ON CONVEX p-VALENT FUNCTIONS MAPPED ONTO THE NEPHROID DOMAIN
Keywords:
Analytic, Univalent, symmetric kidney-shaped function, shell-like curve, geometry
Abstract
p-valent functions serve as natural generalizations of univalent functions and offer a broad platform for studying geometric and functional properties within complex analysis. While coefficient estimation is a core problem in geometric function theory (GFT), the specific bounds and functional determinants for convex p-valent functions associated with the nephroid domain remain under-explored. This research addresses this gap by investigating a new subclass of functions characterized by subordination to a kidney-shaped region, which is motivated by the need to extend existing univalent results to broader p-valent classes. Using the theory of subordination and Taylor–Maclaurin series expansions, the methodology involves comparing the structural coefficients of convex p-valent functions against the nephroid-type mapping P (ξ) = 1+ξ − ξ3/3 . The author establishes estimates for the initial coefficients |a1+p| and |a2+p|, and derives a generalized coefficient bound for |an+p|. These findings are verified for consistency by reducing the results to the specific univalent case where p = 1.
References
[1] Bieberbach, L. (1916). ”ber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln.” Sitzungsberichte der Berliner Akademie, 1916, 940955.
[2] Cho, N. E., Kumar, V., Kumar, S. S. & Ravichandran, V. (2019). ”Radius problems for starlike functions associated with the sine function.” Bull. Iran. Math. Soc., 45, 213232.
[3] Duren, P. L. (1983). ”Univalent Functions.” Springer.
[4] Fagbemiro, O., Sangoniyi, S.O., Raji, M.T., & Olajuwon, B.I. (2024). ”Bi-univalent prob- lem for certain generalized class of analytic functions involving Q-integral operator associ- ated with nephroid domain.” UNILAG J. Maths. Appl., Accepted May, 2025.
[5] Fekete, M., & Szeg, G. (1933). ”Eine Bemerkung ber ungerade schlichte Funktionen.” J. London Math. Soc., 8(2), 8589.
[6] Goodman, A. W. (1946). ”On the coefficients of multivalent functions.” Proc. Amer. Math. Soc., 1(6), 759763.
[7] Goodman, A. W. (1983). Univalent Functions Vols. I & II. Mariner.
[8] Hamzat, J.O., & Fagbemiro, O. (2024). ”Brief Study on Some Properties of Symmetric Cardioid-Bazilevic Functions.”Unilag J. Math. Appl., 4(2), 1-16.
[9] Hamzat, J.O., & El-Ashwah, R.M. (2022). ”Some properties of a generalized Multiplier transform on analytic p-valent functions.” Ukraine J. Math., 74(9), 1274-1283.
[10] Hamzat, J.O., Oladipo, A.T. & Oros G. I. (2022). ”Application of a Multiplier transfor- mation to Libera integral operator associated with generalized distribution.” Symmetry , 14, 1934, 1-14.
[11] Hamzat, J.O., Oladipo, A.T. & Oros G. I. (2022). ”Bi-Univalent Problems Involving Cer- tain New Subclasses of Generalized Multiplier Transform on Analytic Functions Associated with Modified Sigmoid Function.” Symmetry , 14, 1479, 1-13.
[12] Hamzat, J. O. & Adeyemo A. A. (2019). ”New Subclasses of analytic functions with respect to symmetric and conjugate points defined by extended Salagean derivative derivative operator.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2019, no.2, 631-643.
[13] Hamzat, J. O. & Adeleke O. J. (2016). ”A new subclass of p-valent functions defined by Aouf et al. derivative operator.” Int. J. Pure and Appl. Math., vol.110, no.2, 257-264.
[14] Jamiu Olusegun Hamzat, Matthew Olanrewaju Oluwayemi, Alina Alb Lupas & Abbas KareemWanas (2022). ”Bi-Univalent Problems Involving Generalized Multiplier Transform with Respect to Symmetric and Conjugate Points.” Fracta and Fractional , 6, 483, 1-11.
[15] Jamiu Olusegun Hamzat, Matthew Olanrewaju Oluwayemi, Oladipo A. T. & Alina Alb Lupas (2024). ”On alpha- pseudo spiralike functions associated with exponential pareto distribution (EPD) and Libera integral operator.” Mathematics, 12, 1305, 1-10.
[16] Hamzat, J. O. & Sangoniyi S. O. (2021). ”Fekete Szego inequalities associated with certain generalized class of non-Bazilevic functions.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2021, no.2, 874-885.
