BI- UNIVALENT PROBLEM FOR CERTAIN GENERALIZED CLASS OF ANALYTIC FUNCTIONS INVOLVING Q-INTEGRAL OPERATOR ASSOCIATED WITH NEPHROID DOMAIN.
Keywords:
Analytic, Starlike, Bounded Turning, q-Integral Operator, Nephroid Domain
Abstract
The authors in this article study a new subclass of bi-univalent functions involving q-integral operator associated with nephroid domain by using the concept of subordination and bi-linear fractional principles. We further employed our investigation to determine new coefficient bounds and subsequently obtain the Fekete-Szego¨ inequalities for functions belonging to the aforementioned subclass were obtained.
References
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[3] F. I. Akinwale and J.O. Hamzat Coefficient bounds for certain generalized class of analytic functions involving Bazil;evic type function and the Sigmoid function, J. Nig. Math. Phy. Vol. 62, Dec.,(2021), Issue, 1-8.
[4] A. Amourah, B.A. Frasin, T. Abdeijawad. Fekete-Szego¨ Inequality for Analytic and Bi-univalent Functions Subordinate to Gegenbauer Polynomials , J. Funct. Spaces 2021, 2021, 5574673, 1-7.
[5] A. Amourah, B.A. Frasin, M. Ahmad, F. Yousef. Exploiting the Pascal Distribution Series and Gegenbauer Polynomials to Construct and Study a New Subclass of Analytic Bi-Univalent Functions , Symmetry 2022, 14, 147, 1-8.
[6] A. Arai, V. Gupta, and R. P. Agarwal. Applications of q−Calculus in Operator Theory , Springer, New York. NY, USA, (2013), 1 - 13.
[7] O.P. Ahuja, S. Kumar, Ravichandran. Applications of first order differetial subordination for functions with positive real part , Stud. Univ. Babes-Bolyai Math. 63 (2018), no.3, 303 -311.
[8] R.M. Ali, S.K. Lee, V. Ravichandran, and S. Subramaniam. Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions , Appl. Math. Latt.25 (2012), no. 3, 344 351.
[9] A. Akgul, T. G. Shara. u, v- Lucas polynomial coefficient relations of the bi-univalent function class , Commun. Fac. Sci. Univ. Ank. Ser. AI MATH. stat. Volume 71, Number 4, (2022) pages 1121 - 1134. Doi:10.31801/cfsa.smas.1086809.
[10] S. Altinkaya, S. Yal¸cin. Coefficient bounds for a subclass of bi-univalent functions ,6(2015), 180 -185.
[11] S. Altinkaya, S. Yal¸cin. Estimates on Coefficients of a General Subclass of Bi-univalent Functions Associated with Symetric q− Derivative Operator by Means of the Chebyshev Polynomials , Asia Pac. J. Math. (2017), 8, 90 - 99.
[12] L. Bieberbach. Uber koeffizlenten Derjenigen Potenzreihen, Welche Eine Schlichtz Des Einheitskreises Vermitteln , Vol. 38, Sitzungsberichte der PreuBischen Akademie der Wissenschaften zu Berlin, 1916.
[13] S. Bulut. Coefficient Estimates for a Class of Analytic and Bi-univalent Functions ,Novi. Sal.J.Math. 2013, 2013,43, 59-65.
[14] R. Bucar, D. Breaz. On a new class of analytic functions with respect to symmetric points involving the q−derivative operator , In Journal of Physics: Conference Series, IOP Publishing,(2019), 1-8.
[15] T. Bulboara, P. Goswami. On some classes of bi-univalent functions , Studia Universitatis Babas-Bolyai Mathematica, 2(2015), 7 - 13.
[16] P. N. Chidera. New subclasses of the class of close - to - convex functions , Proceedings of the American Mathematical Society, ol. 62, no. 1, (1977), 37 -43.
[17] N.E. Cho, V. Kumar, S.S. Kumar, V. Ravichandran. Radius problems for starlike functions associated with sine functions , Bull. Iran, Math. Soc., 45 (2019), 213 - 232.
