Bi-univalent Function Subclasses with (p, q)-Derivative Operator Linked to Horadam Polynomials
Download PDF
Authors: S. R. SWAMY, D. BREAZ, L-I. COTîRLă AND K. VENUGOPAL
DOI: 10.46793/KgJMat2608.1279S
Abstract:
In the open unit disk {ς ∈ ℂ : |ς| < 1}, two subclasses of bi-univalent functions related to Horadam polynomials are presented and examined in this paper. For functions belonging to the recently established classes, we obtain the estimates of the first two coefficients. Furthermore, an estimate of the Fekete-Szegö problem is provided for functions in these classes. We also provide some observations and draw relevant connections to earlier research.
Keywords:
Bi-univalent functions, (p,q)-derivative operator, subordination, Horodam polynomials, Fekete-Szegö functional.
References:
[1] C. Abirami, N. Magesh, J. Yamini and N. B. Gatti, Horadam polynomial coefficient estimates for the classes of λ-bi-peeudo-starlike and bi-Bazilevic functions, J. Anal. 28 (2020), 951–960.
[2] Q. Z. Ahmad, N. Khan and M. Raza, Certain q-difference operators and their applications to the subclass of meromorphic q-starlike functions, Filomat 33(11) (2019), 3385–3397.
[3] Ş. Altınkaya and S. Yalçın, Certain classes of bi-univalent functions of complex order associated with quasi-subordination involving (p,q)-derivative operator, Kragujevac J. Math. 44(4) (2020), 639–649.
[4] Ş. Altınkaya and S. Yalçın, Lucas polynomials and applications to an unified class of bi-univalent functions equipped with (p,q)-derivative operators, TWMS J. Pure. Appl. Math. 11(1) (2020), 100–108.
[5] S. Araci, U. Duran, M. Acikgoz and H. M. Srivastava, A certain (p,q)-derivative operator and associated divided differences, J. Inequal. Appl. 2016 (2016), Article ID 301.
[6] M. Arik, E. Demircan, T. Turgut, L. Ekinci and M. Mungan, Fibonacci oscillators, Zeitschrift für Physik C Particles and Fields 55 (1992), 89–95.
[7] M. Arif, O. Barkub, H. M. Srivastava, S. Abdullah and S. A. Khan, Some Janowski type harmonic q-starlike functions associated with symmetrical points, Mathematics 8 (2020), Article ID 629.
[8] M. Arif, H. M. Srivastava and S. Uma, Some applications of a q-analogue of the Ruscheweyh type operator for multivalent functions, Rev. Real Acad. Cienc. Exactas Fis. Natur. Ser. A Mat. (RACSAM) 113 (2019), 1211–1221.
[9] D. A. Brannan and J. G. Clunie, Aspects of contemporary complex analysis, Proceedings of the NATO Advanced study institute held at University of Durhary, Newyork, Academic Press, 1979.
[10] D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math. 31(2) (1986), 70–77.
[11] G. Brodimas, A. Jannussis and R. Mignani, Two-parameter Quantum Groups, Universita di Roma “La Sapienza”, Preprint N. 820, 22 Luglio 1991.
[12] R. Chakrabarti and R. A. Jagannathan, (p,q)-oscillator realization of two-parameter quantum algebras, Journal of Phys. A: Math. and Gen. 24 (1991), L711–L718.
[13] L.-I. Cotirlă, New classes of analytic and bi-univalent functions, AIMS Mathematics 6(10) (2021), 10642–10651.
[14] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Ana. 2(1) (2013), 49–60.
[15] U. Duran, M. Acikgoz and S. Aracta, Study on some new results arising from (p,q)-calculus, TWMS J. Pure Appl. Math. 11(1) (2020), 57–71.
[16] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, 1983.
[17] S. M. El-Deeb, T. Bulboacă and B. M. El-Matary, Maclaurin coefficient estimates of bi-univalent functions connected with the q-derivative, Mathematics 8 (2020), Article ID 418.
[18] M. Fekete and G. Szegö, Eine bemerkung über ungerade schlichte funktionen, J. Lond. Math. Soc. 89 (1933), 85–89.
[19] B. A. Frasin and M. Darus, Subclass of analytic functions defined by q-derivative operator associated with Pascal distribution series, AIMS Mathematics 6(5) (2021), 5008–5019. https://doi.org/10.3934/math.2021295
[20] B. A. Frasin, Coefficient bounds for certain classes of bi-univalent functions, Hacet. J. Math. Stat. 43(3) (2014), 383–389.
[21] B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. 24 (2011), 1569–1573.
[22] B. A. Frasin, Y. Sailaja, S.R. Swamy and A. K. Wanas, Coefficients bounds for a family of bi-univalent functions defined by Horadam polynomials, Acta et Communicationes Universitatis Tartuensis de Mathematica 26(1) (2022), 25–32.
[23] S. H. Hadi and M. Darus, (p,q)-Chebyshev polynomials for the families of biunivalent function associating a new integral operator with (p,q)-Hurwitz zeta function, Turk. J. Math. 46 (2022), 2415–2429.
[24] T. Horzum and E. G. Koçer, On some properties of Horadam polynomials, Int. Math. Forum 4 (2009), 1243–1252.
[25] A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quart. 23 (1985), 7–20.
[26] M. E. H. Ismail, E. Merkes and D. Styer, A generalization of starlike functions, Complex Var. Theory Appl. 14 (1990), 77–84.
[27] F. H. Jackson, On q-functions and a certain difference operator, Trans. R. Soc. Edinburgh 46 (1908), 253–281.
[28] R. Jagannathan and K. S. Rao, Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, Proceeding of the International Conference on Number Theory and Mathematical Physics, Srinivasa Ramanujan Centre, Kumbakonam, India, 20–21 December, 2005.