[17] Hamzat, J. O. (2021). ”Some properties of a new subclass of m-fold symmetric bi-Bazilevic functions associated with modified sigmoid function.” Tbilisi Mathematical J., 14(1), 107- 118.
[18] Hamzat, J. O. (2023). ”Estimates of second and third Hankel determinants for Bazilevic functions of order gamma.” Unilag J. Math. Appl., vol.3, 102-112.
[19] Hamzat, J. O. & Makinde D. O. (2018). ”Coefficient bounds for Bazilevic functions involv- ing Logistic Sigmoid function associated with conic Domains.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2018, no.2, 392-400.
[20] Hamzat, J. O., Raji M. T. &Oni A. A. (2018). ”Hankel determinant associated with logis- tic sigmoid functions in the space of Lambda-pseudo-starlike functions.” Asian J. Math. Comput. Research, 25(2), 74-84.
[21] Hamzat, J. O. & Raji M. T. (2018). ”Bessel functions in the space of Lambda-pseudo- starlike functions with respect to other points associated with modified sigmoid function Hankel determinant associated with logistic sigmoid functions in the space of Lambda- pseudo-starlike functions.” Asian J. Math. Comput. Research, 25(1), 38-49.
[22] Hamzat, J.O. & Oladipo, A.T. (2022). ”On a comprehensive class of analytic p-valent functions associated with shell-like curve and modified sigmoid function.” Malaysian J. Comput. , 7(1), 995-1010.
[23] Hamzat, J.O. (2019). ”Coefficient inequalities for bounded turning functions associated with conic domain.” Electronic J. Math. Anal. Appl., 7(2), 73-78.
[24] Hamzat, J. O. & Oni A. A.(2017). ”Hankel Determinant for Certain Subclasses of Analytic Function.” Asian Research J. Math., 5(2), 1-10.
[25] Hamzat, J. O. (2017). ”Coefficient Bounds for Bazilevic Functions Associated with Modi- fied Sigmoid Function.” Asian Research J. Math., 5(3), 1-10.
[26] Hamzat, J. O. & Olayiwola M. A. (2017). ”Initial Coefficients Estimates for Certain Gener- alized Class of Analytic Functions Involving Sigmoid Function.” Asian Research J. Math., 5(2), 1-11.
[27] Hamzat, J. O. (2019). ”Hankel Determinant for Lambda-pseudo-spiralike functions.”
Transaction Nig. Assoc. Math. Phy., 8, 1-4.
[28] Hamzat, J.O, & Fagbemiro, O. (2017). ”Second Hankel determinant for certain generalized subclasses of Modified Bazilevic functions.”J. Nig. Math. Phy., 42, 27-36.
[29] Hamzat, J.O. & Fagbemiro, O. (2017). ”Fekete Szego inequalityy for lambda-pseudo- Bazilevic functions.”J. Nig. Math. Phy., 43, 1-8.
[30] Hamzat, J.O., Oladipo A. T. & Fagbemiro, O. (2018). ”Coefficient bounds for certain new subclass of m-fold symmetric bi-univalent functions associated with conic domains.”FUW Trends Sci. Tech. J., 43, 1-8.
[31] Keogh, F. R. & Merkes, G. P. (1969). ”A coefficient inequality for certain classes of analytic functions.” Proc. Ann. Math. Soc., 20, 812.
[32] Janowski, W. (1973). ”Some extremal problems for certain families of analytic functions.”
Ann. Polon. Math. Soc., 28, 297326.
[33] Janteng, A., Darus, M., & Ibrahim, R. (2007). ”Hankel determinant for starlike and convex functions.” Int. J. Math. Anal., 1(13), 619625.
[34] Pommerenke, C. (1975). ”Univalent functions.” Vanderhoeck & Ruprecht: Gottingen, Ger- many.
[35] Raina, R. K. & Sok, J. (2015). ”Some properties related to a certain class of starlike functions.” C.R. Math. Acad. Sci. Paris, 353, 973-978.
[36] Singh, R., & Singh, S. (1974). ”Coefficient bounds for multivalent functions.” Proc. Amer. Math. Soc., 44(1), 6168.
[37] Wani, L. A. & Swaminathan, A. (2020). ”Starlike and convec functions associated with a nephroid domain.” Bull. Malays. Math. Sci. Soc..
