Research Article
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Year 2022, Volume: 71 Issue: 4, 1121 - 1135, 30.12.2022

Arzu Akgül , Timilehin Shaba

https://doi.org/10.31801/cfsuasmas.1086809
Cited By: 1

Abstract

References

  • Akgül, A., (P,Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class, Turk. J. Math., 43 (2019), 2170–2176. https://doi.org/10.3906/mat-1903-38
  • Akgül, A., On a family of bi-univalent functions related to the Fibonacci numbers, Mathematica Moravica, 26(1) (2022), 103–112. https://doi.org/10.5937/MatMor2201103A
  • Akgül, A., Sakar, F. M., A new characterization of (P,Q)-Lucas polynomial coefficients of the bi-univalent function class associated with q-analogue of Noor integral operator, Afrika Matematika, 33(3) (2022), 1-12.
  • Ali, R. M., Lee, S. K., Ravichandran V., Supramanian S., Coefficient estimates for biunivalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25(3) (2012), 344–351. https://doi.org/10.1016/j.aml.2011.09.012
  • Altınkaya, Ş., Yalçın, S., On the (p, q)-Lucas polynomial coefficient bounds of the biunivalent function class σ, Boletin de la Sociedad Matem´atica Mexicana, 25 (2019), 567-575. https://doi.org/10.1007/s40590-018-0212-z
  • Altınkaya, Ş., Yalçın, S., The (p, q)-Chebyshev polynomial bounds of a general bi-univalent function class, Boletin de la Sociedad Matem´atica Mexicana, 26 (2019), 341–348.
  • Al-Shbeil, I., Shaba, T. G., Cataş, A., Second hankel determinant for the subclass of biunivalent functions using q-Chebyshev polynomial and Hohlov operator, Fractal and Fractional, 6(4) (2022), 186. https://doi.org/10.3390/fractalfract6040186
  • Altinkaya, Ş., Yalçın, S., Some application of the (p, q)-Lucas polynomials to the bi-univalent function class Σ, Mathematical Sciences and Applications E-Notes, 8(1) (2020), 134–141. https://doi.org/10.36753/MATHENOT.650271
  • Babalola, K. O., On U-pseudo-starlike functions, J. Class. Anal., 3(2) (2013), 137-147.
  • Çağlar, M., Deniz, E., Srivastava, H. M., Second Hankel determinant for certain subclasses of bi-univalent functions, Turk. J. Math., 41 (2017), 694-706. https://doi.org/10.3906/mat-1602-25
  • Çağlar, M., Orhan H., Yagmur, N., Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27(7) (2013), 1165–1171. https://doi.org/10.2298/FIL1307165C
  • Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, 1983.
  • Filipponi, P., Horadam, A. F., Second derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Q, 31(3) (1993), 194–204.
  • Goswami, P., Alkahtani, B. S., Bulboaca T., Estimate for initial Maclaurin coefficients of certain subclasses of bi-univalent functions, arXiv:1503.04644v1 (2015).
  • Ibrahim, I. O., Shaba, T. G., Patil, A. B., On some subclasses of m-fold symmetricbi-univalent functions associated with the Sakaguchi type functions, Earthline Journal of Mathematical Sciences, 8(1) (2022), 1-15. https://doi.org/10.34198/ejms.8122.115
  • Khan, B., Liu, Z. G., Shaba, T. G., Araci, S., Khan, N., Khan, M. G., Application of q-derivative operator to the subclass of bi-univalent functions involving q-Chebyshev polynomial, Journal of Mathematics, (2022), Article ID: 8162182. https://doi.org/10.1155/2022/8162182
  • Lee, G. Y., Aşcı M., Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, J. Appl. Math., (2012), Article ID: 264842, 1-18. https://doi.org/10.1155/2012/264842
  • Lewin, M., On a coefficient problem for bi-univalent functions, Proc. Am. Math. Soc., 18(1) (1967), 63–68.
  • Lupas, A., A guide of Fibonacci and Lucas polynomials, Octagon Mathematics Magazine, 7 (1999), 2-12.
  • Murugusundaramoorthy, G., Yalçin, S., On λ-Pseudo bi-starlike functions related (p, q)-Lucas polynomial, Libertas Mathematica (new series), 39(2) (2019), 79-88.
  • Orhan, H., Arıkan H., Lucas polynomial coefficients inequalities of bi-univalent functions defined by the combination of both operators of Al-Oboudi and Ruscheweyh, Afr. Mat., 32(3-4) (2021), 589–598. https://doi.org/10.1007/s13370-020-00847-5
  • Orhan, H., Shaba, T. G., Cağlar, M., (P,Q)-Lucas polynomial coefficient relations of biunivalent functions defined by the combination of Opoola and Babalola differential operator, Afrika Matematika, 33(1), (2022), 1-13. https://doi.org/10.1007/s13370-021-00953-y
  • Özkoç, A., Porsuk, A., A note for the (p, q)-Fibonacci and Lucas quaternion polynomials, Konuralp J. Math., 5(2) (2017), 36-46.
  • Patil, A. B., Shaba, T. G., On sharp Chebyshev polynomial bounds for general subclass of bi-univalent functions, Applied Sciences, 23 (2021), 109–117.