[29] S. Kanas and D. Raducanu, Some class of analytic functions related to conic domains, Math. Slovaca. 64 (2014), 1183–1196.
[30] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
[31] S. Mahmood, Q. Z. Ahmad, H. M. Srivastava, N. Khan, B. Khan and M. Tahir, A certain subclass of meromorphically q-starlike functions associated with the Janowski functions, J. Inequal. Appl. 2019 (2019), Article ID 88.
[32] S. K. Mohapatra and T. Panigrahi, Coefficient estimates for bi-univalent functions defined by (p,q) analogue of the Salagean differential operator related to the Chebyshev polynomials, J. Math. Fund. Sci. 53(1) (2021), 49–66.
[33] A. Motamednezhad and S. Salehian, New subclass of bi-univalent functions by (p,q)-derivative operator, Honam Mathematical J. 41(2) (2019), 381–390. https://doi.org/10.5831/HMJ.2019.41.2.381
[34] H. Orhan, P. K. Mamatha, S. R. Swamy, N. Magesh and J. Yamini, Certain classes of bi-univalent functions associated with the Horadam polynomials, Acta Univ. Sapieniae, Mathematica 13(1) (2021), 258–272.
[35] P. N. Sadjang, On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas, Mathematics (2013).
[36] A. E. Shammaky, B. A. Frasin and S. R. Swamy, Fekete-Szegö inequality for bi-univalent functions subordinate to Horadam polynomials, J. Funct. Spaces (2022), Article ID 9422945, 7 pages.
[37] J. Soontharanon and T. Sitthiwirattham, On fractional (p,q)-calculus, Adv. Difference Equ. 35 (2020), 18 pages. https://doi.org/10.1186/s13662-020-2512-7
[38] H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: H. M. Srivastava, S. Owa (Ed.), Univalent Functions; Fractional Calculus; and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), JohnWiley and Sons, New York, Chichester, Brisbane and Toronto, 1989, 329–354.
[39] H. M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran J. Sci. Technol. Trans. A 44 (2020), 327–344.
[40] H. M. Srivastava, Ş. Altınkaya and S. Yalçın, Certain subclasses of bi-univalent functions associated with the Horadam polynomials, Iran J. Sci. Technol. Trans. A 43 (2019), 1873–1879. https://doi.org/10.1007/s40995-018-0647-0
[41] H. M. Srivastava, S. Gaboury, F. Ghanim, Coefficients estimate for some general subclasses of analytic and bi-univalent functions, Afr. Mat. 28 (2017), 693–706.
[42] H. M. Srivastava, B. Khan, N. Khan and Q. Z. Ahmad, Coefficient inequalities for q-starlike functions associated with the Janowski functions, Hokkaido Math. J. 48 (2019), 407–425.
[43] H. M. Srivastava, B. Khan, N. Khan, Q. Z. Ahmad and M. Tahir, A generalized conic domain and its applications to certain subclasses of analytic functions, Rocky Mountain J. Math. 49 (2019), 2325–2346.
[44] H. M. Srivastava, N. Khan, M. Darus, S. Khan, Q. Z. Ahmad and S. Hussain, Fekete-Szegö type problems and their applications for a subclass of q-starlike functions with respect to symmetrical points, Mathematics 8 (2020), Article ID 842.
[45] H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188–1192.
[46] H. M. Srivastava, N. Raza, E. S. A. AbuJarad, G. Srivastava and M. H. AbuJarad, Fekete-Szegö inequality for classes of (p,q)-starlike and (p,q)-convex functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 113(4) (2019), 3563–3584. https://doi.org/10.1007/s13398-019-00713-5
[47] H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad and N. Khan, Some general families of q-starlike functions associated with the Janowski functions, Filomat 33 (2019), 2613–2626
[48] H. M. Srivastava, A. K. Wanas and R. Srivastava, Applications of the q-Srivastava-Attiya operator involving a family of bi-univalent functions associated with Horadam polynomials, Symmetry 13(7) (2021), Article ID 1230.
[49] S. R. Swamy, Coefficient bounds for Al-Oboudi type bi-univalent functions based on a modified sigmoid activation function and Horadam polynimials, Earthline J. Math. Sci. 7(2) (2021), 251–270.
[50] S. R. Swamy and Y. Sailaja, Horadam polynomial coefficient estimates for two families of holomorphic and bi-univalent functions, International Journal of Mathematical Trends and Technology 66(8) (2020), 131–138.
[51] D. L. Tan, Coefficient estimates for bi-univalent functions, Chin. Ann. Math. Ser. A 5 (1984), 559–568.
[52] H. Tang, G. Deng and S. Li, Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions, J. Inequal. Appl. 2013 (2013), Article ID 317.
[53] A. Tuncer, A. Ali and A. M. Syed, On Kantorovich modification of (p,q)-Baskakov operators, Journal of Inequalities and Applications 2016 (2016), Article ID 98.
[54] M. Wachs and D. White, (p;q)-Stirling numbers and set partition statistics, J. Combin. Theory Ser. A 56(1) (1991), 27–46.
[55] A. K. Wanas and A. A. Lupas, Applications of Horadam polynomials on Bazilevic bi-univalent function satisfying subordinate conditions, IOP Conf. Series: Journal of Physics: Conf. Series 1294, (2019), Article ID 032003. https://doi.org/10.1088/1742-6596/1294/3/032003
[56] X. Zhang, S. Khan, S. Hussain, H. Tang and Z. Shareef, New subclass of q-starlike functions associated with generalized conic domain, AIMS Mathematics 5 (2020), 4830–4848.
Login:
