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Coefficient functionals for non-Bazilevič functions

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Sharp bounds on the second and third order Hermitian-Toeplitz determinants, initial logarithmic and inverse coefficients for functions in the class of non-Bazilevič functions are determined.

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References

  1. Ali, R.M.: Coefficients of the inverse of strongly starlike functions. Bull. Malays. Math. Sci. Soc. (2) 26(1), 63–71 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Ali, M.F., Vasudevarao, A.: On logarithmic coefficients of some close-to-convex functions. Proc. Am. Math. Soc. 146(3), 1131–1142 (2018)

    Article  MathSciNet  Google Scholar 

  3. Campschroer, J.T.P.: Growth of the coefficients of the inverse of a convex function. Nederl. Akad. Wetensch. Indag. Math. 46(1), 1–6 (1984)

    Article  MathSciNet  Google Scholar 

  4. Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: On the third logarithmic coefficient in some subclasses of close-to-convex functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(2), 14 (2020) 52

  5. Cudna, K., Kwon, O.S., Lecko, A., Sim, Y.J., Smiarowska, B.: The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of order \(\alpha \). Bol. Soc. Mat. Mex. (3) 26(2), 361–375 (2020)

    Article  MathSciNet  Google Scholar 

  6. Deng, Q.: On the logarithmic coefficients of Bazilevič functions. Appl. Math. Comput. 217(12), 5889–5894 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Elhosh, M.M.: On the logarithmic coefficients of close-to-convex functions. J. Austral. Math. Soc. Ser. A 60(1), 1–6 (1996)

    Article  MathSciNet  Google Scholar 

  8. Graham, I., Kohr, G.: Geometric function theory in one and higher dimensions, Monographs and Textbooks in Pure and Applied Mathematics, 255. Marcel Dekker Inc, New York (2003)

    Google Scholar 

  9. Girela, D.: Logarithmic coefficients of univalent functions. Ann. Acad. Sci. Fenn. Math. 25(2), 337–350 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Hartman, P., Wintner, A.: The spectra of Toeplitz’s matrices. Am. J. Math. 76, 867–882 (1954)

    Article  MathSciNet  Google Scholar 

  11. Jastrzȩbski, P., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(4), 166 (2020)

    Article  MathSciNet  Google Scholar 

  12. Juneja, O.P., Rajasekaran, S.: Coefficient estimates for inverses of \(\alpha \)-spiral functions. Complex Var. Theory Appl. 6(2–4), 99–108 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Kapoor, G.P., Mishra, A.K.: Coefficient estimates for inverses of starlike functions of positive order. J. Math. Anal. Appl. 329(2), 922–934 (2007)

    Article  MathSciNet  Google Scholar 

  14. Kirwan, W.E., Schober, G.: Inverse coefficients for functions of bounded boundary rotation. J. Anal. Math. 36(1979), 167–178 (1980)

    MathSciNet  MATH  Google Scholar 

  15. Krishna, D.V., Reddy, T.R.: An upper bound to the second Hankel functional for non-Bazilevič functions. Far East J. Math. Sci. 67(2), 187–199 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Krzyż, J.G., Libera, R.J., Złotkiewicz, E.J.: Coefficients of inverses of regular starlike functions. Ann. Univ. Mariae Curie-Skłodowska Sect. A 33(1979), 103–110 (1981)

    MathSciNet  MATH  Google Scholar 

  17. Kucerovsky, D., Mousavand, K., Sarraf, A.: On some properties of Toeplitz matrices, Cogent Math. 3 (2016), Art. ID 1154705, 12 pp

  18. Kumar, S.S., Kumar, V.: Fekete-Szegö problem for a class of analytic functions defined by convolution. Tamkang J. Math. 44(2), 187–195 (2013)

    Article  MathSciNet  Google Scholar 

  19. Libera, R.J., Złotkiewicz, E.J.: Early coefficients of the inverse of a regular convex function. Proc. Am. Math. Soc. 85(2), 225–230 (1982)

    Article  MathSciNet  Google Scholar 

  20. Libera, R.J., Złotkiewicz, E.J.: Coefficient bounds for the inverse of a function with derivative in \({\cal{P}}\). Proc. Am. Math. Soc. 87(2), 251–257 (1983)

    MathSciNet  MATH  Google Scholar 

  21. Libera, R.J., Złotkiewicz, E.J.: Coefficient bounds for the inverse of a function with derivative in \({\cal{P}}\). II. Proc. Am. Math. Soc. 92(1), 58–60 (1984)

