Shweta GANDHİ, Prachi GUPTA, Sumıt NAGPAL, V RAVİCHANDRAN

Starlike functions associated with an epicycloid

Anahtar Kelimeler:

radius problem, starlike functions, cusps, three leaf domain, inclusion relation, coefficient bound, epicycloid, subordination

Starlike functions associated with an epicycloid

For a natural number $n\geq 2$, the function $\phi_{n\mathcal{L}}(z)=1+nz/(n+1)+z^n/(n+1)$ maps the open unit disk onto a domain bounded by an epicycloid with $(n-1)$ cusps. A class of starlike functions associated with $\phi_{n\mathcal{L}}$ is defined in the unit disk and its sharp bounds on initial coefficients, various inclusion relations and radii problems related to the other subclasses of starlike functions are investigated. As an application, the corresponding results are determined in the limiting case for the class of normalized analytic functions $f$ satisfying $|zf'(z)/f(z)-1|<1$ in the unit disk.

Keywords:

radius problem, starlike functions, cusps, three leaf domain, inclusion relation, coefficient bound, epicycloid, subordination,

PDF

___

[1] R. M. Ali, N. K. Jain and V. Ravichandran, Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane, Appl. Math. Comput. 218 (11), 6557–6565, 2012.
[2] A. Alotaibi, M. Arif, M.A. Alghamdi and S. Hussain, Starlikness associated with cosine hyperbolic function, Mathematics 8 (7), 1118, 2020.
[3] N. Bohra, S. Kumar and V. Ravichandran, Some special differential subordinations, Hacet. J. Math. Stat. 48 (4), 1017–1034, 2019
[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 (1), 213–232, 2019.
[5] S. Gandhi, Radius estimates for three leaf function and convex combination of starlike functions, In: Deo N., Gupta V., Acu A., Agrawal P. (eds) Mathematical Analysis I: Approximation Theory. ICRAPAM 2018. Springer Proceedings in Mathematics and Statistics, vol 306. Springer, Singapore, 2020.
[6] P. Goel and S. S. Kumar, Certain Class of Starlike Functions Associated with Modified Sigmoid Function, Bull. Malays. Math. Sci. Soc. 43 (1), 957–991, 2020.
[7] P. Gupta, S. Nagpal and V. Ravichandran, Inclusion relations and radius problems for a subclass of starlike functions, J. Korean Math. Soc. 58 (5), 1147–1180, 2021.
[8] H. Hagen, Curve and surface design, Geometric Design Publications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
[9] R. Kanaga and V. Ravichandran, Starlikeness for certain close-to-star functions, Hacet. J. Math. Stat. 50 (2), 414–432, 2021.
[10] R. Kargar, A. Ebadian and J. Sokół, On Booth lemniscate and starlike functions, Anal. Math. Phys. 9 (1), 143–154, 2019
[11] F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20, 8–12, 1969.
[12] K. Khatter, V. Ravichandran and S. Sivaprasad Kumar, Starlike functions associated with exponential function and the lemniscate of Bernoulli, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (1), 233–253, 2019.
[13] S. S. Kumar and G. Kamaljeet, A cardioid domain and starlike functions, Anal. Math. Phys. 11 (2), 54, 2021.
[14] S. S. Kumar and Kush Arora, Starlike Functions associated with a Petal Shaped Domain, arXiv: 2010.10072, 15 pages, 2020.(accepted in BKMS)
[15] S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40 (2), 199–212, 2016.
[16] J. D. Lawrence, A catalog of special plane curves, Dover Publications, 1972.
[17] W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA. 157–169, Tianjin, 1992.
[18] T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104, 532–537, 1962.
[19] T. H. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14, 514–520, 1963.
[20] T. H. MacGregor, The radius of univalence of certain analytic functions, Proc Amer. Math. Soc. 14, 521–524, 1963.
[21] J. S. Madachy, Madachy’s Mathematical Recreations, New York: Dover, pp. 219-225, 1979.
[22] R. Mendiratta, S. Nagpal and V. Ravichandran, A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli, Internat. J. Math. 25 (9), 1450090, 17 pp ,2014.
[23] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (1), 365–386, 2015.
[24] D. V. Prokhorov and J. Szynal, Inverse coefficients for $(\alpha,\beta )$-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 35 (1981), 125–143, 1984.
[25] R. K. Raina and J. Sokół, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353 (11), 973–978, 2015.
[26] V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353 (6), 505–510, 2015.
[27] K. Sharma, N. K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat. 27 (5-6), 923–939, 2016.
[28] P. Sharma, R. K. Raina and J. Sokół, Certain Ma–Minda type classes of analytic functions associated with the crescent-shaped region, Anal. Math. Phys. 9 (4), 1887– 1903, 2019.
[29] R. Singh, On a class of star-like functions, Compositio Math. 19, 78–82, 1967.
[30] J. Sokół and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19, 101–105, 1996.
[31] J. Sokół, On some subclass of strongly starlike functions, Demonstratio Math. 31 (1), 81–86, 1998.
[32] B. A. Uralegaddi, M. D. Ganigi and S. M. Sarangi, Univalent functions with positive coefficients, Tamkang J. Math. 25 (3), 225–230, 1994.
[33] L. A. Wani and A. Swaminathan, Radius problems for functions associated with a nephroid domain, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (4), 178, 2020.
[34] Y. Yunus, S. A. Halim and A. B. Akbarally, Subclass of starlike functions associated with a limacon, AIP Conference Proceedings, AIP Publishing 1974 (1), 2018.