[18] L. De Branges. A proof of the Bieberbach conjecture , Acta.Math.154,(1985),137-152. http://dx.doi.org/10.1007/BF02392821
[19] K. Dhanalakshmi, S. Poongothai. Third Hankel determinant for a subclass of analytic univalent related to modified Nephroid domain ,International Journal of Mechanical Engineering,ol. 7, N0.4 April, 2022, 1-5. ISSN: 0974-5823.
[20] P.I. Duren, Univalent functions. Grundlehen der Mathematischan Wissenschaften 259: Springer: New York, Ny, USA; Berlin/Heidelberg, Germany;Tokyo,Japan, (1983).
[21] A.H. El-Qadeem, M.A. Mamon, I. S. Elshazly. Application of Einstein function on bi-univalent functions defined on the unit disc, Symmetry, 14, 4(2022),758, 1-11.
[22] Ebrahim Analouei Adegani, Serap Bulut, Ahmad Zireh. Coefficient estimates for a subclass of analytic bi-univalent functions , Bull. Korean Math. Soc. 55 (2018), No. 2, 405 - 413.
[23] B.A. Frasin, M.K. Aouf. New subclasses of Bi-Univalent Functions, Appl. Math. lett (2011), 24, 1569 - 1573.
[24] O. Fagbemiro, J.O. Hamzat, M. T. Raji. A certain class of (j, k)− symmetric function involving sigmoid function defined by using subordination principle ,
Journal of the Nigerian Association of Mathematical Physics(2024), 39 - 46.
[25] O. Fagbemiro, M.T. Raji, J.O. Hamzat, A.T. Oladipo and B.I. Olajuwon. Coefficient bounds for subclass of sigmoid functions involving subordination principle defined by Sa˘la˘gean Differential Operator, Covenant Journal of Physical and Life Sciences, Vol. 13, No. 1, June 2025, 1-16.
[26] M. Fekete, G. Szego¨. Eine Beemer Kung uber ungerade schlichte Functionen, Journal of the London Mathematical Sociebty, Volume 8, 85 89, 1933.
[27] A.W. Goodman. Univalent functions. Vol. 1, Mariner Publishing Co., Inc., Tampa, FL, 1983.
[28] S.P. Goyal and R. Kumar. Coefficient estimates and quasi-subordination properties associated with certain subclasses of analytic and bi-univalent functions , Mathematica Slovaca 65 (2015) No. 3, 533-544.
[29] Huo Tang, Gangadharan Murugusundaramoorthy, Shu-Hai Li, Li-Na Ma. Fekete-Szego¨ and Hankel inequalities for certain class of analytic functions related to the sine function, AIS Mathematics, 7(4) (2022): 6365 - 6380.
[30] J.O. Hamzat, M. O. Oluwayemi, A. A. Lupas and A. K. Wanas. Bi-Univalent Problems Involving Generalized ultiplier Transform with Respect to Symmetric and Conjugate Points , Fractal Fract., 2022, 6 , 483, 1 - 11. https://doi.org/10.3390/fractalfract6090483
[31] J.O. Hamzat, M. O. Oluwayemi, A. T. Oladipo and A. A. Lupas. On alpha-Pseudo Spiralike Function Associated with Exponential Pareto Distribution (EPD) and Libera Integral Operator, Mathematics, 2024, 1-10. https:/doi.org/10.3390/math1010000
[32] J.O. Hamzat, G. Murugusundaramoorthy and M. O. Oluwayemi. Brief Study on a New Family of Analytic Functions, Sahand Commun. Math. Anal. vol. 21, no. 4, 109-122. Doi:1022130/scma.2024.2011065.1457
[33] J.O. Hamzat. Estimate of Second and Third Hankel Determinants for Bazilevic Function of Order Gamma, Unilag Journal of Mathematics and Applications, vol. 3, 2023, 102-112.
[34] J.O. Hamzat. Some Properties of a new subclass of m-fold symmetric bi-Bazilevic Functions Associated with Modified Sigmoid Function, Tbilisi Math. J. 14(1), 2021, 107-118. Doi:10.32513/tmj/1932200819
[35] M.Illafe, A. Amourah and H.Mohd. Inequalities for a Certain Subclass of Analytic and Bi-Univalent Functions , Axioms2022,11,147. https://doi.org//10.3390/axiom 11040147.