  • Sakar, F. M., Estimate for initial Tschebyscheff polynomials coefficients on a certain subclass of bi-univalent functions defined by Sˇalˇagean differential operator, Acta Universitatis Apulensis, 54 (2018), 45-54. https://doi.org/10.17114/j.aua.2018.54.04
  • Shaba, T. G., Subclass of bi-univalent functions satisfying subordinate conditions defined by Frasin differential operator, Turkish Journal of Inequalities, 4(2) (2020), 50-58.
  • Shaba, T. G., Patil, A. B., Coefficient estimates for certain subclasses of m-fold symmetric bi-univalent functions associated with pseudo-starlike functions, Earthline Journal of Mathematical Sciences, 6(2) (2021), 2581-8147. https://doi.org/10.34198/ejms.6221.209223
  • Shaba, T. G., On some subclasses of bi-pseudo-starlike functions defined by Salagean differential operator, Asia Pac. J. Math., 8(6) (2021), 1–11. https://doi:10.28924/apjm/8-6
  • Shaba, T. G., Wanas, A. K., Coefficient bounds for a new family of bi-univalent functions associated with (U, V )−Lucas polynomials, International Journal of Nonlinear Analysis and Applications, 13(1), (2022), 615-626. http://dx.doi.org/10.22075/ijnaa.2021.23927.2639
  • Srivastava, H. M., Bulut, S., Cağlar, M., Yağmur, N., Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27 (2013), 831–842. https://doi.org/10.2298/FIL1305831S
  • Srivastava, H. M., Eker, S. S., Hamidi, S. G., Jahangiri, J. M., Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator, Bull. Iran. Math. Soc., 44 (2018), 149–157. https://doi.org/10.1007/s41980-018-0011-3
  • Srivastava, H. M., Khan, S., Ahmad, Q. Z, Khan, N., Hussain, S., The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain q-integral operator, Stud. Univ. Babe¸s- Bolyai Math., 63 (2018), 419–436.
  • Srivastava, H. M., Mishra, A. K., Das, M. K., The Fekete-Szego problem for a subclass of close-to-convex functions, Complex Variables Theory Appl., 44 (2001), 145–163. https://doi.org/10.1080/17476930108815351
  • Srivastava, H. M., Eker, S. S., Some applications of a subordination theorem for a class of analytic functions, Appl. Math. Lett., 21 (2008), 394–399. https://doi.org/10.1016/j.aml.2007.02.032
  • Srivastava, H. M., Mishra, A. K., Gochhayat, P., Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23(10) (2010), 1188–1192. https://doi.org/10.1016/j.aml.2010.05.009
  • Srivastava, H. M., Eker, S. S., Ali, R. M., Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29(8) (2015), 1839–1845. https://doi.org/10.2298/FIL1508839S
  • Srivastava, H. M., Magesh, N., Yamini, J., Initial coefficient estimates for bi-λ-convex and bi-μ-starlike functions connected with arithmetic and geometric means, Electron. J. Math. Anal. Appl., 2(2) (2014), 152–162.
  • Srivastava, H. M., Sakar, F. M., Güney, H. Ö., Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination, Filomat, 34 (2018), 1313–1322. https://doi.org/10.2298/FIL1804313S
  • Srivastava, H. M., Gaboury, S., Ghanim, F., Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat., 28 (2017), 693–706. https://doi.org/10.1007/s13370-016-0478-0
  • Srivastava, H. M., Altinkaya, S¸., Yalçın, S., Certain subclasses of bi-univalent functions associated with the Horadam polynomials, Iran. J. Sci. Technol. Trans. Sci., 43 (2019), 1873–1879. https://doi.org/10.1007/s40995-018-0647-0
  • Tang, H., Srivastava, H. M., Sivasubramanian, S., Gurusamy, P., The Fekete-Szegö functional problems for some subclasses of m-fold symmetric bi-univalent functions, J. Math. Inequal., 10(4) (2016), 1063–1092. https://doi.org/10.7153/jmi-10-85
  • Vellucci, P., Bersani, A. M., The class of Lucas-Lehmer polynomials, Rendiconti di Matematica, 37 (2016), 43-62 .
  • Wang, T., Zhang, W., Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Roum., 5(1) (2012), 95-103
  • Wanas, A. K., Application of (M,N)-Lucas polynomials for holomorphic and bi-univalent functions, Filomat, 39(10) (2020), 3361–3368. https://doi.org/10.2298/FIL2010361W
  • Wanas, A. K., Shaba, T. G., Horadam polynomials and their applications to certain family of bi-univalent functions defined by Wanas operator, General Mathematics, (2022), 103. https://doi.org/10.2478/gm-2021-0009
  • Yalçın, S., Muthunagai, K., Saravanan, G., A subclass with bi-univalent involving (p, q)-Lucas polynomials and its coefficient bounds, Boletin de la Sociedad Matematica Mexicana, 26 (2020), 1015-1022. https://doi.org/10.1007/s40590-020-00294-z

