DOI QR코드

DOI QR Code

SUFFICIENT CONDITIONS AND RADII PROBLEMS FOR A STARLIKE CLASS INVOLVING A DIFFERENTIAL INEQUALITY

  • Received : 2019.12.12
  • Accepted : 2020.04.23
  • Published : 2020.11.30

Abstract

Let 𝒜n be the class of analytic functions f(z) of the form f(z) = z + ∑k=n+1 αkzk, n ∈ ℕ defined on the open unit disk 𝔻, and let $${\Omega}_n:=\{f{\in}{\mathcal{A}}_n:\|zf^{\prime}(z)-f(z)\|<{\frac{1}{2}},\;z{\in}{\mathbb{D}}\}$$. In this paper, we make use of differential subordination technique to obtain sufficient conditions for the class Ωn. Writing Ω := Ω1, we obtain inclusion properties of Ω with respect to functions which map 𝔻 onto certain parabolic regions and as a consequence, establish a relation connecting the parabolic starlike class 𝒮P and the uniformly starlike UST. Various radius problems for the class Ω are considered and the sharpness of the radii estimates is obtained analytically besides graphical illustrations.

Keywords

References

  1. R. M. Ali and V. Ravichandran, Uniformly convex and uniformly starlike functions, Math. Newsletter 21 (2011), no. 1, 16-30.
  2. R. Bharati, R. Parvatham, and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math. 28 (1997), no. 1, 17-32. https://doi.org/10.5556/j.tkjm.28.1997.4330
  3. T. Bulboaca, Differential Subordinations and Superordinations: Recent Results, House of Scientific Book Publ., Cluj-Napoca, 2005.
  4. N. E. Cho, V. Kumar, S. S. Kumar, and V. Ravichandran, Radius problems for starlike functions associated with the sine function, Bull. Iranian Math. Soc. 45 (2019), no. 1, 213-232. https://doi.org/10.1007/s41980-018-0127-5
  5. P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.
  6. P. Goel and S. Sivaprasad Kumar, Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 1, 957-991. https://doi.org/10.1007/s40840-019-00784-y
  7. A. W. Goodman, Univalent Functions. Vol. II, Mariner Publishing Co., Inc., Tampa, FL, 1983.
  8. A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), no. 2, 364-370. https://doi.org/10.1016/0022-247X(91)90006-L
  9. S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), no. 1-2, 327-336. https://doi.org/10.1016/S0377-0427(99)00018-7
  10. S. Kanas and A. Wisniowska, Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl. 45 (2000), no. 4, 647-657 (2001).
  11. 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.
  12. 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
  13. E. Merkes and M. Salmassi, Subclasses of uniformly starlike functions, Internat. J. Math. Math. Sci. 15 (1992), no. 3, 449-454. https://doi.org/10.1155/S0161171292000607
  14. S. S. Miller and P. T. Mocanu, Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker, Inc., New York, 2000.
  15. M. Obradovic and Z. Peng, Some new results for certain classes of univalent functions, Bull. Malays. Math. Sci. Soc. 41 (2018), no. 3, 1623-1628. https://doi.org/10.1007/s40840-017-0546-0
  16. Z. Peng and M. Obradovic, New Results for a Class of Univalent Functions, Acta Math. Sci. Ser. B (Engl. Ed.) 39 (2019), no. 6, 1579-1588. https://doi.org/10.1007/s10473-019-0609-4
  17. Z. Peng and G. Zhong, Some properties for certain classes of univalent functions defined by differential inequalities, Acta Math. Sci. Ser. B (Engl. Ed.) 37 (2017), no. 1, 69-78. https://doi.org/10.1016/S0252-9602(16)30116-3
  18. F. Ronning, A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 47 (1993), 123-134.
  19. F. Ronning, 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
  20. St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Polya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135. https://doi.org/10.1007/BF02566116
  21. T. N. Shanmugam and V. Ravichandran, Certain properties of uniformly convex functions, in Computational methods and function theory 1994 (Penang), 319-324, Ser. Approx. Decompos., 5, World Sci. Publ., River Edge, NJ, 1995.
  22. K. Sharma, N. K. Jain, and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat. 27 (2016), no. 5-6, 923-939. https://doi.org/10.1007/s13370-015-0387-7
  23. P. Sharma, R. K. Raina, and J. Sokol, Certain Ma-Minda type classes of analytic functions associated with the crescent-shaped region, Anal. Math. Phys. 9 (2019), no. 4, 1887-1903. https://doi.org/10.1007/s13324-019-00285-y