************************************************************
The aim of this paper is to find an upper bound of the fourth Hankel determinant $H_{4}(1)$ for a subclass of analytic functions associated with the right half of the Bernoulli's lemniscate of the form $\left(x^{2}+y^{2}\right) ^{2}-2\left( x^{2}-y^{2}\right) =0$. 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 $\mathcal{P}$ of functions with positive real part.***************************************************************************************************************************
ARİF, M., UMAR, S., RAZA, M., BULBOACA, T., FAROOQ, M. U., KHAN, H. (2020). On fourth Hankel determinant for functions associated with Bernoulli's lemniscate. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 5, 1777–1787. https://ajindex.com/articles/669271
R.M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. 26 (1), 63–71, 2003
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
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
M. Arif, L. Rani, M. Raza and P. Zaprawa, Fourth Hankel determinant for a set of starlike function, submitted
K.O. Babalola, On $H_{3}(1)$ Hankel determinant for some classes of univalent functions, Inequal. Theory Appl. 6, 1–7, 2007
C. Carathéodory, Über den variabilitätsbereich der koeffizienten von potenzreihen die gegebene werte nicht annehmen, Math. Ann. 64, 95–115, 1907
C. Carathéodory, Über den variabilitätsbereich der fourier’schen konstanten von pos- itiven harmonischen funktionen, Rend. Circ. Mat. Palermo, 32, 193-127, 1911
E. Deniz and L. Budak, Second Hankel determinat for certain analytic functions satisfying subordinate condition, Math. Slovaca, 68 (2), 463–471, 2018
M. Fekete and G. Szegö, Eine bemerkung uber ungerade schlichte funktionen, J. Lon- don Math. Soc. 8, 85–89, 1933
S.A. Halim and R. Omar, Applications of certain functions associated with lemniscate Bernoulli, J. Indones. Math. Soc. 18 (2), 93–99, 2012
A. Janteng, S.A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (13), 619–625, 2007
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
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
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
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
J. Sokól, On application of certain sufficient condition for starlikeness, J. Math. Appl. 30, 40–53, 2008