Estimate on Logarithmic Co-Efficients of Sokol-Stankiewicz Type Star-Like Function Associated with Caratheodory Functions

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DOI: https://doi.org/10.28924/2291-8639-23-2025-191
Published: Aug 15, 2025
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M. Nandeesh, M. Ruby Salestina, Archana, G. Murugusundaramoorthy, Estimate on Logarithmic Co-Efficients of Sokol-Stankiewicz Type Star-Like Function Associated with Caratheodory Functions, Int. J. Anal. Appl., 23 (2025), 191.

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M. Nandeesh, M. Ruby Salestina, Archana, G. Murugusundaramoorthy

Abstract

The fundamental focus of researching coefficient problems for various families of univalent functions involves characterizing the coefficients of functions within a particular family based on the coefficients of Caratheodory functions. Consequently, by employing known inequalities for the class of Caratheodory functions, coefficient functionals can be scrutinized. This study will tackle several coefficient problems by applying the methodology to the aforementioned family of functions. Our investigation centers on the family of Sokol-Stankiewicz star-like functions which is defined in the open unit disk D. We explore the bounds of certain initial coefficients, including the Fekete-Szego inequality and other results concerning logarithmic coefficients for functions within this class.

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References

  1. W. Ma, A Unified Treatment of Some Special Classes of Univalent Functions, in: Proceedings of the Conference on Complex Analysis, 1992. https://cir.nii.ac.jp/crid/1570572700543766144.
  2. N.E. Cho, V. Kumar, S.S. Kumar, V. Ravichandran, Radius Problems for Starlike Functions Associated with the Sine Function, Bull. Iran. Math. Soc. 45 (2018), 213–232. https://doi.org/10.1007/s41980-018-0127-5.
  3. L.A. Wani, A. Swaminathan, Starlike and Convex Functions Associated with a Nephroid Domain, Bull. Malays. Math. Sci. Soc. 44 (2020), 79–104. https://doi.org/10.1007/s40840-020-00935-6.
  4. J. Sokol, J. Stankiewicz, Radius of Convexity of Some Subclasses of Strongly Starlike Functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19 (1996), 101–105.
  5. K. Sharma, N.K. Jain, V. Ravichandran, Starlike Functions Associated With a Cardioid, Afr. Mat. 27 (2015), 923–939. https://doi.org/10.1007/s13370-015-0387-7.
  6. R. Mendiratta, S. Nagpal, V. Ravichandran, On a Subclass of Strongly Starlike Functions Associated Exponential Function, Bull. Malays. Math. Sci. Soc. 38 (2015), 365–386.
  7. R.K. Raina, J. Sokol, On Coefficient Estimates for a Certain Class of Starlike Functions, Hacettepe J. Math. Stat. 44 (2015), 1427–1433.
  8. H. Tang, M. Arif, M. Haq, N. Khan, M. Khan, K. Ahmad, B. Khan, Fourth Hankel Determinant Problem Based on Certain Analytic Functions, Symmetry 14 (2022), 663. https://doi.org/10.3390/sym14040663.
  9. N.E. Cho, S. Kumar, V. Kumar, V. Ravichandran, H.M. Srivastava, Starlike Functions Related to the Bell Numbers, Symmetry 11 (2019), 219. https://doi.org/10.3390/sym11020219.
  10. J. Dziok, R.K. Raina, J. Sokół, Certain Results for a Class of Convex Functions Related to a Shell-Like Curve Connected With Fibonacci Numbers, Comp. Math. Appl. 61 (2011), 2605–2613. https://doi.org/10.1016/j.camwa.2011.03.006.
  11. S. Kanas, D. R ˘aducanu, Some Class of Analytic Functions Related to Conic Domains, Math. Slovaca 64 (2014), 1183–1196. https://doi.org/10.2478/s12175-014-0268-9.
  12. J. Sokół, J. Stankiewicz, Radius of Convexity of Some Subclasses of Strongly Starlike Functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19 (1996), 101–105.
