Bi-univalent Function Subfamilies Associated with the (p,q)-derivative Operator Subordinate to Lucas-Balancing Polynomials
Abstract
In the open disc $\{\zeta\in\mathbb{C}:|\zeta| <1\}$ $=\mathfrak{D}$, we present a family of bi-univalent functions $g(\zeta)=\zeta+\sum\limits_{j=2}^{\infty}d_j\zeta^j$ associated with the $(p,q)$-derivative operator and Lucas-Balancing polynomials. For members of this family, we obtain the upper bounds for $|d_2|$, $|d_3|$, and $|d_3-\xi d_2^2|$, $\xi \in\mathbb{R}$. The new implications of the main results are also discussed, along with relevant connections to earlier research.
Downloads
References
Amourah, A., Frasin, B. A., Swamy, S. R., & Sailaja, Y. (2022). Coefficient bounds for Al-Oboudi type bi-univalent functions connected with a modified sigmoid activation function and k-Fibonacci numbers. Journal of Mathematical and Computer Science, 27, 105-117. https://doi.org/10.22436/jmcs.027.02.02
Altınkaya, Ş., & Yalçın, S. (2020). Lucas polynomials and applications to a unified class of bi-univalent functions equipped with (P,Q)-derivative operators. TWMS Journal of Pure and Applied Mathematics, 11 (1), 100-108.
Altınkaya, Ş., & Yalçın, S. (2020). Certain classes of bi-univalent functions of complex order associated with quasi-subordination involving (p,q)-derivative operator. Kragujevac Journal of Mathematics, 44(4), 639-649. https://doi.org/10.46793/KgJMat2004.639A
Al-Oboudi, F. M. (2004). On univalent functions defined by a generalized Salagean operator. International Journal of Mathematics and Mathematical Sciences, 27, 1429-1436. https://doi.org/10.1155/S0161171204108090
Araci, S., Duran, U., Acikgoz, M., & Srivastava, H. M. (2016). A certain (p,q)-derivative operator and associated divided differences. Journal of Inequalities and Applications, 2016(301), 8 pp. https://doi.org/10.1186/s13660-016-1240-8
Arik, M., Demircan, E., Turgut, T., Ekinci, L., & Mungan, M. (1992). Fibonacci oscillators. Zeitschrift für Physik C Particles and Fields, 55, 89-95. https://doi.org/10.1007/BF01558292
Behera, A., & Panda, G. K. (1999). On the sequence of roots of triangular numbers. The Fibonacci Quarterly, 37(2), 98-105. https://doi.org/10.1080/00150517.1999.12428864
Berczes, A., Liptai, K., & Pink, I. (2020). On generalized balancing sequences. The Fibonacci Quarterly, 48(2), 121-128. https://doi.org/10.1080/00150517.2010.12428112
Brannan, D. A., & Taha, T. S. (1985). On some classes of bi-univalent functions. In S. M. Mazhar, A. Hamoui, & N. S. Faour (Eds.), Mathematical Analysis and its Applications (pp. 53-60). Kuwait: KFAS Proceedings Series, Vol. 3. Pergamon Press (Elsevier Science Limited), Oxford; see also Studia Universitatis Babeş-Bolyai Mathematica, 31 (2), 70-77. https://doi.org/10.1016/B978-0-08-031636-9.50012-7
Brannan, D. A., & Clunie, J. G. (1979). Aspects of contemporary complex analysis. Proceedings of the NATO Advanced Study Institute held at University of Durham. New York: Academic Press.
Brodimas, G., Jannussis, A., & Mignani, R. (1991). Two-parameter quantum groups. Dipartimento di Fisica Università di Roma "La Sapienza" I.N.F.N. - Sezione di Roma, preprint ID No. 820.
Bukweli-Kyemba, J. D., & Hounkonnou, M. N. (2013). Quantum deformed algebras: coherent states and special functions. arXiv preprint, arXiv:1301.0116.
Bulut, S. (2014). Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions. C. R. Acad. Sci. Paris Sér. I, 35, 479-484. https://doi.org/10.1016/j.crma.2014.04.004
Çağlar, M., Deniz, E., & Srivastava, H. M. (2017). Second Hankel determinant for certain subclasses of bi-univalent functions. Turkish Journal of Mathematics, 41, 694-706. https://doi.org/10.3906/mat-1602-25
Cătaş, A. (2007). On certain class of p-valent functions defined by new multiplier transformations. In A. Cătaş (Ed.), Proceedings book of the international symposium on geometric function theory and applications, August 20-24, 2007, TC Istanbul Kultur University, Turkey (pp. 241-250).