[38] Wang, B., Srivastava, R., & Liu, J. L. (2021). ”A certain subclass of multivalent analytic functions defined by the q-difference operator related to the Janowski functions.” Mathe- matics, 9, 1706.
[39] Zaprawa, P. (2017). ”Hankel determinants for starlike and convex functions associated with symmetric domains.” Acta Univ. Apulensis, 51, 149160.
[2] Cho, N. E., Kumar, V., Kumar, S. S. & Ravichandran, V. (2019). ”Radius problems for starlike functions associated with the sine function.” Bull. Iran. Math. Soc., 45, 213232.
[3] Duren, P. L. (1983). ”Univalent Functions.” Springer.
[4] Fagbemiro, O., Sangoniyi, S.O., Raji, M.T., & Olajuwon, B.I. (2024). ”Bi-univalent prob- lem for certain generalized class of analytic functions involving Q-integral operator associ- ated with nephroid domain.” UNILAG J. Maths. Appl., Accepted May, 2025.
[5] Fekete, M., & Szeg, G. (1933). ”Eine Bemerkung ber ungerade schlichte Funktionen.” J. London Math. Soc., 8(2), 8589.
[6] Goodman, A. W. (1946). ”On the coefficients of multivalent functions.” Proc. Amer. Math. Soc., 1(6), 759763.
[7] Goodman, A. W. (1983). Univalent Functions Vols. I & II. Mariner.
[8] Hamzat, J.O., & Fagbemiro, O. (2024). ”Brief Study on Some Properties of Symmetric Cardioid-Bazilevic Functions.”Unilag J. Math. Appl., 4(2), 1-16.
[9] Hamzat, J.O., & El-Ashwah, R.M. (2022). ”Some properties of a generalized Multiplier transform on analytic p-valent functions.” Ukraine J. Math., 74(9), 1274-1283.
[10] Hamzat, J.O., Oladipo, A.T. & Oros G. I. (2022). ”Application of a Multiplier transfor- mation to Libera integral operator associated with generalized distribution.” Symmetry , 14, 1934, 1-14.
[11] Hamzat, J.O., Oladipo, A.T. & Oros G. I. (2022). ”Bi-Univalent Problems Involving Cer- tain New Subclasses of Generalized Multiplier Transform on Analytic Functions Associated with Modified Sigmoid Function.” Symmetry , 14, 1479, 1-13.
[12] Hamzat, J. O. & Adeyemo A. A. (2019). ”New Subclasses of analytic functions with respect to symmetric and conjugate points defined by extended Salagean derivative derivative operator.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2019, no.2, 631-643.
[13] Hamzat, J. O. & Adeleke O. J. (2016). ”A new subclass of p-valent functions defined by Aouf et al. derivative operator.” Int. J. Pure and Appl. Math., vol.110, no.2, 257-264.
[14] Jamiu Olusegun Hamzat, Matthew Olanrewaju Oluwayemi, Alina Alb Lupas & Abbas KareemWanas (2022). ”Bi-Univalent Problems Involving Generalized Multiplier Transform with Respect to Symmetric and Conjugate Points.” Fracta and Fractional , 6, 483, 1-11.
[15] Jamiu Olusegun Hamzat, Matthew Olanrewaju Oluwayemi, Oladipo A. T. & Alina Alb Lupas (2024). ”On alpha- pseudo spiralike functions associated with exponential pareto distribution (EPD) and Libera integral operator.” Mathematics, 12, 1305, 1-10.
[16] Hamzat, J. O. & Sangoniyi S. O. (2021). ”Fekete Szego inequalities associated with certain generalized class of non-Bazilevic functions.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2021, no.2, 874-885.
[17] Hamzat, J. O. (2021). ”Some properties of a new subclass of m-fold symmetric bi-Bazilevic functions associated with modified sigmoid function.” Tbilisi Mathematical J., 14(1), 107- 118.
[18] Hamzat, J. O. (2023). ”Estimates of second and third Hankel determinants for Bazilevic functions of order gamma.” Unilag J. Math. Appl., vol.3, 102-112.
[19] Hamzat, J. O. & Makinde D. O. (2018). ”Coefficient bounds for Bazilevic functions involv- ing Logistic Sigmoid function associated with conic Domains.” Int. J. Math. Anal. Opt.: Theory and Appl., vol.2018, no.2, 392-400.
[20] Hamzat, J. O., Raji M. T. &Oni A. A. (2018). ”Hankel determinant associated with logis- tic sigmoid functions in the space of Lambda-pseudo-starlike functions.” Asian J. Math. Comput. Research, 25(2), 74-84.
