Geometric perspectives on the variability of spiralike functions with respect to a boundary point in relation to Janowski functions

Bilal Khan; Wafa F. Alfwzan; Khadijah M. Abualnaja; Manuela Oliveira; Bilal Khan; Wafa F. Alfwzan; Khadijah M. Abualnaja; Manuela Oliveira

doi:10.3934/math.2025585

AIMS Mathematics

2025, Volume 10, Issue 6: 13006-13024. doi: 10.3934/math.2025585

Previous Article Next Article

Research article

Geometric perspectives on the variability of spiralike functions with respect to a boundary point in relation to Janowski functions

1.
Institute of Mathematics, Henan Academy of Sciences NO. 228, Chongshi Village, Zhengdong New District, Zhengzhou 450046, Henan, China
2.
Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics and Statistics, Collage of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia
4.
Department of Mathematics and CIMA-Center for Research on Mathematics and its Applications, University of Evora, Evora, Portugal

Received: 14 February 2025 Revised: 16 April 2025 Accepted: 21 April 2025 Published: 06 June 2025
MSC : 30C45, 30C50, 30C80

Investigating the variability domain in the geometric function theory yields profound insights into the behavior of geometric functions, thereby facilitating the examination of extremal problems and the derivation of bounds and inequalities. While the previous literature has examined similar classes, our approach offers significant advantages through a more generalized framework. Our study considers normalized analytic functions with specific positivity conditions which involve complex parameters. This investigation extends a previous work by analyzing a broader set of non-vanishing analytic functions. Unlike earlier studies that focused on specific parameter values, our approach allows for wider applications across multiple subclasses through the incorporation of additional parameters. We aim to determine the variability domain for the logarithm of these functions at fixed points within the unit disk as the functions range over a particular class defined by the specific parameter constraints. This generalized approach unifies several known results and provides a comprehensive framework to solve previously intractable boundary problems in the geometric function theory.
- Janowski functions,
- spiralike functions,
- boundary point,
- starlike functions,
- variability region
Citation: Bilal Khan, Wafa F. Alfwzan, Khadijah M. Abualnaja, Manuela Oliveira. Geometric perspectives on the variability of spiralike functions with respect to a boundary point in relation to Janowski functions[J]. AIMS Mathematics, 2025, 10(6): 13006-13024. doi: 10.3934/math.2025585

Related Papers:

Abstract

Investigating the variability domain in the geometric function theory yields profound insights into the behavior of geometric functions, thereby facilitating the examination of extremal problems and the derivation of bounds and inequalities. While the previous literature has examined similar classes, our approach offers significant advantages through a more generalized framework. Our study considers normalized analytic functions with specific positivity conditions which involve complex parameters. This investigation extends a previous work by analyzing a broader set of non-vanishing analytic functions. Unlike earlier studies that focused on specific parameter values, our approach allows for wider applications across multiple subclasses through the incorporation of additional parameters. We aim to determine the variability domain for the logarithm of these functions at fixed points within the unit disk as the functions range over a particular class defined by the specific parameter constraints. This generalized approach unifies several known results and provides a comprehensive framework to solve previously intractable boundary problems in the geometric function theory.

