On Strongly Starlike Functions Related to the Bernoulli Lemniscate

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Vali Soltani Masih
Ali Ebadian
Janusz Sokol


Let $\mathcal{S}^{\ast}_{L}(\lambda)$ be the class of functions $f$, analytic in the unit disc $\Delta=\{z:|z|<1\}$, with the normalization $f(0)=f'(0)-1=0$, which satisfy the condition
\frac{zf'(z)}{f(z)}\prec \left(1+z\right)^{\lambda},
where $\prec$ is the subordination relation. The class $\mathcal{S}^{\ast}_{L}(\lambda)$ is a subfamily of the known class of strongly starlike functions of order $\lambda$. In this paper,
the relations between $\mathcal{S}^{\ast}_{L}(\lambda)$ and other classes geometrically defined are considered. Also, we obtain some characteristics such as, bounds for coefficients, radius of convexity, the Fekete-Szeg\"{o} inequality, logarithmic coefficients and the second Hankel determinant inequality for functions belonging to this class. The univalent functions $f$ which satisfy the condition
(z \in \Delta)\end{equation*}
are also considered here.

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How to Cite
Masih, V. S., Ebadian, A., & Sokol, J. (2022). On Strongly Starlike Functions Related to the Bernoulli Lemniscate. Tamkang Journal of Mathematics, 53(3), 187–199. https://doi.org/10.5556/j.tkjm.53.2022.3234


R. M. Ali, N. E. Cho, N. K. Jain and V. Ravichandran, Radii of starlikeness and convexity for functions with fixed second coefficient defined by subordination. Filomat 26(3), 553–561 (2012)

M. K. Aouf, J. Dziok and J. Sokół, On a subclass of strongly starlike functions. Appl. Math. Comput. 24(1), 27–32 (2011)

R. M. Ali, N. K. Jain and V. Ravichandran, Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane. Appl. Math. Comput. 218(11), 6557–6565 (2012)

D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions. J. London Math. Soc. 2(1), 431–443 (1969)

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Vol. 259. Springer, New York (1983)

R. J. Libera and E. J. Złotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87(2)(1983) 251–257.

F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions. Proc. Amer. Math. Soc. 20(1), 8–12 (1969)

K. Kuroki and S. Owa, Notes on new class for certain analytic functions. Adv. Math. Sci. J. 1(2), 127–131 (2012)

W. Ma and D. Minda, A unied treatment of some special classes of univalent functions, in Proc. Conf. on Complex Analysis, Tianjin, 1992, Conference Proceedings and Lecture Notes in Analysis, Vol. 1 (International Press, Cambridge, MA, 1994) 157-169.

S. S. Miller and P. T. Mocanu, Differential subordinations : theory and applications, in : Series of Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York, Basel, (2000)

J. W. Noonan and D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions. Trans. Amer. Math. Soc. 223, 337–346 (1976)

M. Obradović, S. Ponnusamy and K. J. Wirths, Coefficient characterizations and sections for some univalent functions. Sib. Math. J. 54(4), 679–696 (2013)

E. Paprocki and J. Sokół, The extermal problems in some subclasses of strongly functions. Folia Scient. Univ. Tech. Resoviensis 20, 89–94 (1996)

M. I. Robertson, On the theory of univalent functions. Appl. Math. 374–408 (1936)

W. Rogosinski, On the coefficients of subordinate functions. Proc. Lond. Math. Soc. 2(1), 48–82 (1945)

J. Sokół, On application of certain sufficient condition for starlikeness. J. Math. Appl. 30, 131– 135 (2008)

J. Sokół, On some subclass of strongly starlike functions. Demonstr. Math. 31(1), 81–86 (1998)

J. Sokół, Coefficient Estimates in a Class of Strongly Starlike Functions. Kyungpook Math. J. 49(2), 349-353 (2009)

J. Sokół and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions. Folia Scient. Univ. Tech. Resoviensis 19, 101–105 (1996)

J. Sokół and D. K. Thomas, Further Results on a Class of Starlike Functions Related to the Bernoulli Leminscate. Houston J. Math. 44, 83–95 (2018)