(U, V )-Lucas polynomial coefficient relations of the bi-univalent function class

Year 2022, Volume: 71 Issue: 4, 1121 - 1135, 30.12.2022

Arzu Akgül , Timilehin Shaba

https://doi.org/10.31801/cfsuasmas.1086809
Cited By: 1

Abstract

In geometric function theory, Lucas polynomials and other special polynomials have recently gained importance. In this study, we develop a new family of bi-univalent functions. Also we examined coefficient inequalities and Fekete-Szegö problem for this new family via these polynomials.

Keywords

(U, V)-Lucas polynomial bi-univalent analytic function subordination

References

  • Akgül, A., (P,Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class, Turk. J. Math., 43 (2019), 2170–2176. https://doi.org/10.3906/mat-1903-38
  • Akgül, A., On a family of bi-univalent functions related to the Fibonacci numbers, Mathematica Moravica, 26(1) (2022), 103–112. https://doi.org/10.5937/MatMor2201103A
  • Akgül, A., Sakar, F. M., A new characterization of (P,Q)-Lucas polynomial coefficients of the bi-univalent function class associated with q-analogue of Noor integral operator, Afrika Matematika, 33(3) (2022), 1-12.
  • Ali, R. M., Lee, S. K., Ravichandran V., Supramanian S., Coefficient estimates for biunivalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25(3) (2012), 344–351. https://doi.org/10.1016/j.aml.2011.09.012
  • Altınkaya, Ş., Yalçın, S., On the (p, q)-Lucas polynomial coefficient bounds of the biunivalent function class σ, Boletin de la Sociedad Matem´atica Mexicana, 25 (2019), 567-575. https://doi.org/10.1007/s40590-018-0212-z
  • Altınkaya, Ş., Yalçın, S., The (p, q)-Chebyshev polynomial bounds of a general bi-univalent function class, Boletin de la Sociedad Matem´atica Mexicana, 26 (2019), 341–348.
  • Al-Shbeil, I., Shaba, T. G., Cataş, A., Second hankel determinant for the subclass of biunivalent functions using q-Chebyshev polynomial and Hohlov operator, Fractal and Fractional, 6(4) (2022), 186. https://doi.org/10.3390/fractalfract6040186
  • Altinkaya, Ş., Yalçın, S., Some application of the (p, q)-Lucas polynomials to the bi-univalent function class Σ, Mathematical Sciences and Applications E-Notes, 8(1) (2020), 134–141. https://doi.org/10.36753/MATHENOT.650271
  • Babalola, K. O., On U-pseudo-starlike functions, J. Class. Anal., 3(2) (2013), 137-147.
  • Çağlar, M., Deniz, E., Srivastava, H. M., Second Hankel determinant for certain subclasses of bi-univalent functions, Turk. J. Math., 41 (2017), 694-706. https://doi.org/10.3906/mat-1602-25
  • Çağlar, M., Orhan H., Yagmur, N., Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27(7) (2013), 1165–1171. https://doi.org/10.2298/FIL1307165C
  • Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, 1983.
  • Filipponi, P., Horadam, A. F., Second derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Q, 31(3) (1993), 194–204.
  • Goswami, P., Alkahtani, B. S., Bulboaca T., Estimate for initial Maclaurin coefficients of certain subclasses of bi-univalent functions, arXiv:1503.04644v1 (2015).
  • Ibrahim, I. O., Shaba, T. G., Patil, A. B., On some subclasses of m-fold symmetricbi-univalent functions associated with the Sakaguchi type functions, Earthline Journal of Mathematical Sciences, 8(1) (2022), 1-15. https://doi.org/10.34198/ejms.8122.115
  • Khan, B., Liu, Z. G., Shaba, T. G., Araci, S., Khan, N., Khan, M. G., Application of q-derivative operator to the subclass of bi-univalent functions involving q-Chebyshev polynomial, Journal of Mathematics, (2022), Article ID: 8162182. https://doi.org/10.1155/2022/8162182
  • Lee, G. Y., Aşcı M., Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, J. Appl. Math., (2012), Article ID: 264842, 1-18. https://doi.org/10.1155/2012/264842
  • Lewin, M., On a coefficient problem for bi-univalent functions, Proc. Am. Math. Soc., 18(1) (1967), 63–68.
  • Lupas, A., A guide of Fibonacci and Lucas polynomials, Octagon Mathematics Magazine, 7 (1999), 2-12.
  • Murugusundaramoorthy, G., Yalçin, S., On λ-Pseudo bi-starlike functions related (p, q)-Lucas polynomial, Libertas Mathematica (new series), 39(2) (2019), 79-88.
  • Orhan, H., Arıkan H., Lucas polynomial coefficients inequalities of bi-univalent functions defined by the combination of both operators of Al-Oboudi and Ruscheweyh, Afr. Mat., 32(3-4) (2021), 589–598. https://doi.org/10.1007/s13370-020-00847-5
  • Orhan, H., Shaba, T. G., Cağlar, M., (P,Q)-Lucas polynomial coefficient relations of biunivalent functions defined by the combination of Opoola and Babalola differential operator, Afrika Matematika, 33(1), (2022), 1-13. https://doi.org/10.1007/s13370-021-00953-y
  • Özkoç, A., Porsuk, A., A note for the (p, q)-Fibonacci and Lucas quaternion polynomials, Konuralp J. Math., 5(2) (2017), 36-46.
