Differential Subordination for Starlike Functions
Main Article Content
Abstract
A normalized analytic function, $f$ defined on the open unit disk, is starlike of order $\alpha$ if $RE(zf'(z)/f(z))>\alpha$, and is said to be reciprocal starlike of order $\alpha$ if $RE(f(z)/zf'(z))>\alpha$. Such functions are univalent and, therefore we find sufficient conditions for functions to be starlike and reciprocal starlike. We prove a general differential subordination theorem and sufficient conditions in terms of $zf'(z)/f(z)$ and $1+zf''(z)/f'(z)$ for functions to be starlike. Further, we prove sufficient conditions for the reciprocal starlikeness of functions and integral operators.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.
References
[1] R. Aghalary and P. Arjomandinia, On a first order strong differential subordination and application to univalent functions, Commun. Korean Math. Soc. 37 (2022), 445–454. Google Scholar
[2] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. Math. 17 (1915), 12–22. https://doi.org/10.2307/2007212 Google Scholar
[3] R. M. Ali, V. Ravichandran, and N. Seenivasagan, Subordination and superordination on Schwarzian derivatives, J. Inequal. Appl. 2008 (2008), Article ID 712328, 18 pp. Google Scholar
[4] B. A. Frasin, Y. Talafha, and T. Al-Hawary, Subordination results for classes of functions of reciprocal order, Tamsui Oxf. J. Inf. Math. Sci. 30 (2014), 81–89. Google Scholar
[5] B. A. Frasin and M. A. Sabri, Sufficient conditions for starlikeness of reciprocal order, Eur. J. Pure Appl. Math. 10 (2017), 871–876. Google Scholar
[6] T. Al-Hawary and B. A. Frasin, Coefficient estimates and subordination properties for certain classes of analytic functions of reciprocal order, Stud. Univ. Babeș-Bolyai Math. 63 (2018), 203–212. Google Scholar
[7] N. Jose, V. Ravichandran, and A. Das, Differential subordination for bounded turning functions using pre-Schwarzian and the Schwarzian derivatives, Anal. Math. Phys. 14 (2024), Paper No. 113, 17 pp. Google Scholar
[8] V. Kumar, S. Kumar, and N. E. Cho, Coefficient functionals for starlike functions of reciprocal order, Thai J. Math. 20 (2022), 1183–1197. Google Scholar
[9] A. Lecko, A new boundary approach for differential subordinations, Proc. Amer. Math. Soc. 153 (2025), 1969–1983. Google Scholar
[10] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755–758. Google Scholar
[11] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (1981), 157–172. Google Scholar
[12] S. S. Miller and P. T. Mocanu, On some classes of first-order differential subordinations, Michigan Math. J. 32 (1985), 185–195. Google Scholar
[13] S. S. Miller and P. T. Mocanu, Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker, Inc., New York, 2000. Google Scholar
[14] P. T. Mocanu and I. Șerb, A sharp simple criterion for a subclass of starlike functions, Complex Variables Theory Appl. 32 (1997), 161–168. https://doi.org/10.1080/17476939708814986 Google Scholar
[15] M. Nunokawa et al., Sufficient conditions for starlikeness, Chinese J. Math. 24 (1996), 265–271. Google Scholar
[16] S. Madhumitha and V. Ravichandran, Sufficient conditions for starlikeness of reciprocal order, Korean J. Math. 31 (2023), 243–258. Google Scholar
[17] M. Nunokawa et al., Differential subordination and argumental property, Comput. Math. Appl. 56 (2008), 2733–2736. Google Scholar
[18] M. Obradović and N. Tuneski, On the starlike criteria defined by Silverman, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 24 (2000), 59–64. Google Scholar
[19] V. Ravichandran, C. Selvaraj, and R. Rajalaksmi, Sufficient conditions for starlike functions of order α, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), Article 81, 6 pp. Google Scholar
[20] M. S. Robertson, Schlicht solutions of W"+pW=0, Trans. Amer. Math. Soc. 76 (1954), 254–274. https://doi.org/10.2307/1990768 Google Scholar
[21] A. Saliu, K. Jabeen, and V. Ravichandran, Differential subordination for certain strongly starlike functions, Rend. Circ. Mat. Palermo (2) 73 (2024), 1–18. Google Scholar
[22] H. Silverman, Convex and starlike criteria, Int. J. Math. Math. Sci. 22 (1999), 75–79. Google Scholar
[23] S. Yadav and V. Ravichandran, Sufficient conditions for analytic functions to be starlike of reciprocal order, Honam Math. J. 46 (2024), 120–135. https://doi.org/10.5831/HMJ.2024.46.1.120 Google Scholar