[36] F.H. Jackson. On q−functions and a certain difference operator, Trans. R. Soc. Ediah, 1908,46, 253-281.
[37] F.H. Jackson. On q−definte integrals, Q.J. Pure Appl. Math.1910, 41, 193-203.
[38] K.A. Karthikeyan, G. Murugusundaramoorthy, S.D. Purohit and D.I. Suthar. Certain class of Analytic Functions with respect to the Symmetric Points defined by Q−Calculus ,Journal of Mathematics, volume 2021,Article 105298848, 9 pages.
[39] F.R. Keogh, E.P. Merkes. A coefficient inequality for certain classes of analytic functions , Proceedings of the American Mathematical Society, Vol, 20, 969, 8 - 12.
[40] S. Kumar, V. Ravichandran. Subordination for functions with positive real part, Complex Anal. Open Theory 12 (2018), no.5, 1179 - 1191.
[41] M. Lewin. On a Coefficient Problem for Bi-Univalent Functions ,Proc. Math. Soc. 1967, 18, 63 -68.
[42] A.Y. Lashin. On certain subclasses of analytic and bi-univalent functions, Journal of the Egyptian Mathematical Society 24 (2016), no. 2, 220 - 225.
[43] A. M. Lashin, A. Badghaish and B. Algethami. Certain subclasses of analytic functions and bi-univalent functions defined on the unite disc, Research Square, 2023, 1 - 12. Doi:https:https://doi.org/10.21203/rs.3.rs-2800195/v1
[44] K. Lo˙ wner. UntersuchungenU¨ ber schlichte Konforme Abbildungen des Einheitskreises, I, Mathematische Annalen,V ol. 89, no. 1-2, (1923), 103 -121.
[45] Mamoun Harayzeh Al-Abbadi, Maslina Darus. The Fekete-Szego Theorem for certain class of analytic functions, General Mathematics Vol. 19, No. 3 (2011), 41 -51.
[46] B.S. Mehrok, G. Singh. Fekete-Szego¨ coefficient functional for Certain subclasses of close-to-star functions , Hindawi Publishing Corporation, Germany, Volume 2013, Article ID 642142, 6 pages.
[47] G. Murugusundaramoorthy, N. Magesh, V. Prameela. Coefficient Bounds for Certain Subclasses of Bi-univalent Function , Abstr. Appl. (2013), 2013, 573017, 1-3.
[48] C. Pommerenke. Univalent Functions , Vandenhoeck and Rupercht: Go˙ttingen, Germany,1975.
[49] B.Seker, V. Mehmetogia. Coefficient bounds for bi-univalent functions , New Trends in Mathematical Sciences, 4(2016), 197 -203.
[50] H.M. Srivastava, A.K. Mistra, P. Gochhayat. Certain Subclasses of Analytic and Bi-Univalent Functions , Appl. Math. Lett. 2010,23,1188 - 1192.
[51] L.A. Wani, A. Swaminathan. Starlike and Convex functions associated with a Nephroid domain having CUSPS on the real axis, arxiv: 1912.05767v1[math.V] 12 Dec 2019, pp 1-23.
[52] T. Yavuz, S. Altinkaya. Notes on some classes of spiralike functions associated with the q−integral operator ,Hacet.J.Math.Stat. Volume 53 (1) 2024, 53 - 61.
[53] F. Yousef, S. Alroud, M. IIIafe. New Subclasses of Analytic and Bi-univalent Functions Endowed with Coefficient Estimate Problems , Anal. Math. Phys. 2021, 11: 58, 1-12.
[54] F. Yousef, T. Al-Hawary, G. Murugusundarmoorthy. Fekete-Szego¨ Functional Problems for some Subclasses of Bi-univalent Functions Defined by Frasin Differential Operator,Afr. Mat. 2019, 30, 495 - 503.
[2] E. A. Adegani, S. Bulut and A. Zireh. Cofficient estimates for a subclass of analytivs bi-univalent functions, Bull. Korean Math. Soc. 55 (2018); No 2, 405 - 413. https://doi. org/10.4134/BKMS.6170061.