[21] Hamzat, J. O. & Raji M. T. (2018). ”Bessel functions in the space of Lambda-pseudo- starlike functions with respect to other points associated with modified sigmoid function Hankel determinant associated with logistic sigmoid functions in the space of Lambda- pseudo-starlike functions.” Asian J. Math. Comput. Research, 25(1), 38-49.
[22] Hamzat, J.O. & Oladipo, A.T. (2022). ”On a comprehensive class of analytic p-valent functions associated with shell-like curve and modified sigmoid function.” Malaysian J. Comput. , 7(1), 995-1010.
[23] Hamzat, J.O. (2019). ”Coefficient inequalities for bounded turning functions associated with conic domain.” Electronic J. Math. Anal. Appl., 7(2), 73-78.
[24] Hamzat, J. O. & Oni A. A.(2017). ”Hankel Determinant for Certain Subclasses of Analytic Function.” Asian Research J. Math., 5(2), 1-10.
[25] Hamzat, J. O. (2017). ”Coefficient Bounds for Bazilevic Functions Associated with Modi- fied Sigmoid Function.” Asian Research J. Math., 5(3), 1-10.
[26] Hamzat, J. O. & Olayiwola M. A. (2017). ”Initial Coefficients Estimates for Certain Gener- alized Class of Analytic Functions Involving Sigmoid Function.” Asian Research J. Math., 5(2), 1-11.
[27] Hamzat, J. O. (2019). ”Hankel Determinant for Lambda-pseudo-spiralike functions.”
Transaction Nig. Assoc. Math. Phy., 8, 1-4.
[28] Hamzat, J.O, & Fagbemiro, O. (2017). ”Second Hankel determinant for certain generalized subclasses of Modified Bazilevic functions.”J. Nig. Math. Phy., 42, 27-36.
[29] Hamzat, J.O. & Fagbemiro, O. (2017). ”Fekete Szego inequalityy for lambda-pseudo- Bazilevic functions.”J. Nig. Math. Phy., 43, 1-8.
[30] Hamzat, J.O., Oladipo A. T. & Fagbemiro, O. (2018). ”Coefficient bounds for certain new subclass of m-fold symmetric bi-univalent functions associated with conic domains.”FUW Trends Sci. Tech. J., 43, 1-8.
[31] Keogh, F. R. & Merkes, G. P. (1969). ”A coefficient inequality for certain classes of analytic functions.” Proc. Ann. Math. Soc., 20, 812.
[32] Janowski, W. (1973). ”Some extremal problems for certain families of analytic functions.”
Ann. Polon. Math. Soc., 28, 297326.
[33] Janteng, A., Darus, M., & Ibrahim, R. (2007). ”Hankel determinant for starlike and convex functions.” Int. J. Math. Anal., 1(13), 619625.
[34] Pommerenke, C. (1975). ”Univalent functions.” Vanderhoeck & Ruprecht: Gottingen, Ger- many.
[35] Raina, R. K. & Sok, J. (2015). ”Some properties related to a certain class of starlike functions.” C.R. Math. Acad. Sci. Paris, 353, 973-978.
[36] Singh, R., & Singh, S. (1974). ”Coefficient bounds for multivalent functions.” Proc. Amer. Math. Soc., 44(1), 6168.
[37] Wani, L. A. & Swaminathan, A. (2020). ”Starlike and convec functions associated with a nephroid domain.” Bull. Malays. Math. Sci. Soc..
[38] Wang, B., Srivastava, R., & Liu, J. L. (2021). ”A certain subclass of multivalent analytic functions defined by the q-difference operator related to the Janowski functions.” Mathe- matics, 9, 1706.
[39] Zaprawa, P. (2017). ”Hankel determinants for starlike and convex functions associated with symmetric domains.” Acta Univ. Apulensis, 51, 149160.
Published
2026-04-16
How to Cite
ARIKEWUYO, M. A. (2026). ON CONVEX p-VALENT FUNCTIONS MAPPED ONTO THE NEPHROID DOMAIN. Unilag Journal of Mathematics and Applications, 6(2), 60 - 71. Retrieved from https://lagjma.unilag.edu.ng/article/view/3038
Section
Articles
Copyright (c) 2026 MOJEED ADELEKE ARIKEWUYO
This work is licensed under a Creative Commons Attribution 4.0 International License.
