Skip to main content
SpringerLink
Log in

Sharp Coefficients Bounds for Starlike Functions Associated with Gregory Coefficients

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

In this paper we introduced the class \({{\mathcal {S}}}_{G}^{*}\) of analytic functions which is related with starlike functions and generating function of Gregory coefficients. By using bounds on some coefficient functionals for the family of functions with positive real part, we obtain several sharp coefficient bounds on the first six coefficients and also further bounds on the corresponding Hankel determinants for functions in the class \({{\mathcal {S}}}_{G}^{*}\). Additionally, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the considered class \({{\mathcal {S}}}_{G}^{*}\) were estimated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ali, R.M., Chandrashekar, R., Ravichandran, V.: Janowski starlikeness for a class of analytic functions. Appl. Math. Lett. 24(4), 501–505 (2011)

    Article  MathSciNet  Google Scholar 

  2. Ali, R.M., Ravichandran, V., Seenivasagan, N.: Sufficient conditions for Janowski starlikeness. Int. J. Math. Math. Sci. Art. ID 62925: 7pp (2007)

  3. Alimohammadi, D., Analouei Adegani, E., Bulboacă, T., Cho, N.E.: Logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions. J. Funct. Spaces 2021, Article ID 6690027, 7 pages (2021)

  4. Andreev, V.V., Duren, P.L.: Inequalities for logarithmic coefficients of univalent functions and their derivatives. Indiana Univ. Math. J. 37(4), 721–733 (1988)

    Article  MathSciNet  Google Scholar 

  5. Avkhadiev, F.G., Wirths, K.J.: Schwarz-Pick type inequalities. Springer Science & Business Media, Berlin/Heidelberg, Germany (2009)

    Book  Google Scholar 

  6. Berezin, I.S., Zhidkov, N.P.: Computing Methods. Pergamon (1965)

  7. Cantor, D.G.: Power series with integral coefficients. Bull. Am. Math. Soc. 69, 362–366 (1963)

    Article  MathSciNet  Google Scholar 

  8. Carathéodory, C.: Über den Variabilitötsbereich der Fourier’schen Konstanten von positiven harmonischen Funktionen. Rendiconti del Circolo Matematico di Palermo 32(1), 193–217 (1911)

    Article  Google Scholar 

  9. Cho, N.E., Kumar, S., Kumar, V., Ravichandran, V.: Differential subordination and radius estimates for starlike functions associated with the Booth lemniscate. Turk. J. Math. 42(3), 1380–1399 (2018)

    MathSciNet  Google Scholar 

  10. De Branges, L.: A proof of the Bieberbach conjecture. Acta Mathematics 154(1–2), 137–152 (1985)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  12. Deniz, E., Budak, L.: Second Hankel determinat for certain analytic functions satisfying subordinate condition. Math. Slovaca 68(2), 463–471 (2018)

    Article  MathSciNet  Google Scholar 

  13. Deniz, E.: Sharp coefficients bounds for starlike functions associated with generalized telephone numbers. Bull. Malaysian Math. Sci. Soc. 44(3), 1525–1542 (2021)

    Article  MathSciNet  Google Scholar 

  14. Dziok, J., Raina, R.K., Sokół, J.: Certain results for a class of convex functions related to a shell-like curve connected with Fibonacci numbers. Comput. Math. Appl. 61(9), 2605–2613 (2011)

    Article  MathSciNet  Google Scholar 

  15. Dziok, J., Raina, R.K., Sokół, J.: On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers. Math. Comput. Model. 57(5–6), 1203–1211 (2013)

    Article  MathSciNet  Google Scholar 

  16. FitzGerald, C.H., Pommerenke, C.: The De Branges theorem on univalent functions. Trans. Am. Math. Soc. 290(2), 683–690 (1985)

    Article  MathSciNet  Google Scholar 

  17. FitzGerald, C.H., Pommerenke, C.: A theorem of De Branges on univalent functions. Serdica 13(1), 21–25 (1987)

    MathSciNet  Google Scholar 

  18. Girela, D.: Logarithmic coefficients of univalent functions. In Annales-Academiae Scientiarum Fennicae Series A1 Mathematica 25(2), 337–350 (2000)

    MathSciNet  Google Scholar 

  19. Goel, P., Kumar, S.S.: Certain class of starlike functions associated with modified sigmoid function. Bull. Malaysian Math. Sci. Soc. 43(1), 957–991 (2020)

    Article  MathSciNet  Google Scholar 

  20. Janowski, W.: Some extremal problems for certain families of analytic functions I. Ann. Polon. Math. 28(3), 297–326 (1973)

    Article  MathSciNet  Google Scholar 

  21. Janteng, A., Halim, S.A., Darus, M.: Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 1(13), 619–625 (2007)

    MathSciNet  Google Scholar 

  22. Kayumov, I.P.: On Brennan’s conjecture for a special class of functions. Math. Notes 78(3–4), 498–502 (2005)

    Article  MathSciNet  Google Scholar 

  23. Kazımoğlu, S., Deniz, E.: Fekete-Szegö problem for generalized bi-subordinate functions of complex order. Hacet. J. Math. Stat. 49(5), 1695–1705 (2020)

    Article  MathSciNet  Google Scholar 

  24. Keogh, F.R., Merkes, E.P.: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 20(1), 8–12 (1969)

    Article  MathSciNet  Google Scholar 

  25. Kowalczyk, B., Lecko, A., Thomas, D.: The sharp bound of the third Hankel determinant for starlike functions. Forum Math. 34(5), 1249–1254 (2022)

    MathSciNet  Google Scholar 

  26. Kumar, V., Cho, N.E., Ravichandran, V., Srivastava, H.M.: Sharp Coefficient bounds for starlike functions associated with the Bell numbers. Math. Slovaca 69(5), 1053–1064 (2019)