[3] F. I. Akinwale and J.O. Hamzat Coefficient bounds for certain generalized class of analytic functions involving Bazil;evic type function and the Sigmoid function, J. Nig. Math. Phy. Vol. 62, Dec.,(2021), Issue, 1-8.
[4] A. Amourah, B.A. Frasin, T. Abdeijawad. Fekete-Szego¨ Inequality for Analytic and Bi-univalent Functions Subordinate to Gegenbauer Polynomials , J. Funct. Spaces 2021, 2021, 5574673, 1-7.
[5] A. Amourah, B.A. Frasin, M. Ahmad, F. Yousef. Exploiting the Pascal Distribution Series and Gegenbauer Polynomials to Construct and Study a New Subclass of Analytic Bi-Univalent Functions , Symmetry 2022, 14, 147, 1-8.
[6] A. Arai, V. Gupta, and R. P. Agarwal. Applications of q−Calculus in Operator Theory , Springer, New York. NY, USA, (2013), 1 - 13.
[7] O.P. Ahuja, S. Kumar, Ravichandran. Applications of first order differetial subordination for functions with positive real part , Stud. Univ. Babes-Bolyai Math. 63 (2018), no.3, 303 -311.
[8] R.M. Ali, S.K. Lee, V. Ravichandran, and S. Subramaniam. Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions , Appl. Math. Latt.25 (2012), no. 3, 344 351.
[9] A. Akgul, T. G. Shara. u, v- Lucas polynomial coefficient relations of the bi-univalent function class , Commun. Fac. Sci. Univ. Ank. Ser. AI MATH. stat. Volume 71, Number 4, (2022) pages 1121 - 1134. Doi:10.31801/cfsa.smas.1086809.
[10] S. Altinkaya, S. Yal¸cin. Coefficient bounds for a subclass of bi-univalent functions ,6(2015), 180 -185.
[11] S. Altinkaya, S. Yal¸cin. Estimates on Coefficients of a General Subclass of Bi-univalent Functions Associated with Symetric q− Derivative Operator by Means of the Chebyshev Polynomials , Asia Pac. J. Math. (2017), 8, 90 - 99.
[12] L. Bieberbach. Uber koeffizlenten Derjenigen Potenzreihen, Welche Eine Schlichtz Des Einheitskreises Vermitteln , Vol. 38, Sitzungsberichte der PreuBischen Akademie der Wissenschaften zu Berlin, 1916.
[13] S. Bulut. Coefficient Estimates for a Class of Analytic and Bi-univalent Functions ,Novi. Sal.J.Math. 2013, 2013,43, 59-65.
[14] R. Bucar, D. Breaz. On a new class of analytic functions with respect to symmetric points involving the q−derivative operator , In Journal of Physics: Conference Series, IOP Publishing,(2019), 1-8.
[15] T. Bulboara, P. Goswami. On some classes of bi-univalent functions , Studia Universitatis Babas-Bolyai Mathematica, 2(2015), 7 - 13.
[16] P. N. Chidera. New subclasses of the class of close - to - convex functions , Proceedings of the American Mathematical Society, ol. 62, no. 1, (1977), 37 -43.
[17] N.E. Cho, V. Kumar, S.S. Kumar, V. Ravichandran. Radius problems for starlike functions associated with sine functions , Bull. Iran, Math. Soc., 45 (2019), 213 - 232.
[18] L. De Branges. A proof of the Bieberbach conjecture , Acta.Math.154,(1985),137-152. http://dx.doi.org/10.1007/BF02392821
[19] K. Dhanalakshmi, S. Poongothai. Third Hankel determinant for a subclass of analytic univalent related to modified Nephroid domain ,International Journal of Mechanical Engineering,ol. 7, N0.4 April, 2022, 1-5. ISSN: 0974-5823.
[20] P.I. Duren, Univalent functions. Grundlehen der Mathematischan Wissenschaften 259: Springer: New York, Ny, USA; Berlin/Heidelberg, Germany;Tokyo,Japan, (1983).