    MathSciNet  MATH  Google Scholar 

  22. Livingston, A.E.: The coefficients of a class of meromorphic functions and their inverses. Complex Var. Theory Appl. 3(1–3), 191–204 (1984)

    MathSciNet  MATH  Google Scholar 

  23. Löwner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89(1–2), 103–121 (1923)

    Article  MathSciNet  Google Scholar 

  24. Obradovič, M.: A class of univalent functions. Hokkaido Math. J. 27(2), 329–335 (1998)

    Article  MathSciNet  Google Scholar 

  25. Obradovič, M.: A class of univalent functions. II. Hokkaido Math. J 28(3), 557–562 (1999)

    Article  MathSciNet  Google Scholar 

  26. Obradovič, M., Tuneski, N.: On certain properties of a class of univalent functions. Filomat No. 13, 59–65 (1999)

    MathSciNet  MATH  Google Scholar 

  27. Park, J.H., Kumar, V., Cho, N.E.: Sharp coefficient estimates for non-Bazilevič functions. J. Comput. Anal. Appl. 27(7), 1103–1113 (2019)

    Google Scholar 

  28. Prokhorov, D.V., Szynal, J.: Inverse coefficients for \((\alpha,\beta )\)-convex functions. Ann. Univ. Mariae Curie-Skłodowska Sect. A 35, 125–143 (1981)

    MathSciNet  MATH  Google Scholar 

  29. Ravichandran, V., Verma, S.: Bound for the fifth coefficient of certain starlike functions. C. R. Math. Acad. Sci. Paris 353(6), 505–510 (2015)

    Article  MathSciNet  Google Scholar 

  30. Sahoo, P., Mohapatra, R. N.: A survey on some special classes of Bazilevič functions and related function classes, in Current topics in pure and computational complex analysis, 63–88, Trends Math, Birkhäuser/Springer, New Delhi

  31. Silverman, H.: Coefficient bounds for inverses of classes of starlike functions. Complex Var. Theory Appl. 12(1–4), 23–31 (1989)

    MathSciNet  MATH  Google Scholar 

  32. Srivastava, H.M., Mishra, A.K., Kund, S.N.: Coefficient estimates for the inverses of starlike functions represented by symmetric gap series. PanAm. Math. J. 21(4), 105–123 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Thomas, D.K.: On the coefficients of strongly starlike functions. Indian J. Math. 58(2), 135–146 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Thomas, D.K.: On the coefficients of Bazilevič functions with logarithmic growth. Indian J. Math. 57(3), 403–418 (2015)

    MathSciNet  MATH  Google Scholar 

  35. Thomas, D.K.: On the logarithmic coefficients of close to convex functions. Proc. Am. Math. Soc. 144(4), 1681–1687 (2016)

    Article  MathSciNet  Google Scholar 

  36. Tuneski, N., Darus, M.: Fekete-Szegö functional for non-Bazilevič functions. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 18(2), 63–65 (2002)

    MathSciNet  MATH  Google Scholar 

  37. Vasilév, A.: Univalent functions in the dynamics of viscous flows. Comput. Methods Funct. Theory 1(2), 311–337 (2001)

    Article  MathSciNet  Google Scholar 

  38. Vasilév, A.: Univalent functions in two-dimensional free boundary problems. Acta Appl. Math. 79(3), 249–280 (2003)

    Article  MathSciNet  Google Scholar 

  39. Wang, Z., Gao, C., Liao, M.: On certain generalized class of non-Bazilevič functions. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 21(2), 147–154 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Wang, Z.-G., Wang, H.-T., Sun, Y.: A class of multivalent non-Bazilevič functions involving the Cho-Kwon-Srivastava operator. Tamsui Oxf. J. Math. Sci. 26(1), 1–19 (2010)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are thankful to the referee for his/ her valuable comments. The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861).

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Authors and Affiliations

  1. Department of Mathematics, Ramanujan College, University of Delhi, Kalkaji, New Delhi, 110019, India

    Virendra Kumar & Sumit Nagpal

  2. Department of Applied Mathematics, Pukyong National University, Busan, 48513, Korea

    Nak Eun Cho

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  1. Virendra Kumar
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  2. Sumit Nagpal
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  3. Nak Eun Cho
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Correspondence to Nak Eun Cho.

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Kumar, V., Nagpal, S. & Cho, N.E. Coefficient functionals for non-Bazilevič functions. RACSAM 116, 44 (2022). https://doi.org/10.1007/s13398-021-01185-2

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