  • Patil, A. B., Shaba, T. G., On sharp Chebyshev polynomial bounds for general subclass of bi-univalent functions, Applied Sciences, 23 (2021), 109–117.
  • Sakar, F. M., Estimate for initial Tschebyscheff polynomials coefficients on a certain subclass of bi-univalent functions defined by Sˇalˇagean differential operator, Acta Universitatis Apulensis, 54 (2018), 45-54. https://doi.org/10.17114/j.aua.2018.54.04
  • Shaba, T. G., Subclass of bi-univalent functions satisfying subordinate conditions defined by Frasin differential operator, Turkish Journal of Inequalities, 4(2) (2020), 50-58.
  • Shaba, T. G., Patil, A. B., Coefficient estimates for certain subclasses of m-fold symmetric bi-univalent functions associated with pseudo-starlike functions, Earthline Journal of Mathematical Sciences, 6(2) (2021), 2581-8147. https://doi.org/10.34198/ejms.6221.209223
  • Shaba, T. G., On some subclasses of bi-pseudo-starlike functions defined by Salagean differential operator, Asia Pac. J. Math., 8(6) (2021), 1–11. https://doi:10.28924/apjm/8-6
  • Shaba, T. G., Wanas, A. K., Coefficient bounds for a new family of bi-univalent functions associated with (U, V )−Lucas polynomials, International Journal of Nonlinear Analysis and Applications, 13(1), (2022), 615-626. http://dx.doi.org/10.22075/ijnaa.2021.23927.2639
  • Srivastava, H. M., Bulut, S., Cağlar, M., Yağmur, N., Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27 (2013), 831–842. https://doi.org/10.2298/FIL1305831S
  • Srivastava, H. M., Eker, S. S., Hamidi, S. G., Jahangiri, J. M., Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator, Bull. Iran. Math. Soc., 44 (2018), 149–157. https://doi.org/10.1007/s41980-018-0011-3
  • Srivastava, H. M., Khan, S., Ahmad, Q. Z, Khan, N., Hussain, S., The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain q-integral operator, Stud. Univ. Babe¸s- Bolyai Math., 63 (2018), 419–436.
  • Srivastava, H. M., Mishra, A. K., Das, M. K., The Fekete-Szego problem for a subclass of close-to-convex functions, Complex Variables Theory Appl., 44 (2001), 145–163. https://doi.org/10.1080/17476930108815351
  • Srivastava, H. M., Eker, S. S., Some applications of a subordination theorem for a class of analytic functions, Appl. Math. Lett., 21 (2008), 394–399. https://doi.org/10.1016/j.aml.2007.02.032
  • Srivastava, H. M., Mishra, A. K., Gochhayat, P., Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23(10) (2010), 1188–1192. https://doi.org/10.1016/j.aml.2010.05.009
  • Srivastava, H. M., Eker, S. S., Ali, R. M., Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29(8) (2015), 1839–1845. https://doi.org/10.2298/FIL1508839S
  • Srivastava, H. M., Magesh, N., Yamini, J., Initial coefficient estimates for bi-λ-convex and bi-μ-starlike functions connected with arithmetic and geometric means, Electron. J. Math. Anal. Appl., 2(2) (2014), 152–162.
  • Srivastava, H. M., Sakar, F. M., Güney, H. Ö., Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination, Filomat, 34 (2018), 1313–1322. https://doi.org/10.2298/FIL1804313S
  • Srivastava, H. M., Gaboury, S., Ghanim, F., Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat., 28 (2017), 693–706. https://doi.org/10.1007/s13370-016-0478-0
  • Srivastava, H. M., Altinkaya, S¸., Yalçın, S., Certain subclasses of bi-univalent functions associated with the Horadam polynomials, Iran. J. Sci. Technol. Trans. Sci., 43 (2019), 1873–1879. https://doi.org/10.1007/s40995-018-0647-0
  • Tang, H., Srivastava, H. M., Sivasubramanian, S., Gurusamy, P., The Fekete-Szegö functional problems for some subclasses of m-fold symmetric bi-univalent functions, J. Math. Inequal., 10(4) (2016), 1063–1092. https://doi.org/10.7153/jmi-10-85
  • Vellucci, P., Bersani, A. M., The class of Lucas-Lehmer polynomials, Rendiconti di Matematica, 37 (2016), 43-62 .
  • Wang, T., Zhang, W., Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Roum., 5(1) (2012), 95-103
  • Wanas, A. K., Application of (M,N)-Lucas polynomials for holomorphic and bi-univalent functions, Filomat, 39(10) (2020), 3361–3368. https://doi.org/10.2298/FIL2010361W
  • Wanas, A. K., Shaba, T. G., Horadam polynomials and their applications to certain family of bi-univalent functions defined by Wanas operator, General Mathematics, (2022), 103. https://doi.org/10.2478/gm-2021-0009
  • Yalçın, S., Muthunagai, K., Saravanan, G., A subclass with bi-univalent involving (p, q)-Lucas polynomials and its coefficient bounds, Boletin de la Sociedad Matematica Mexicana, 26 (2020), 1015-1022. https://doi.org/10.1007/s40590-020-00294-z
There are 46 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Arzu Akgül 0000-0001-7934-0339