  13. M. Raza, S.N. Malik, Upper Bound of the Third Hankel Determinant for a Class of Analytic Functions Related With Lemniscate of Bernoulli, J. Ineq. Appl. 2013 (2013), 412. https://doi.org/10.1186/1029-242x-2013-412.
  14. A.K. Sahoo, J. Patel, Hankel Determinant for a Class of Analytic Functions Related With Lemniscate of Bernoulli, Int. J. Anal. Appl. 6 (2014), 170–177.
  15. K. Bano, M. Raza, Starlike Functions Associated with Cosine Functions, Bull. Iran. Math. Soc. 47 (2020), 1513–1532. https://doi.org/10.1007/s41980-020-00456-9.
  16. A. Alotaibi, M. Arif, M.A. Alghamdi, S. Hussain, Starlikness Associated with Cosine Hyperbolic Function, Mathematics 8 (2020), 1118. https://doi.org/10.3390/math8071118.
  17. F.M. Al-Oboudi, n-Bazilevic Functions, Abstr. Appl. Anal. 2012 (2012), 383592. https://doi.org/10.1155/2012/383592.
  18. B. Pinchuk, A Variational Method for Functions of Bounded Boundary Rotation, Trans. Amer. Math. Soc. 138 (1969), 107–113. https://doi.org/10.1090/s0002-9947-1969-0237761-8.
  19. P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, 1983.
  20. D. Alimohammadi, E. Analouei Adegani, T. Bulboac ˘a, N.E. Cho, Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions, J. Function Spaces 2021 (2021), 6690027. https://doi.org/10.1155/2021/6690027.
  21. V.V. Andreev, P.L. Duren, Inequalities for Logarithmic Coefficients of Univalent Functions and their Derivatives, Indiana Univ. Math. J. 37 (1988), 721–733. https://www.jstor.org/stable/24895353.
  22. Q. Deng, On the Logarithmic Coefficients of Bazilevic Functions, Appl. Math. Comp. 217 (2011), 5889–5894. https://doi.org/10.1016/j.amc.2010.12.075.
  23. D. Girela, Logarithmic Coefficients of Univalent Functions, Ann. Acad. Sci. Fenn. Math. 25 (2000), 337–350.
  24. M. Obradovi´c, S. Ponnusamy, K.J. Wirths, Logarithmic Coefficients and a Coefficient Conjecture for Univalent Functions, Monatsh Math. 185 (2017), 489–501. https://doi.org/10.1007/s00605-017-1024-3.
  25. O. Roth, A sharp inequality for the logarithmic coefficients of univalent functions, Proc. Amer. Math. Soc. 135 (2007), 2051–2054. https://doi.org/10.1090/s0002-9939-07-08660-1.
  26. D.K. Thomas, On the Logarithmic Coefficients of Close to Convex Functions, Proc. Amer. Math. Soc. 144 (2015), 1681–1687. https://doi.org/10.1090/proc/12921.
  27. J.W. Noonan, D.K. Thomas, On the Second Hankel Determinant of Areally Mean p-Valent Functions, Trans. Amer. Math. Soc. 223 (1976), 337–346. https://doi.org/10.2307/1997533.
  28. K.I. Noor, Hankel Determinant Problem for the Class of Functions with Bounded Boundary Rotation, Rev. Roum. Math. Pures Appl. 28 (1983), 731–739.
  29. H.M. Srivastava, N. Khan, M. Darus, S. Khan, Q.Z. Ahmad, S. Hussain, Fekete-Szegö Type Problems and Their Applications for a Subclass of q-Starlike Functions with Respect to Symmetrical Points, Mathematics 8 (2020), 842. https://doi.org/10.3390/math8050842.
  30. H.M. Srivastava, M. Kamalı, A. Urdaletova, A Study of the Fekete-Szegö Functional and Coefficient Estimates for Subclasses of Analytic Functions Satisfying a Certain Subordination Condition and Associated With the Gegenbauer Polynomials, AIMS Math. 7 (2022), 2568–2584. https://doi.org/10.3934/math.2022144.
  31. H.M. Srivastava, T.G. Shaba, G. Murugusundaramoorthy, A.K. Wanas, G.I. Oros, The Fekete-Szegö Functional and the Hankel Determinant for a Certain Class of Analytic Functions Involving the Hohlov Operator, AIMS Math. 8 (2023), 340–360. https://doi.org/10.3934/math.2023016.
  32. H.M. Srivastava, K. Alshammari, M. Darus, A New q-Fractional Integral Operator and Its Applications to the Coefficient Problem Involving the Second Hankel Determinant for q-Starlike and q-Convex Functions, J. Nonlinear Var. Anal. 7 (2023), 985–994. https://doi.org/10.23952/jnva.7.2023.6.07.