Chakrabarti, R., & Jagannathan, R. (1991). A (p,q)-oscillator realization of two-parameter quantum. Journal of Physics A: Mathematical and General, 24 (13), L711. https://doi.org/10.1088/0305-4470/24/13/002
Cotîrlă, L.-I. (2021). New classes of analytic and bi-univalent functions. AIMS Mathematics, 6(10), 10642-10651. https://doi.org/10.3934/math.2021618
Davala, R. K., & Panda, G. K. (2015). On sum and ratio formulas for balancing numbers. Journal of the Indian Mathematical Society, 82(1-2), 23-32.
Durani, U., Acikgoz, M., & Araci, S. (2020). A study on some new results arising from (p,q)-calculus. TWMS Journal of Pure and Applied Mathematics, 11(1), 57-71.
Duren, P. L. (1983). Univalent functions. Grundlehren der Mathematischen Wissenschaften, Band 259. Springer-Verlag, New York.
Fekete, M., & Szegő, G. (1933). Eine Bemerkung über ungerade schlichte Funktionen. Journal of the London Mathematical Society, 89, 85-89. https://doi.org/10.1112/jlms/s1-8.2.85
Frasin, B. A., & Aouf, M. K. (2011). New subclasses of bi-univalent functions. Applied Mathematics, 24, 1569-1573. https://doi.org/10.1016/j.aml.2011.03.048
Frasin, B. A., Swamy, S. R., Amourah, A., Salah, J., & Maheshwarappa, R. H. (2023). A family of bi-univalent functions defined by (p,q)-derivative operator subordinate to a generalized bivariate Fibonacci polynomials. European Journal of Pure and Applied Mathematics, 17(4), 3801-3814. https://doi.org/10.29020/nybg.ejpam.v1714.5526
Frontczak, R. (2019). On balancing polynomials. Applied Mathematical Sciences, 13, 57-66. https://doi.org/10.12988/ams.2019.812183
Frontczak, R. (2018). A note on hybrid convolutions involving Balancing and Lucas-Balancing numbers. Applied Mathematical Sciences, 12, 1201-1208. https://doi.org/10.12988/ams.2018.87111
Frontczak, R., & Baden-Württemberg, L. (2018). A note on hybrid convolutions involving balancing and Lucas-Balancing numbers. Applied Mathematical Sciences, 12, 2001-2008.
Hussen, A., & Illafe, M. (2023). Coefficient bounds for certain subclasses of bi-univalent functions associated with Lucas-Balancing polynomials. Mathematics, 11, 4941. https://doi.org/10.3390/math11244941
Jackson, F. H. (1909). On q-functions and a certain difference operator. Earth and Environmental Science Transactions of the Royal Society of Edinburgh, 46(2), 253-281. https://doi.org/10.1017/S0080456800002751
Jackson, F. H. (1910). q-difference equations. American Journal of Mathematics, 32, 305-314. https://doi.org/10.2307/2370183
Jagannathan, R., & Rao, K. S. (2005). Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. In Proceedings of the International Conference on Number Theory and Mathematical Physics (pp. 20-21). Srinivasa Ramanujan Centre, Kumbakonam, India.
Lewin, M. (1967). On a coefficient problem for bi-univalent functions. Proceedings of the American Mathematical Society, 18, 63-68. https://doi.org/10.1090/S0002-9939-1967-0206255-1
Liptai, K., Luca, F., Pintér, A., & Szalay, L. (2009). Generalized balancing numbers. Indian Mathematics (N.S.), 20, 87-100. https://doi.org/10.1016/S0019-3577(09)80005-0
Malyali, D. (2020). (p,q)-Hahn Difference Operator. (Master's thesis). Submitted to the Institute of Graduate Studies and Research, Eastern Mediterranean University, Gazimağusa, North Cyprus.
Motamednezhad, A., & Salehian, S. (2019). New subclasses of bi-univalent functions by (p; q)-derivative operator. Honam Mathematical Journal, 41 (2), 381-390. https://doi.org/10.5831/HMJ.2019.41.2.381
Panda, C. K., Komatsu, T., & Davala, R. K. (2018). Reciprocal sums of sequences involving balancing and Lucas-balancing numbers. Mathematical Reports, 20, 201-214.