[21] A.H. El-Qadeem, M.A. Mamon, I. S. Elshazly. Application of Einstein function on bi-univalent functions defined on the unit disc, Symmetry, 14, 4(2022),758, 1-11.
[22] Ebrahim Analouei Adegani, Serap Bulut, Ahmad Zireh. Coefficient estimates for a subclass of analytic bi-univalent functions , Bull. Korean Math. Soc. 55 (2018), No. 2, 405 - 413.
[23] B.A. Frasin, M.K. Aouf. New subclasses of Bi-Univalent Functions, Appl. Math. lett (2011), 24, 1569 - 1573.
[24] O. Fagbemiro, J.O. Hamzat, M. T. Raji. A certain class of (j, k)− symmetric function involving sigmoid function defined by using subordination principle ,
Journal of the Nigerian Association of Mathematical Physics(2024), 39 - 46.
[25] O. Fagbemiro, M.T. Raji, J.O. Hamzat, A.T. Oladipo and B.I. Olajuwon. Coefficient bounds for subclass of sigmoid functions involving subordination principle defined by Sa˘la˘gean Differential Operator, Covenant Journal of Physical and Life Sciences, Vol. 13, No. 1, June 2025, 1-16.
[26] M. Fekete, G. Szego¨. Eine Beemer Kung uber ungerade schlichte Functionen, Journal of the London Mathematical Sociebty, Volume 8, 85 89, 1933.
[27] A.W. Goodman. Univalent functions. Vol. 1, Mariner Publishing Co., Inc., Tampa, FL, 1983.
[28] S.P. Goyal and R. Kumar. Coefficient estimates and quasi-subordination properties associated with certain subclasses of analytic and bi-univalent functions , Mathematica Slovaca 65 (2015) No. 3, 533-544.
[29] Huo Tang, Gangadharan Murugusundaramoorthy, Shu-Hai Li, Li-Na Ma. Fekete-Szego¨ and Hankel inequalities for certain class of analytic functions related to the sine function, AIS Mathematics, 7(4) (2022): 6365 - 6380.
[30] J.O. Hamzat, M. O. Oluwayemi, A. A. Lupas and A. K. Wanas. Bi-Univalent Problems Involving Generalized ultiplier Transform with Respect to Symmetric and Conjugate Points , Fractal Fract., 2022, 6 , 483, 1 - 11. https://doi.org/10.3390/fractalfract6090483
[31] J.O. Hamzat, M. O. Oluwayemi, A. T. Oladipo and A. A. Lupas. On alpha-Pseudo Spiralike Function Associated with Exponential Pareto Distribution (EPD) and Libera Integral Operator, Mathematics, 2024, 1-10. https:/doi.org/10.3390/math1010000
[32] J.O. Hamzat, G. Murugusundaramoorthy and M. O. Oluwayemi. Brief Study on a New Family of Analytic Functions, Sahand Commun. Math. Anal. vol. 21, no. 4, 109-122. Doi:1022130/scma.2024.2011065.1457
[33] J.O. Hamzat. Estimate of Second and Third Hankel Determinants for Bazilevic Function of Order Gamma, Unilag Journal of Mathematics and Applications, vol. 3, 2023, 102-112.
[34] J.O. Hamzat. Some Properties of a new subclass of m-fold symmetric bi-Bazilevic Functions Associated with Modified Sigmoid Function, Tbilisi Math. J. 14(1), 2021, 107-118. Doi:10.32513/tmj/1932200819
[35] M.Illafe, A. Amourah and H.Mohd. Inequalities for a Certain Subclass of Analytic and Bi-Univalent Functions , Axioms2022,11,147. https://doi.org//10.3390/axiom 11040147.
[36] F.H. Jackson. On q−functions and a certain difference operator, Trans. R. Soc. Ediah, 1908,46, 253-281.
[37] F.H. Jackson. On q−definte integrals, Q.J. Pure Appl. Math.1910, 41, 193-203.
[38] K.A. Karthikeyan, G. Murugusundaramoorthy, S.D. Purohit and D.I. Suthar. Certain class of Analytic Functions with respect to the Symmetric Points defined by Q−Calculus ,Journal of Mathematics, volume 2021,Article 105298848, 9 pages.
