DOI QR코드

DOI QR Code

APPLICATIONS OF DIFFERENTIAL SUBORDINATIONS TO CERTAIN CLASSES OF STARLIKE FUNCTIONS

  • Banga, Shagun (Department of Applied Mathematics Delhi Technological University) ;
  • Kumar, S. Sivaprasad (Department of Applied Mathematics Delhi Technological University)
  • Received : 2019.01.17
  • Accepted : 2019.08.26
  • Published : 2020.03.01

Abstract

Let p be an analytic function defined on the open unit disk 𝔻. We obtain certain differential subordination implications such as ψ(p) := pλ(z)(α+βp(z)+γ/p(z)+δzp'(z)/pj(z)) ≺ h(z) (j = 1, 2) implies p ≺ q, where h is given by ψ(q) and q belongs to 𝒫, by finding the conditions on α, β, γ, δ and λ. Further as an application of our derived results, we obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy |log(zf'(z)/f(z))| < 1, |(zf'(z)/f(z))2 - 1| < 1 and zf'(z)/f(z) lying in the parabolic region v2 < 2u - 1.

Keywords

Acknowledgement

The work presented here was supported by a Research Fellowship from the Department of Science and Technology, New Delhi.

References

  1. D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. Lond. Math. Soc. (2) 1 (1969), 431-443. https://doi.org/10.1112/jlms/s2-1.1.431
  2. N. E. Cho and J. A. Kim, Angular estimations of certain analytic functions, J. Korean Math. Soc. 34 (1997), no. 2, 427-436.
  3. N. E. Cho and J. A. Kim, On a sufficient condition and an angular estimation for $\Phi$-like functions, Taiwanese J. Math. 2 (1998), no. 4, 397-403. https://doi.org/10.11650/twjm/1500407012
  4. N. E. Cho, H. J. Lee, J. H. Park, and R. Srivastava, Some applications of the first-order differential subordinations, Filomat 30 (2016), no. 6, 1465-1474. https://doi.org/10.2298/FIL1606465C
  5. W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970/1971), 159-177. https://doi.org/10.4064/ap-23-2-159-177
  6. W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185 (1953). http://projecteuclid.org/euclid.mmj/1028988895
  7. J.-L. Liu and R. Srivastava, The order of starlikeness of some classes of strongly starlike functions, Quaest. Math. 41 (2018), no. 5, 707-718. https://doi.org/10.2989/16073606.2017.1398194
  8. W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157-169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994.
  9. A. Marx, Untersuchungen uber schlichte Abbildungen, Math. Ann. 107 (1933), no. 1, 40-67. https://doi.org/10.1007/BF01448878
  10. R. Mendiratta, S. Nagpal, and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 1, 365-386. https://doi.org/10.1007/s40840-014-0026-8
  11. S. S. Miller and P. T. Mocanu, Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker, Inc., New York, 2000.
  12. P. T. Mocanu, Une propriete de convexite generalisee dans la theorie de la representation conforme, Mathematica (Cluj) 11 (34) (1969), 127-133.
  13. M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 7, 234-237. http://projecteuclid.org/euclid.pja/1195511343 https://doi.org/10.3792/pjaa.69.234
  14. M. Nunokawa, S. Owa, N. Takahashi, and H. Saitoh, Sufficient conditions for Caratheodory functions, Indian J. Pure Appl. Math. 33 (2002), no. 9, 1385-1390.
  15. M. Obradowic and N. Tuneski, On the starlike criteria defined by Silverman, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 24 (2000), 59-64 (2001).
  16. V. Ravichandran and S. S. Kumar, On sufficient conditions for starlikeness, Southeast Asian Bull. Math. 29 (2005), no. 4, 773-783.
  17. M. S. Robertson, Certain classes of starlike functions, Michigan Math. J. 32 (1985), no. 2, 135-140. https://doi.org/10.1307/mmj/1029003181
  18. F. Rnning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), no. 1, 189-196. https://doi.org/10.2307/2160026
  19. K. Sharma and V. Ravichandran, Applications of subordination theory to starlike functions, Bull. Iranian Math. Soc. 42 (2016), no. 3, 761-777.
  20. J. Soko l and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19 (1996), 101-105.
  21. H. M. Srivastava and S. S. Eker, Some applications of a subordination theorem for a class of analytic functions, Appl. Math. Lett. 21 (2008), no. 4, 394-399. https://doi.org/10.1016/j.aml.2007.02.032
  22. J. Stankiewicz, Quelques problemes extremaux dans les classes des fonctions ${\alpha}$-angulairement etoilees, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 20 (1966), 59-75 (1971).
  23. E. Strohhacker, Beitrage zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), no. 1, 356-380. https://doi.org/10.1007/BF01474580
  24. B. A. Uralegaddi, M. D. Ganigi, and S. M. Sarangi, Univalent functions with positive coefficients, Tamkang J. Math. 25 (1994), no. 3, 225-230. https://doi.org/10.5556/j.tkjm.25.1994.4448