    Article  MathSciNet  Google Scholar 

  27. Kwon, O.S., Lecko, A., Sim, Y.J.: The bound of the Hankel determinant of the third kind for starlike functions. Bull. Malaysian Math. Sci. Soc. 42, 767–780 (2019)

    Article  MathSciNet  Google Scholar 

  28. Lee, S.K., Ravichandran, V., Supramaniam, S.: Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl. 2013, 1–17 (2013)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Ma, W.C., Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis (Z. Li, F. Ren, L. Yang, S. Zhang, eds.), Tianjin, China, 1992, Conference Proceedings and Lecture Notes in Analysis, Vol. I, International Press, Cambridge, Massachusetts 157-169 (1994)

  32. Mendiratta, R., Nagpal, S., Ravichandran, V.: On a subclass of strongly starlike functions associated with exponential function. Bull. Malaysian Math. Sci. Soc. 38, 365–386 (2015)

    Article  MathSciNet  Google Scholar 

  33. Noonan, J.W., Thomas, D.K.: On the second Hankel determinant of areally mean p-valent functions. Trans. Am. Math. Soc. 223, 337–346 (1976)

    MathSciNet  Google Scholar 

  34. Obradović, M., Ponnusamy, S., Wirths, K.J.: Logarithmic coefficients and a coefficient conjecture for univalent functions. Monatshefte für Mathematik 185(3), 489–501 (2018)

    Article  MathSciNet  Google Scholar 

  35. Ohno, R., Sugawa, T.: Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions. Kyoto J. Math. 58(2), 227–241 (2018)

    Article  MathSciNet  Google Scholar 

  36. Phillips, G.M.: Gregory’s method for numerical integration. Amer. Math. Monthly 79(3), 270–274 (1972)

    Article  MathSciNet  Google Scholar 

  37. Pommerenke, C.: Univalent Function. Math, Lehrbucher, vandenhoeck and Ruprecht, Gottingen, (1975)

  38. Ravichandran, V., Verma, S.: Bound for the fifth coefficient of certain starlike functions. Comptes Rendus de l’Académie des Sciences-Series I - Mathematics 353(6), 505–510 (2015)

    MathSciNet  Google Scholar 

  39. Roth, O.: A sharp inequality for the logarithmic coefficients of univalent functions. Proc. Am. Math. Soc. 135(7), 2051–2054 (2017)

    Article  MathSciNet  Google Scholar 

  40. Rønning, F.: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 118(1), 189–196 (1993)

    Article  MathSciNet  Google Scholar 

  41. Shi, L., Srivastava, H.M., Rafiq, A., Arif, M., Ihsan, M.: Results on Hankel determinants for the inverse of certain analytic functions subordinated to the exponential function. Mathematics 2022; 10, Article ID 3429, 1–15 (2022)

  42. Srivastava, H.M., Kaur, G., Singh, G.: Estimates of the fourth Hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains. J. Nonlinear Convex Anal. 22(3), 511–526 (2021)

    MathSciNet  Google Scholar 

  43. Srivastava, H.M., Murugusundaramoorthy, G., Bulboacă, T.: The second Hankel determinant for subclasses of bi-univalent functions associated with a nephroid domain. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM) 116 Article ID 145, 1–21 (2022)

  44. Srivastava, H.M., Kumar, S., Kumar, V., Cho, N.E.: Hermitian-Toeplitz and Hankel determinants for starlike functions associated with a rational function. J. Nonlinear Convex Anal. 23(12), 2815–2833 (2022)

    MathSciNet  Google Scholar 

  45. Srivastava, H.M., Shaba, T.G., Murugusundaramoorthy, G., Wanas, A.K., Oros, G.I.: The Fekete-Szegö functional and the Hankel determinant for a certain class of analytic functions involving the Hohlov operator. AIMS Math. 8, 340–360 (2023)

    Article  MathSciNet  Google Scholar 

  46. Yang, S.S.: Estimates for coefficients of univalent functions from integral means and Grunsky inequalities. Israel J. Math. 87, 129–142 (1994)

    Article  MathSciNet  Google Scholar 

  47. Ye, Z.: The logarithmic coefficients of close-to-convex functions. Bull. Inst. Math. Academia Sinica (New Series) 3(3), 445–452 (2008)

    MathSciNet  Google Scholar 

  48. Zaprawa, P.: Third Hankel determinants for subclasses of univalent functions. Med. J. Math.14(1) Art. ID 19, 1–10 (2017)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Letters, Kafkas University, Kars, Turkey

    Sercan Kazımoğlu & Erhan Deniz

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4, Canada

    H. M. Srivastava

  3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan, People’s Republic of China

    H. M. Srivastava

  4. Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Republic of Korea

    H. M. Srivastava

  5. Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007, Baku, Azerbaijan

    H. M. Srivastava

  6. Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City, 320314, Taiwan, Republic of China

    H. M. Srivastava

  7. Section of Mathematics, International Telematic University Uninettuno, I-00186, Rome, Italy

    H. M. Srivastava

Authors
  1. Sercan Kazımoğlu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erhan Deniz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. M. Srivastava
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors actively worked on the research contained in the paper. All authors reviewed the manuscript.

Corresponding author

Correspondence to Sercan Kazımoğlu.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Communicated by Heinrich Begehr

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini and Daniele Struppa.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kazımoğlu, S., Deniz, E. & Srivastava, H.M. Sharp Coefficients Bounds for Starlike Functions Associated with Gregory Coefficients. Complex Anal. Oper. Theory 18, 6 (2024). https://doi.org/10.1007/s11785-023-01445-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11785-023-01445-6

Keywords

Mathematics Subject Classification

Search

Navigation