[39] F.R. Keogh, E.P. Merkes. A coefficient inequality for certain classes of analytic functions , Proceedings of the American Mathematical Society, Vol, 20, 969, 8 - 12.
[40] S. Kumar, V. Ravichandran. Subordination for functions with positive real part, Complex Anal. Open Theory 12 (2018), no.5, 1179 - 1191.
[41] M. Lewin. On a Coefficient Problem for Bi-Univalent Functions ,Proc. Math. Soc. 1967, 18, 63 -68.
[42] A.Y. Lashin. On certain subclasses of analytic and bi-univalent functions, Journal of the Egyptian Mathematical Society 24 (2016), no. 2, 220 - 225.
[43] A. M. Lashin, A. Badghaish and B. Algethami. Certain subclasses of analytic functions and bi-univalent functions defined on the unite disc, Research Square, 2023, 1 - 12. Doi:https:https://doi.org/10.21203/rs.3.rs-2800195/v1
[44] K. Lo˙ wner. UntersuchungenU¨ ber schlichte Konforme Abbildungen des Einheitskreises, I, Mathematische Annalen,V ol. 89, no. 1-2, (1923), 103 -121.
[45] Mamoun Harayzeh Al-Abbadi, Maslina Darus. The Fekete-Szego Theorem for certain class of analytic functions, General Mathematics Vol. 19, No. 3 (2011), 41 -51.
[46] B.S. Mehrok, G. Singh. Fekete-Szego¨ coefficient functional for Certain subclasses of close-to-star functions , Hindawi Publishing Corporation, Germany, Volume 2013, Article ID 642142, 6 pages.
[47] G. Murugusundaramoorthy, N. Magesh, V. Prameela. Coefficient Bounds for Certain Subclasses of Bi-univalent Function , Abstr. Appl. (2013), 2013, 573017, 1-3.
[48] C. Pommerenke. Univalent Functions , Vandenhoeck and Rupercht: Go˙ttingen, Germany,1975.
[49] B.Seker, V. Mehmetogia. Coefficient bounds for bi-univalent functions , New Trends in Mathematical Sciences, 4(2016), 197 -203.
[50] H.M. Srivastava, A.K. Mistra, P. Gochhayat. Certain Subclasses of Analytic and Bi-Univalent Functions , Appl. Math. Lett. 2010,23,1188 - 1192.
[51] L.A. Wani, A. Swaminathan. Starlike and Convex functions associated with a Nephroid domain having CUSPS on the real axis, arxiv: 1912.05767v1[math.V] 12 Dec 2019, pp 1-23.
[52] T. Yavuz, S. Altinkaya. Notes on some classes of spiralike functions associated with the q−integral operator ,Hacet.J.Math.Stat. Volume 53 (1) 2024, 53 - 61.
[53] F. Yousef, S. Alroud, M. IIIafe. New Subclasses of Analytic and Bi-univalent Functions Endowed with Coefficient Estimate Problems , Anal. Math. Phys. 2021, 11: 58, 1-12.
[54] F. Yousef, T. Al-Hawary, G. Murugusundarmoorthy. Fekete-Szego¨ Functional Problems for some Subclasses of Bi-univalent Functions Defined by Frasin Differential Operator,Afr. Mat. 2019, 30, 495 - 503.
Published
2025-12-23
How to Cite
FAGBEMIRO , O., SANGONIYI , S. O., RAJI , M. T., & OLAJUWON , B. I. (2025). BI- UNIVALENT PROBLEM FOR CERTAIN GENERALIZED CLASS OF ANALYTIC FUNCTIONS INVOLVING Q-INTEGRAL OPERATOR ASSOCIATED WITH NEPHROID DOMAIN. Unilag Journal of Mathematics and Applications, 4(2), 106 - 125. Retrieved from https://lagjma.unilag.edu.ng/article/view/2754
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Copyright (c) 2024 Olalekan Fagbemiro, Sunday Oloruntoyin Sangoniyi , Musiliu Tayo Raji , Bakai Ishola Olajuwon
This work is licensed under a Creative Commons Attribution 4.0 International License.
