Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SHARP ESTIMATES ON THE THIRD ORDER HERMITIAN-TOEPLITZ DETERMINANT FOR SAKAGUCHI CLASSES

  • Kumar, Sushil (Bharati Vidyapeeth's College of Engineering) ;
  • Kumar, Virendra (Department of Mathematics Ramanujan College University of Delhi)
  • Received : 2021.10.08
  • Accepted : 2022.01.25
  • Published : 2022.10.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, sharp lower and upper bounds on the third order Hermitian-Toeplitz determinant for the classes of Sakaguchi functions and some of its subclasses related to right-half of lemniscate of Bernoulli, reverse lemniscate of Bernoulli and exponential functions are investigated.

Keywords

References

  1. R. Chand and P. Singh, On certain schlicht mappings, Indian J. Pure Appl. Math. 10 (1979), no. 9, 1167-1174.
  2. N. E. Cho, S. Kumar, and V. Kumar, Hermitian-Toeplitz and Hankel determinants for certain starlike functions, Asian-European J. Math. (2021). https://doi.org/10.1142/S1793557122500425
  3. K. Cudna, O. S. Kwon, A. Lecko, Y. J. Sim, and B. Smiarowska, The second and thirdorder Hermitian Toeplitz determinants for starlike and convex functions of order α, Bol. Soc. Mat. Mex. (3) 26 (2020), no. 2, 361-375. https://doi.org/10.1007/s40590-019-00271-1
  4. R. N. Das and P. Singh, On subclasses of schlicht mapping, Indian J. Pure Appl. Math. 8 (1977), no. 8, 864-872.
  5. P. L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.
  6. R. M. Goel and B. S. Mehrok, A subclass of starlike functions with respect to symmetric points, Tamkang J. Math. 13 (1982), no. 1, 11-24.
  7. P. Hartman and A. Wintner, The spectra of Toeplitz's matrices, Amer. J. Math. 76 (1954), 867-882. https://doi.org/10.2307/2372661
  8. P. Jastrz,ebski, B. Kowalczyk, O. S. Kwon, A. Lecko, and Y. J. Sim, Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions, Rev. R. Acad. Cienc. Exactas F'is. Nat. Ser. A Mat. RACSAM 114 (2020), no. 4, Paper No. 166, 14 pp. https://doi.org/10.1007/s13398-020-00895-3
  9. K. Khatter, V. Ravichandran, and S. Sivaprasad Kumar, Estimates for initial coefficients of certain starlike functions with respect to symmetric points, in Applied analysis in biological and physical sciences, 385-395, Springer Proc. Math. Stat., 186, Springer, New Delhi, 2016. https://doi.org/10.1007/978-81-322-3640-5_24
  10. B. Kowalczyk, O. S. Kwon, A. Lecko, Y. J. Sim, and B. Smiarowska, The third-order Hermitian Toeplitz determinant for classes of functions convex in one direction, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 4, 3143-3158. https://doi.org/10.1007/s40840-019-00859-w
  11. B. Kowalczyk, A. Lecko, and B. Smiarowska, Sharp inequalities for Hermitian Toeplitz determinants for strongly starlike and strongly convex functions, J. Math. Inequal. 15 (2021), no. 1, 323-332. https://doi.org/10.7153/jmi-2021-15-24
  12. D. Kucerovsky, K. Mousavand, and A. Sarraf, On some properties of Toeplitz matrices, Cogent Math. 3 (2016), Art. ID 1154705, 12 pp. https://doi.org/10.1080/23311835.2016.1154705
  13. V. Kumar, Hermitian-Toeplitz determinants for certain classes of close-to-convex functions, Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-021-00564-0
  14. V. Kumar and N. E. Cho, Hermitian-Toeplitz determinants for functions with bounded turning, Turkish J. Math. 45 (2021), no. 6, 2678-2687. https://doi.org/10.3906/mat2104-104
  15. V. Kumar and S. Kumar, Bounds on Hermitian-Toeplitz and Hankel determinants for strongly starlike functions, Bol. Soc. Mat. Mex. (3) 27 (2021), no. 2, Paper No. 55, 16 pp. https://doi.org/10.1007/s40590-021-00362-y
  16. V. Kumar, S. Kumar, and V. Ravichandran, Third Hankel determinant for certain classes of analytic functions, in Mathematical analysis. I. Approximation theory, 223-231, Springer Proc. Math. Stat., 306, Springer, Singapore, 2020. https://doi.org/10.1007/978-981-15-1153-0_19
  17. V. Kumar, S. Nagpal, and N. E. Cho, Coefficient functionals for non-Bazilevic functions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 116 (2022), no. 1, Paper No. 44, 14 pp. https://doi.org/10.1007/s13398-021-01185-2
  18. V. Kumar, R. Srivastava, and N. E. Cho, Sharp estimation of Hermitian-Toeplitz determinants for Janowski type starlike and convex functions, Miskolc Math. Notes 21 (2020), no. 2, 939-952. https://doi.org/10.18514/mmn.2020.3361
  19. R. J. Libera and E. J. Z lotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (1983), no. 2, 251-257. https://doi.org/10.2307/2043698
  20. 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 (2014), no. 9, 1450090, 17 pp. https://doi.org/10.1142/S0129167X14500906
  21. 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
  22. P. T. Mocanu, Certain classes of starlike functions with respect to symmetric points, Mathematica (Cluj) 32(55) (1990), no. 2, 153-157.
  23. V. Ravichandran, Starlike and convex functions with respect to conjugate points, Acta Math. Acad. Paedagog. Nyh'azi. (N.S.) 20 (2004), no. 1, 31-37.
  24. K. B. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72-75. https://doi.org/10.2969/jmsj/01110072
  25. J. Sok'o l and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19 (1996), 101-105.
  26. H. M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A Sci. 44 (2020), no. 1, 327-344. https://doi.org/10.1007/s40995-019-00815-0
  27. H. M. Srivastava, Q. Z. Ahmad, N. Khan, and B. Khan, Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain, Mathematics 7 (2019), Article ID 181, 1-15.
  28. H. M. Srivastava, B. Khan, N. Khan, M. Tahir, S. Ahmad, and N. Khan, Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function, Bull. Sci. Math. 167 (2021), Paper No. 102942, 16 pp. https://doi.org/10.1016/j.bulsci.2020.102942
  29. J. Thangamani, On starlike functions with respect to symmetric points, Indian J. Pure Appl. Math. 11 (1980), no. 3, 392-405.
  30. D. Vamshee Krishna, B. Venkateswarlu, and T. RamReddy, Third Hankel determinant for starlike and convex functions with respect to symmetric points, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 70 (2016), no. 1, 37-45. https://doi.org/10.17951/a.2016.70.1.37