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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 54 Issue: 4

Radii of convexity associated with various subclasses of analytic functions for functions related to Kaplan classes

Jananı B B(NATIONAL INSTITUTE OF TECHNOLOGY, TIRUCHIRAPPALLI)0009-0000-0857-2650
V Ravichandran(NATIONAL INSTITUTE OF TECHNOLOGY TIRUCHIRAPPALLI)0000-0002-3632-7529
Nisha Bohra(Sri Venkateswara College, University of Delhi)0000-0002-8916-5789
Pages
—
Publication Date
2025-08-29
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The results obtained are sharp.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:A normalized analytic function $f$ defined on the open unit disc $\mathbb{D}$ is called Ma-Minda convex if $ 1+zf''(z)/f'(z)$ is subordinate to the function $\varphi$.
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Key Findings & Data:For $ 0\leqslant\alpha\leqslant\beta$, the Kaplan class $\mathcal{K}(\alpha,\beta)$ of type $\alpha$ and $\beta$ consists of normalized analytic functions of the form $p^\alpha g$ defined on $\mathbb{D}$ where $p$ with $p(0)=1$ is an analytic function taking values in the right half-plane and $g$ is an analytic function with $g(0)=1$ satisfying Re$(zg'(z)/g(z))>(\alpha-\beta)/2$.
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Conclusion & Impact:The results obtained are sharp.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:univalent functions · starlike functions · convex functions

Abstract

A normalized analytic function $f$ defined on the open unit disc $\mathbb{D}$ is called Ma-Minda convex if $ 1+zf''(z)/f'(z)$ is subordinate to the function $\varphi$. For $ 0\leqslant\alpha\leqslant\beta$, the Kaplan class $\mathcal{K}(\alpha,\beta)$ of type $\alpha$ and $\beta$ consists of normalized analytic functions of the form $p^\alpha g$ defined on $\mathbb{D}$ where $p$ with $p(0)=1$ is an analytic function taking values in the right half-plane and $g$ is an analytic function with $g(0)=1$ satisfying Re$(zg'(z)/g(z))>(\alpha-\beta)/2$. For functions $f$ with $f'\in \mathcal{K}(\alpha,\beta)$, we obtain the radius of Ma-Minda convexity for various choices of $\varphi$. The radius of lemniscate convexity, lune convexity, nephroid convexity, exponential convexity and several other radius estimates are examined. The results obtained are sharp.

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Cite This Article

B, J. B., Ravichandran, V., Bohra, N. (2025). Radii of convexity associated with various subclasses of analytic functions for functions related to Kaplan classes. Hacettepe Journal of Mathematics and Statistics, Volume: 54 Issue: 4. https://doi.org/10.15672/hujms.1471427

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References (29)

  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, no. 11, 6557–6565, 2012.
  2. K. Arora and S. S. Kumar, Starlike functions associated with a petal shaped domain, Bull. Korean Math. Soc. 59, no. 4, 993–1010, 2022.
  3. 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, no. 1, 213– 232, 2019.
  4. S. Gandhi and V. Ravichandran, Starlike functions associated with a lune, Asian-Eur. J. Math. 10, no. 4, 1750064, 12 pp, 2017.
  5. K. Gangania and S. S. Kumar, $\mathcal{S}^*(\phi)$ and $\mathcal{C}(\phi)$-radii for some special functions, Iran. J. Sci. Technol. Trans. A Sci. 46, no. 3, 955–966, 2022.
  6. P. Goel and S. Sivaprasad Kumar, Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc. 43, no. 1, 957–991, 2020.
  7. A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56, no. 1, 87–92, 1991.
  8. P. Gupta, S. Nagpal and V. Ravichandran, Inclusion relations and radius problems for a subclass of starlike functions, J. Korean Math. Soc. 58, no. 5, 1147–1180, 2021.
  9. W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1, 169–185, 1952.
  10. S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40, no. 2, 199–212, 2016.
  11. A. Lecko, V. Ravichandran and A. Sebastian, Starlikeness of certain non-univalent functions, Anal. Math. Phys. 11, no. 4, Paper No. 163, 23 pp, 2021.
  12. A. Lecko and B. Śmiarowska, Classes of analytic functions related to Blaschke products, Filomat 32, no. 18, 6289–6309, 2018.
  13. S. K. Lee, K. Khatter and V. Ravichandran, Radius of starlikeness for classes of analytic functions, Bull. Malays. Math. Sci. Soc. 43, no. 6, 4469–4493, 2020.
  14. W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157–169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA.
  15. T. H. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14, 514–520, 1963.
  16. R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38, no. 1, 365–386, 2015.
  17. V. Paatero, Uber die Konforme Abbildungen von gebieten, deren rander von beschrakter Drehung Sind, Ann. Acad. Sci. Fenn. Ser. A 33, 177, 1931.
  18. V. Paatero, Uber gebiete von beschrankter, Ann. Acad. Sci. Fenn. Ser. A 37, 120, 1933.
  19. R. K. Raina and J. Sokół, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353, no. 11, 973–978, 2015.
  20. V. Ravichandran, F. Rønning and T. N. Shanmugam, Radius of convexity and radius of starlikeness for some classes of analytic functions, Complex Variables Theory Appl. 33, no. 1-4, 265–280, 1997.
  21. S. Ruscheweyh, Convolutions in geometric function theory, Séminaire de Mathématiques Supérieures, 83, Presses de l’Université de Montréal, Montreal, QC, 1982.
  22. A. Sebastian and V. Ravichandran, Radius of starlikeness of certain analytic functions, Math. Slovaca 71, no. 1, 83–104, 2021.
  23. T. N. Shanmugam and V. Ravichandran, Certain properties of uniformly convex functions, in Computational methods and function theory 1994 (Penang), 319–324, Ser. Approx. Decompos., 5, World Sci. Publ., River Edge, NJ.
  24. K. Sharma, N. K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat. 27, no. 5-6, 923–939, 2016.
  25. T. B. Sheil-Small, The Hadamard product and linear transformations of classes of analytic functions, J. Analyse Math. 34, 204–239, 1979.
  26. J. Sokół and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19, 101–105, 1996.
  27. 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, no. 4, Paper No. 178, 20 pp, 2020.
  28. L. A. Wani and A. Swaminathan, Starlike and convex functions associated with a nephroid domain, Bull. Malays. Math. Sci. Soc. 44, no. 1, 79–104, 2021.
  29. 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, no. 1, 213– 232, 2019.