References

[1]	A. Lyzzaik, On a conjecture of M. S. Robertson, Proc. Amer. Math. Soc., 91 (1984), 108–110. https://doi.org/10.2307/2045280 doi: 10.2307/2045280
[2]	H. Silverman, E. M. Silvia, Subclasses of univalent functions starlike with respect to a boundary point, Houston J. Math., 16 (1990), 289–299.
[3]	M. S. Robertson, Univalent functions starlike with respect to a boundary point, J. Math. Anal. Appl., 81 (1981), 327–345. https://doi.org/10.1016/0022-247X(81)90067-6 doi: 10.1016/0022-247X(81)90067-6
[4]	M. Elin, S. Reich, D. Shoikhet, Dynamics of inequalities in geometric function theory, J. Inequal. Appl., 6 (2001), 651–664. https://doi.org/10.1155/S1025583401000406 doi: 10.1155/S1025583401000406
[5]	A. Lecko, On the class of functions starlike functions with respect to the boundary point, J. Math. Anal. Appl., 261 (2001), 649–664. https://doi.org/10.1006/jmaa.2001.7564 doi: 10.1006/jmaa.2001.7564
[6]	H. Grunsky, Koeffizientenbedingungen für schlicht abbildende meromorphe funktionen, Math. Z., 45 (1939), 29–61. https://doi.org/10.1007/BF01580272 doi: 10.1007/BF01580272
[7]	H. Yanagihara, Regions of variability for functions of bounded derivatives, Kodai Math. J., 28 (2005), 452–462. https://doi.org/10.2996/kmj/1123767023 doi: 10.2996/kmj/1123767023
[8]	S. Ponnusamy, A. Vasudevarao, H. Yanagihara, Region of variability for close-to-convex functions, Complex Var. Elliptic, 53 (2008), 709–716. https://doi.org/10.1080/17476930801996346 doi: 10.1080/17476930801996346
[9]	H. Yanagihara, Variability regions for families of convex functions, Comput. Methods Funct. Theory, 10 (2010), 291–302. https://doi.org/10.1007/BF03321769 doi: 10.1007/BF03321769
[10]	S. Ponnusamy, A. Vasudevarao, Region of variability of two subclasses of univalent functions, J. Math. Anal. Appl., 332 (2007), 1323–1334. https://doi.org/10.1016/j.jmaa.2006.11.019 doi: 10.1016/j.jmaa.2006.11.019
[11]	S. Ponnusamy, A. Vasudevarao, H. Yanagihara, Region of variability of univalent functions $f$ for which $zf_{0}$ is spirallike, Houston J. Math., 34 (2008), 1037–1048.
[12]	S. Ponnusamy, A. Vasudevarao, M. Vuorinen, Region of variability for spiral-like functions with respect to a boundary point, Colloquium Mathematicum, 116 (2009), 31–46.
[13]	S. Ponnusamy, A. Vasudevarao, Region of variability for functions with positive real part, Ann. Pol. Math., 99 (2010), 225–245. https://doi.org/10.4064/ap99-3-2 doi: 10.4064/ap99-3-2
[14]	S. Chen, A. W. Huang, Region of variabilty for generalized -convex and -starlike functions and their extreme points, Commun. Korean Math. S., 25 (2010), 557–569. https://doi.org/10.4134/CKMS.2010.25.4.557 doi: 10.4134/CKMS.2010.25.4.557
[15]	S. Ponnusamy, A. Vasudevarao, M. Vuorinen, Region of variability for exponentially convex univalent functions, Complex Anal. Oper. Theory, 5 (2011), 955–966. https://doi.org/10.1007/s11785-010-0089-y doi: 10.1007/s11785-010-0089-y
[16]	M. Raza, W. Ul-Haq, S. Noreen, Regions of variablity of variability for Janowski functions, Miskolc Mathematical Notes, 16 (2015), 1117–1127. https://doi.org/10.18514/MMN.2015.1344 doi: 10.18514/MMN.2015.1344
[17]	W. Ul-Haq, Variability regions for Janowski convex functions, Complex Var. Elliptic, 59 (2014), 355–361. https://doi.org/10.1080/17476933.2012.725164 doi: 10.1080/17476933.2012.725164
[18]	M. Raza, W. Ul Haq, J. L. Liu, S. Noreen, Regions of variability for a subclass of analytic functions, AIMS Mathematics, 5 (2020), 3365–3377. https://doi.org/10.3934/math.2020217 doi: 10.3934/math.2020217
[19]	S. Z. H. Bukhari, A. K. Wanas, M. Abdalla, S. Zafar, Region of variablity for Bazilevic functions, AIMS Mathematics, 8 (2023), 25511–25527. https://doi.org/10.3934/math.20231302 doi: 10.3934/math.20231302
[20]	D. Aharonov, M. Elin, D. Shoikhet, Spiral-like functions with respect to a boundary point, J. Math. Anal. Appl., 280 (2003), 17–29. https://doi.org/10.1016/S0022-247X(02)00615-7 doi: 10.1016/S0022-247X(02)00615-7
[21]	H. Yanagihara, Regions of variability for functions of bounded derivatives, Kodai Math. J., 28 (2005), 452–462. https://doi.org/10.2996/kmj/1123767023 doi: 10.2996/kmj/1123767023
[22]	D. Chalishajar, M. Somasundaram, P. Sethuraman, Analyticity of weighted composition semigroups on the space of holomorphic functions, Bull. Iran. Math. Soc., 51 (2025), 15. https://doi.org/10.1007/s41980-024-00923-7 doi: 10.1007/s41980-024-00923-7
[23]	R. Kasinathan, R. Kasinathan, D. Chalishajar, Exponential decay in mean square of mean-field neutral stochastic integrodifferential evolution equations: global attracting set and fractional Brownian motion, Stochastics, 97 (2025), 287–298. https://doi.org/10.1080/17442508.2024.2430579 doi: 10.1080/17442508.2024.2430579
[24]	D. Chalishajar, D. Kasinathan, R. Kasinathan, R. Kasinathan, T-Controllability of higher-order fractional stochastic delay system via integral contractor, Journal of Control and Decision, 2024 (2024), 1–24. https://doi.org/10.1080/23307706.2024.2379993 doi: 10.1080/23307706.2024.2379993
[25]	V. Sandrasekaran, R. Kasinathan, R. Kasinathan, D. Chalishajar, D. Kasinathan, Fractional stochastic Schrödinger evolution system with complex potential and poisson jumps: Qualitative behavior and T-controllability, Partial Differential Equations in Applied Mathematics, 10 (2024), 100713. https://doi.org/10.1016/j.padiff.2024.100713 doi: 10.1016/j.padiff.2024.100713
[26]	M. G. Khan, W. K. Mashwani, L. Shi, S. Araci, B. Ahmad, B. Khan, Hankel inequalities for bounded turning functions in the domain of cosine Hyperbolic function, AIMS Mathematics, 8 (2023), 21993–22008. https://doi.org/10.3934/math.20231121 doi: 10.3934/math.20231121
[27]	M. G. Khan, B. Khan, F. M. O. Tawfiq, J.-S. Ro, Zalcman functional and majorization results for certain subfamilies of holomorphic functions, Axioms, 12 (2023), 868. https://doi.org/10.3390/axioms12090868 doi: 10.3390/axioms12090868
[28]	B. Khan, J. Gong, M. G. Khan, F. Tchier, Sharp coefficient bounds for a class of symmetric starlike functions involving the balloon shape domain, Heliyon, 10 (2024), e38838. https://doi.org/10.1016/j.heliyon.2024.e38838 doi: 10.1016/j.heliyon.2024.e38838
[29]	S. Ponnusamy, Foundations of complex analysis, Math. Gaz., 83 (2005), 183–183.
[30]	S. Dineen, The Schwarz lemma, Oxford: Clarendon Press, 1989.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)