Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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First Order Differential Subordinations for Carathéodory Functions

  • Gandhi, Shweta (Department of Mathematics, University of Delhi) ;
  • Kumar, Sushil (Bharati Vidyapeeth's College of Engineering) ;
  • Ravichandran, V. (Department of Mathematics, University of Delhi)
  • Received : 2017.04.19
  • Accepted : 2017.09.09
  • Published : 2018.06.23
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Abstract

The well-known theory of differential subordination developed by Miller and Mocanu is applied to obtain several inclusions between $Carath{\acute{e}}odory$ functions and starlike functions. These inclusions provide sufficient conditions for normalized analytic functions to belong to certain class of Ma-Minda starlike functions.

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References

  1. R. M. Ali, N. E. Cho, V. Ravichandran and S. S. Kumar, Differential subordination for functions associated with the lemniscate of Bernoulli, Taiwanese J. Math., 16(3)(2012), 1017-1026. https://doi.org/10.11650/twjm/1500406676
  2. R. M. Ali, V. Ravichandran and N. Seenivasagan, Sufficient conditions for Janowski starlikeness, Int. J. Math. Math. Sci., 2007(2007), Art. ID 62925, 7 pp.
  3. A. A. Attiya and M. F. Yassen, Majorization for some classes of analytic functions associated with the Srivastava-Attiya operator, Filomat, 30(7)(2016), 2067-2074. https://doi.org/10.2298/FIL1607067A
  4. N. E. Cho, V. Kumar, S. Sivaprasad Kumar and V. Ravichandran, Radius problems for starlike functions associated with sine function, preprint.
  5. N. E. Cho, H. J. Lee, J. H. Park and R. Srivastava, Some applications of the first-order differential subordinations, Filomat, 30(6)(2016), 1465-1474. https://doi.org/10.2298/FIL1606465C
  6. P. L. Duren, Univalent Functions, Fundamental Principles of Mathematical Sciences 259, Springer, New York, 1983.
  7. G. M. Goluzin, On the majorization principle in function theory, Dokl. Akad. Nauk. SSSR, 42(1935), 647-650.
  8. A. Gori and F. Vlacci, Starlikeness for functions of a hypercomplex variable, Proc. Amer. Math. Soc., 145(2)(2017), 791-804.
  9. S. Hussain, Z. Shareef and M. Darus, On a class of functions associated with Salagean operator, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 111(1)(2017), 213-222.
  10. 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.
  11. S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math., 40(2)(2016), 199-212.
  12. S. Kumar and V. Ravichandran, Subordinations for functions with positive real part, Complex Anal. Oper. Theory, 2017. https://doi.org/10.1007/s11785-017-0690-4
  13. 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.
  14. R. Mendiratta, S. Nagpal, and V. Ravichandran, On a subclass of strongly star-like functions associated with exponential function, Bull. Malays. Math. Sci. Soc., 38(1)(2015), 365-386. https://doi.org/10.1007/s40840-014-0026-8
  15. S. S. Miller and P. T. Mocanu, On some classes of first-order differential subordinations, Michigan Math. J., 32(2)(1985), 185-195. https://doi.org/10.1307/mmj/1029003185
  16. S. S. Miller and P. T. Mocanu, Differential Subordinations, Dekker, New York, 2000.
  17. M. Nunokawa, M. Obradovic and S. Owa, One criterion for univalency, Proc. Amer. Math. Soc., 106(4)(1989), 1035-1037.
  18. M. Nunokawa, S. Owa, N. Takahashi and H. Saitoh, Sufficient conditions for Caratheodory functions, Indian J. Pure Appl. Math., 33(9)(2002), 1385-1390.
  19. K. S. Padmanabhan and R. Parvatham, Some applications of differential subordination, Bull. Austral. Math. Soc., 32(3)(1985), 321-330. https://doi.org/10.1017/S0004972700002410
  20. R. K. Raina and J. Soko l, On coefficient estimates for a certain class of starlike functions, Hacet. J. Math. Stat., 44(6)(2015), 1427-1433.
  21. V. Ravichandran and K. Sharma, Sufficient conditions for starlikeness, J. Korean Math. Soc., 52(4)(2015), 727-749. https://doi.org/10.4134/JKMS.2015.52.4.727
  22. T. N. Shanmugam, Convolution and differential subordination, Internat. J. Math. Math. Sci., 12(2)(1989), 333-340. https://doi.org/10.1155/S0161171289000384
  23. K. Sharma, N. K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat., 27(5-6)(2016), 923-939. https://doi.org/10.1007/s13370-015-0387-7
  24. S. Sivaprasad Kumar, V. Kumar, V. Ravichandran and N. E. Cho, Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli, J. Inequal. Appl., 2013(176)(2013), 13 pp. https://doi.org/10.1186/1029-242X-2013-13
  25. J. Soko l and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat., 19(1996), 101-105.
  26. N. Tuneski, T. Bulboaca, and B. Jolevska-Tunesk, Sharp results on linear combination of simple expressions of analytic functions, Hacet. J. Math. Stat., 45(1)(2016), 121-128.