Timilehin Shaba 0000-0001-5881-9260

Publication Date December 30, 2022
Submission Date March 12, 2022
Acceptance Date June 8, 2022
Published in Issue Year 2022 Volume: 71 Issue: 4

Cite

APA Akgül, A., & Shaba, T. (2022). (U, V )-Lucas polynomial coefficient relations of the bi-univalent function class. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4), 1121-1135. https://doi.org/10.31801/cfsuasmas.1086809
AMA Akgül A, Shaba T. (U, V )-Lucas polynomial coefficient relations of the bi-univalent function class. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2022;71(4):1121-1135. doi:10.31801/cfsuasmas.1086809
Chicago Akgül, Arzu, and Timilehin Shaba. “(U, V )-Lucas Polynomial Coefficient Relations of the Bi-Univalent Function Class”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 4 (December 2022): 1121-35. https://doi.org/10.31801/cfsuasmas.1086809.
EndNote Akgül A, Shaba T (December 1, 2022) (U, V )-Lucas polynomial coefficient relations of the bi-univalent function class. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 4 1121–1135.
IEEE A. Akgül and T. Shaba, “(U, V )-Lucas polynomial coefficient relations of the bi-univalent function class”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 4, pp. 1121–1135, 2022, doi: 10.31801/cfsuasmas.1086809.
ISNAD Akgül, Arzu - Shaba, Timilehin. “(U, V )-Lucas Polynomial Coefficient Relations of the Bi-Univalent Function Class”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/4 (December 2022), 1121-1135. https://doi.org/10.31801/cfsuasmas.1086809.
JAMA Akgül A, Shaba T. (U, V )-Lucas polynomial coefficient relations of the bi-univalent function class. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:1121–1135.
MLA Akgül, Arzu and Timilehin Shaba. “(U, V )-Lucas Polynomial Coefficient Relations of the Bi-Univalent Function Class”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 4, 2022, pp. 1121-35, doi:10.31801/cfsuasmas.1086809.
Vancouver Akgül A, Shaba T. (U, V )-Lucas polynomial coefficient relations of the bi-univalent function class. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(4):1121-35.

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