Patel, B. K., Irmak, N., & Ray, P. K. (2018). Incomplete balancing and Lucas-balancing numbers. Mathematical Reports, 20, 59-72.
Ray, P. K. (2015). Balancing and Lucas-balancing sums by matrix method. Mathematical Reports, 17, 225-233.
Ray, P. K., & Sahu, J. (2016). Generating functions for certain balancing and Lucas-balancing numbers. Palestine Journal of Mathematics, 5(2), 122-129. https://doi.org/10.12697/ACUTM.2016.20.14
Sadjang, P. N. (2013). On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas. arXiv preprint, arXiv:1309.3934v1, August 22.
Santhiya, S., & Thilagavathi, K. (2023). Geometric properties of analytic functions defined by the (p, q)-derivative operator involving the Poisson distribution. Journal of Mathematics, 2023, Article ID 2097976, 12 pages. https://doi.org/10.1155/2023/2097976
Selvaraj, C., Thirupathi, G., & Umadevi, E. (2017). Certain classes of analytic functions involving a family of generalized differential operators. Transylvanian Journal of Mathematics and Mechanics, 9(1), 51-61.
Srivastava, H. M., Mishra, A. K., & Gochhayat, P. (2010). Certain subclasses of analytic and bi-univalent functions. Applied Mathematics Letters, 23, 1188-1192. https://doi.org/10.1016/j.aml.2010.05.009
Srivastava, H. M., Raza, N., AbuJarad, E. S. A., Srivastava, G., & AbuJarad, M. H. (2019). Fekete-Szegö inequality for classes of (p,q)-Starlike and (p,q)-convex functions. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas, 113(4), 3563-3584. https://doi.org/10.1007/s13398-019-00713-5
Swamy, S. R. (2021). Coefficient bounds for Al-Oboudi type bi-univalent functions based on a modified sigmoid activation function and Horadam polynomials. Earthline Journal of Mathematical Sciences, 7(2), 251-270. https://doi.org/10.34198/ejms.7221.251270
Swamy, S. R. (2012). Inclusion properties of certain subclasses of analytic functions. International Mathematical Forum, 7(36), 1751-1760.
Swamy, S. R. (2012). Inclusion properties for certain subclasses of analytic functions defined by a generalized multiplier transformation. International Journal of Mathematical Analysis, 6(32), 1553-1564.
Swamy, S. R., & Atshan, W. G. (2022). Coefficient bounds for regular and bi-univalent functions linked with Gegenbauer polynomials. Gulf Journal of Mathematics, 13(2), 67-77. https://doi.org/10.15393/j3.art.2022.10351
Swamy, S. R., Breaz, D., Kala, V., Mamatha, P. K., Cotîrlą, L.-I., & Răpeanu, E. (2024). Initial coefficient bounds analysis for novel subclasses of bi-univalent functions linked with Lucas-Balancing polynomials. Mathematics, 12, 1325. https://doi.org/10.3390/math12091325
Swamy, S. R., Frasin, B. A., Breaz, D., & Cotîrlą, L.-I. (2024). Two families of bi-univalent functions associating the (p, q)-derivative with generalized bivariate Fibonacci polynomials. Mathematics, 12, 3933. https://doi.org/10.3390/math12243933
Swamy, S. R., Mamatha, P. K., Magesh, N., & Yamini, J. (2020). Certain subclasses of bi-univalent functions defined by Salagean operator with the (p, q)-Lucas polynomials. Advances in Mathematics: Scientific Journal, 9(8), 6017-6025. https://doi.org/10.37418/amsj.9.8.70
Tan, D. L. (1984). Coefficient estimates for bi-univalent functions. Chinese Annals of Mathematics, Series A, 5, 559-568.
Tuncer, A., Ali, A., & Abdul, M. S. (2016). On Kantorovich modification of (p, q)-Baskakov operators. Journal of Inequalities and Applications, 2016, Article 98. https://doi.org/10.1186/s13660-016-1045-9
Vijayalakshmi, S. P., Sudharsan, T. V., & Bulboaca, T. (2024). Symmetric Toeplitz determinants for classes defined by post-quantum operators subordinated to the limacon function. Studia Universitatis Babeş-Bolyai Mathematica, 69(2), 299-316. https://doi.org/10.24193/subbmath.2024.2.04
Wachs, M., & White, D. (1991). (p,q)-Stirling numbers and set partition statistics. Journal of Combinatorial Theory, Series A, 56(1), 27-46. https://doi.org/10.1016/0097-3165(91)90020-H
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
