New subclasses of Ozaka's convex functions

Document Type : Research Paper


Department of Mathematics, Faculty of Sciences, Urmia University, P. O. Box 165, Urmia, Iran


Let $\mathcal{S}^{\ast}_{L}(\uplambda)$ and $\mathcal{CV}_L(\uplambda)$ be the classes of functions
$f$, analytic in the unit disc $\Updelta=\{z\colon|z|<1\}$, with the
normalization $f(0)=f'(0)-1=0$, which satisfies the conditions
\frac{zf'(z)}{f(z)}\prec \left(1+z\right)^{\uplambda}\quad\text{and}\quad \left(1+\frac{zf''(z)}{f'(z)}\right)\prec \left(1+z\right)^{\uplambda}
\qquad \left(0<\uplambda\le 1 \right),
where $\prec$ is the subordination relation, respectively. The classes
$\mathcal{S}^{\ast}_{L}(\uplambda)$ and $\mathcal{CV}_L(\uplambda)$ are subfamilies of the known classes of strongly starlike and convex functions of order $\uplambda$. We consider  the relations between $\mathcal{S}^{\ast}_{L}(\uplambda)$, $\mathcal{CV}_L(\uplambda)$ and other classes geometrically defined. Also, we obtain the sharp radius of convexity for functions belonging to $\mathcal{S}^{\ast}_{L}(\uplambda)$ class. Furthermore,  the norm of pre-Schwarzian derivatives  and univalency of functions $f$ which satisfy the condition
\myp{z \in \Updelta},
are considered.


[1] 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 (2012), no. 3, 553–561.
[2] M.K. Aouf, J. Dziok and J. Sok´o l, On a subclass of strongly starlike functions, Appl. Math. Comput. 24 (2011), no. , 27–32.
[3] 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 (2012), no. 1, 6557–6565.
[4] R.M. Ali, V. Ravichandran and N. Seenivasagan, Coefficient bounds for p-valent functions, Appl. Math. Comput. 187 (2007), no. 1, 35–46.
[5] D.A. Brannan and W.E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. 2 (1969), no. 1, 431–443.
[6] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Vol. 259. Springer, New York (1983)
[7] K. Kuroki and S. Owa, Notes on new class for certain analytic functions, Adv. Math. Sci. J. 1 (2012), no. 1, 127–131.
[8] 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.
[9] S.S. Miller and P.T. Mocanu, Differential Subordinations: Theory and Applications, Series of Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York, Basel, 2000.
[10] J.W. Noonan and D.K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc. 223 (1976), 337–346.
[11] S. Ozaki, On the theory of multivalent functions. II, Sci. Rep. Tokyo Bunrika Daigaku. Sect. A. 4 (1941), 45–87.
[12] M. Obradovi´c, S. Ponnusamy and K.-J. Wirths, Coefficient characterizations and sections for some univalent functions, Sib. Math. J. 54 (2013), 679–696.
[13] E. Paprocki and J. Sok´o l, The extermal problems in some subclasses of strongly functions, Folia Scient. Univ. Tech. Resov. 20 (1996), 89–94.
[14] M.I. Robertson, On the theory of univalent functions, Ann. Math. 37 (1936), no. 2, 374–408.
[15] J. Sok´o l, On application of certain sufficient condition for starlikeness, J. Math. Appl. 30 (2008), 131–135.
[16] J. Sok´o l, On some subclass of strongly starlike functions, Demonstr. Math. 31 (1998), no. 1, 81–86.
[17] J. Sok´o l, Coefficient Estimates in a Class of Strongly Starlike Functions, Kyungpook Math. J. 49 (2009), no. 2, 349–353.
[18] J. Sok´o l and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Folia Scient. Univ. Tech. Resov. 19 (1996), 101–105.
[19] J. Sok´o l and D. K. Thomas, Further Results on a Class of Starlike Functions Related to the Bernoulli Lemniscate, Houston J. Math. 44 (2018), 83–95.
[20] T. Umezawa, Analytic functions convex in one direction, J. Math. Soc. Japan 4 (1952), 194–202.
Volume 14, Issue 3
March 2023
Pages 189-199
  • Receive Date: 18 July 2022
  • Revise Date: 30 May 2022
  • Accept Date: 12 October 2022
  • First Publish Date: 01 March 2023