Sufficient conditions and radius problems for the Silverman class
Abstract
UDC 517.5
For $0<\alpha\leq1$ and $\lambda>0,$ let \begin{equation}\label{1} G_{\lambda,\alpha}=\left\{f\in\mathcal{A}\colon \left|\frac{1-\alpha+\alpha zf''(z)/f'(z)}{z f'(z)/f(z)}-(1-\alpha)\right|<\lambda,\ z\in\mathbb{D}\right\}.\tag{0.1}\end{equation} The general form of the Silverman class introduced by Tuneski and Irmak [Int. J. Math. and Math. Sci., {\bf 2006}, Article~ID 38089 (2006)]. Our differential inequality formulation lays out several sufficient conditions for this class. Further, we consider a class $\Omega$ given by \begin{equation}\label{omega}\Omega=\left\{f\in\mathcal{A}\colon |zf'(z)-f(z)|<\frac{1}{2},\ z\in\mathbb{D}\right\}.\tag{0.2}\end{equation} For these two classes, we establish inclusion relations involving some well-known subclasses of $\mathcal{S}^*$ and compute radius estimates featuring various pairings of these classes.
References
T. Bulboacǎ, N. Tuneski, New criteria for starlikeness and strongly starlikeness, Mathematica, 43(66), № 1, 11–22 (2003) (2001).
N. E. Cho, V. Kumar, S. S. Kumar, V. Ravichandran, Radius problems for starlike functions associated with the sine function, Bull. Iranian Math. Soc., 45, № 1, 213–232 (2019). DOI: https://doi.org/10.1007/s41980-018-0127-5
J. Dieudonné, Recherches sur quelques problèmes relatifs aux polynômes et aux fonctions bornées d'une variable complexe, Ann. Sci. Ècole Norm. Supér (3), 48, 247–358 (1931). DOI: https://doi.org/10.24033/asens.812
P. Goel, S. Sivaprasad Kumar, Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc., 43, № 1, 957–991 (2020). DOI: https://doi.org/10.1007/s40840-019-00784-y
S. S. Kumar, G. Kamaljeet, A cardioid domain and starlike functions, Anal. Math. Phys., 11, № 2, Article 54 (2021). DOI: https://doi.org/10.1007/s13324-021-00483-7
S. G. Krantz, Handbook of complex variables, Birkhäuser Boston, Inc., Boston, MA (1999). DOI: https://doi.org/10.1007/978-1-4612-1588-2
W. C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin), Lecture Notes Anal., I, Int. Press, Cambridge, MA (1992), p. 157–169.
R. Mendiratta, S. Nagpal, V. Ravichandran, A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli, Internat. J. Math., 25, № 9, Article 1450090 (2014). DOI: https://doi.org/10.1142/S0129167X14500906
R. Mendiratta, S. Nagpal, V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc., 38, № 1, 365–386 (2015). DOI: https://doi.org/10.1007/s40840-014-0026-8
S. S. Miller, P. T. Mocanu, Differential subordinations, Monogr. and Textbooks Pure and Appl. Math., 225, Marcel Dekker, Inc., New York (2000). DOI: https://doi.org/10.1201/9781482289817
M. Obradowič, N. Tuneski, On the starlike criteria defined by Silverman, Zeszyty Nauk. Politech. Rzeszowskiej Mat., № 24, 59–64 (2001).
Z. Peng, M. Obradović, New results for a class of univalent functions, Acta Math. Sci. Ser. B (Engl. Ed.), 39, № 6, 1579–1588 (2019). DOI: https://doi.org/10.1007/s10473-019-0609-4
Z. Peng, G. Zhong, Some properties for certain classes of univalent functions defined by differential inequalities, Acta Math. Sci. Ser. B (Engl. Ed.), 37, № 1, 69–78 (2017). DOI: https://doi.org/10.1016/S0252-9602(16)30116-3
K. Sharma, N. K. Jain, V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat., 27, № 5-6, 923–939 (2016). DOI: https://doi.org/10.1007/s13370-015-0387-7
P. Sharma, R. K. Raina, J. Sokół, Certain Ma–Minda type classes of analytic functions associated with the crescent-shaped region, Anal. Math. Phys., 9, № 4, 1887–1903 (2019). DOI: https://doi.org/10.1007/s13324-019-00285-y
H. Silverman, Convex and starlike criteria, Int. J. Math. and Math. Sci., 22, № 1, 75–79 (1999). DOI: https://doi.org/10.1155/S0161171299220753
J. Sokół, Radius problems in the class ${SL}^*,$ Appl. Math. and Comput., 214, № 2, 569–573 (2009). DOI: https://doi.org/10.1016/j.amc.2009.04.031
A. Swaminathan, L. A. Wani, Sufficient conditions and radii problems for a starlike class involving a differential inequality, Bull. Korean Math. Soc., 57, № 6, 1409–1426 (2020).
N. Tuneski, H. Irmak, Starlikeness and convexity of a class of analytic functions, Int. J. Math. and Math. Sci., 2006, Article ID 38089 (2006). DOI: https://doi.org/10.1155/IJMMS/2006/38089
L. A. Wani, A. Swaminathan, Starlike and convex functions associated with a nephroid domain, Bull. Malays. Math. Sci. Soc., 44, № 1, 79–104 (2021). DOI: https://doi.org/10.1007/s40840-020-00935-6
Copyright (c) 2024 Priyanka Goel, S. Sivaprasad Kumar
This work is licensed under a Creative Commons Attribution 4.0 International License.