  33. A. Naik, T. Panigrahi, Upper Bound on Hankel Determinant for Bounded Turning Function Associated With S ˘al ˘agean-Difference Operator, Surv. Math. Appl. 15 (2020), 525–543.
  34. L. Shi, H.M. Srivastava, A. Rafiq, M. Arif, M. Ihsan, Results on Hankel Determinants for the Inverse of Certain Analytic Functions Subordinated to the Exponential Function, Mathematics 10 (2022), 3429. https://doi.org/10.3390/math10193429.
  35. L. Shi, M. Arif, H.M. Srivastava, M. Ihsan, Sharp bounds on the Hankel determinant of the inverse functions for certain analytic functions, J. Math. Ineq. 17 (2023), 1129–1143. https://doi.org/10.7153/jmi-2023-17-73.
  36. H.M. Srivastava, G. Kaur, G. Singh, Estimates of the fourth Hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains, J. Nonlinear Convex Anal. 22 (2021), 511–526.
  37. H.M. Srivastava, G. Murugusundaramoorthy, T. Bulboac ˘a, The Second Hankel Determinant for Subclasses of BiUnivalent Functions Associated with a Nephroid Domain, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116 (2022), 145. https://doi.org/10.1007/s13398-022-01286-6.
  38. H.M. Srivastava, B. Rath, K.S. Kumar, D.V. Krishna, Some Sharp Bounds of the Third-Order Hankel Determinant for the Inverses of the Ozaki Type Close-to-Convex Functions, Bull. Sci. Math. 191 (2024), 103381. https://doi.org/10.1016/j.bulsci.2023.103381.
  39. R.J. Libera, E.J. Złotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (1983), 251–257. https://doi.org/10.1090/s0002-9939-1983-0681830-8.
  40. C. Pommerenke, Univalent Function, Vandenhoeck & Ruprecht, Gottingen, 1975.
  41. V. Ravichandran, S. Verma, Bound for the Fifth Coefficient of Certain Starlike Functions, Compt. Rend. Math. 353 (2015), 505–510. https://doi.org/10.1016/j.crma.2015.03.003.
  42. N.E. Cho, V. Kumar, S.S. Kumar, V. Ravichandran, Radius Problems for Starlike Functions Associated with the Sine Function, Bull. Iran. Math. Soc. 45 (2018), 213–232. https://doi.org/10.1007/s41980-018-0127-5.
  43. R.J. Libera, E.J. Złotkiewicz, Early Coefficients of the Inverse of a Regular Convex Function, Proc. Amer. Math. Soc. 85 (1982), 225–230. https://doi.org/10.1090/s0002-9939-1982-0652447-5.
  44. M. Arif, O.M. Barukab, S. Afzal Khan, M. Abbas, The Sharp Bounds of Hankel Determinants for the Families of Three-Leaf-Type Analytic Functions, Fractal Fract. 6 (2022), 291. https://doi.org/10.3390/fractalfract6060291.
  45. K. Arora, S.S. Kumar, Starlike Functions Associated With a Petal Shaped Domain, Bull. Korean Math. Soc. 59 (2022), 993–1010. https://doi.org/10.4134/BKMS.B210602.
  46. V. Ravichandran, S. Verma, Bound for the Fifth Coefficient of Certain Starlike Functions, Compt. Rend. Math. 353 (2015), 505–510. https://doi.org/10.1016/j.crma.2015.03.003.
  47. K. Bano, M. Raza, Starlike Functions Associated with Cosine Functions, Bull. Iran. Math. Soc. 47 (2020), 1513–1532. https://doi.org/10.1007/s41980-020-00456-9.
  48. K. Ullah, S. Zainab, M. Arif, M. Darus, M. Shutaywi, Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function, J. Function Spaces 2021 (2021), 9967640. https://doi.org/10.1155/2021/9967640.
  49. A. Alotaibi, M. Arif, M.A. Alghamdi, S. Hussain, Starlikness Associated with Cosine Hyperbolic Function, Mathematics 8 (2020), 1118. https://doi.org/10.3390/math8071118.
  50. W. Janowski, Extremal Problems for a Family of Functions With Positive Real Part and for Some Related Families, Ann. Pol. Math. 23 (1970), 159–177.
  51. A.S. Alshehry, R. Shah, A. Bariq, The Second Hankel Determinant of Logarithmic Coefficients for Starlike and Convex Functions Involving Four-Leaf-Shaped Domain, Journal of Function Spaces 2022 (2022), 2621811. https://doi.org/10.1155/2022/2621811.
  52. L. Branges, A Proof of the Bieberbach Conjecture, Acta Math. 154 (1985), 137–152. https://doi.org/10.1007/bf02392821.