Research Article
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Year 2022, Volume: 10 Issue: 1, 59 - 68, 15.04.2022

Timilehin Shaba , Abbas Kareem

Abstract

References

  • [1] R. M. Ali, S. K. Lee, V. Ravichandran, S. Subramanian, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions. Appl. Math. Lett., 2012, 25: 344–351.
  • [2] S. Altinkaya, S. Yalcin, Coefficient Bounds for Certain Subclasses of m-Fold Symmetric Bi-Univalent Functions, Journal of Mathematics, Art. ID 241683, (2015), 1-5.
  • [3] S. Altinkaya, S. Yalcin, On Some Subclasses of m-Fold Symmetric Bi-Univalent Functions, Commun. Fac. Sci. Univ. Ank., Series A1, 67(1), (2018), 29-36.
  • [4] A. A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat, 30 ( 7) (2016), 2075-2081.
  • [5] K. O. Babalola, On λ-pseudo-starlike functions, J. Class. Anal., 3(2), (2013), 137–147.
  • [6] D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Universitatis Babes¸-Bolyai, Mathematica, 31(2) (1986) 70-7.
  • [7] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenchaften, Springer, New York, NY, USA, 1983.
  • [8] S. S. Eker, Coefficient Bounds for Subclasses of m-Fold Symmetric Bi-Univalent Functions, Turk. J. Math., 40, (2016), 641-646.
  • [9] B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24, (2004), 1529-1573.
  • [10] V. B. Girgaonkar, S. B. Joshi, P. P. Yadav, Certain special subclasses of analytic function associated with bi-univalent functions. Palestine Journal of Mathematics, 2017, 6(2): 617-623.
  • [11] D. Guo, H. Tang, E. Ao, L. Xiong, Coeffient estimates for a class of m-fold symmetric bi-univalent function defined by subordination. Communication in Mathematical Research, 2019, 35(1): 57–64.
  • [12] W. Koepf, Coefficients of Symmetric Functions of Bounded Boundary Rotations, Proc. Amer. Math. Soc., 105, (1989), 324-329.
  • [13] M. Lewin, On a coefficients problem of bi-univalent functions, Proc. Am. Math. Soc., 18 (1967), 63-68.
  • [14] G. M. Mittag-Leffler, Sur lanouvelle function, Comptes Rendus de l’Academie des Sciences Paris, 137 (1903), 554-558.
  • [15] G. M. Mittag-Leffler, Sur la representation analytique d’une function monogene (cinquieme note), Acta Mathematica, 29 (1905), 101-181.
  • [16] F. Muge Sakar, M. O. G ¨ uney, Coefficient estimates for certain subclasses of ¨ m-mold symmetric bi-univalent functions defined by the q-derivative operator, Konuralp Journal of Mathematics, 6(2) (2018), 279-285.
  • [17] C. Pommerenke, On the Coefficients of Close-to-Convex Functions, Michigan Math. J., 9, (1962), 259-269.
  • [18] H. M. Srivastava, S. Sivasubramanian, R. Sivakumar, Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions. Tbilisi Math. J., 2014, 7(2): 1–10.
  • [19] H.M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23(10), (2010), 1188-1192.
  • [20] H. M. Srivastava, S. Gaboury, F. Ghanim, Coefficient Estimates for Some Subclasses of m-Fold Symmetric Bi-Univalent Functions, Acta Universitatis Apulensis, 41, (2015), 153-164.
  • [21] H. M. Srivastava, S. Gaboury, F. Ghanim, Initial Coefficient Estimates for Some Subclasses of m-Fold Symmetric Bi-Univalent Functions, Acta Mathematica Scientia, 36B(3), (2016), 863-871.
  • [22] H. M. Srivastava, A. Motamednezhad and E. A. Adegani, Faber polynomial coefficient estimates for bi-univalent functions defined by using differential subordination and a certain fractional derivative operator, Mathematics, 8 (2020), Article ID 172, 1-12.
  • [23] H. M. Srivastava, A. K. Wanas, Initial Maclaurin coefficient bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination, Kyungpook Math. J., 59, (2019), 493-503.
  • [24] H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernal, Applied Mathematics and Computation, 211 (2009), 198-210.
  • [25] T. G. Shaba, A. B. Patil, 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.
  • [26] T. G. Shaba, Subclass of bi-univalent functions satisfying subordinate conditions defined by Frasin differential operator, Turkish Journal of Inequalities, 4 (2) (2020), 50–58.
  • [27] T. G. Shaba, On some subclasses of bi-pseudo-starlike functions defined by Salagean differential operator, Asia Pac. J. Math., 8 (6) (2021), 1–11; Available online at https://doi:10.28924/apjm/8-6
  • [28] T. G. Shaba, On some new subclass of bi-univalent functions associated with the Opoola differential operator, Open J. Math. Anal., 4 (2), (2020), 74–79. [29] H. Tang, H. M. Srivastava, S. Sivasubramanian, P. Gurusamy, The Fekete-Szego Functional Problems for Some Subclasses of m-Fold Symmetric
  • Bi-Univalent Functions, J. Math. Ineq., 10, (2016), 1063-1092. [30] A. k. Wanas, Coefficient Estimates for Bazilevic Functions of Bi-Prestarlike Functions, Miskolc Mathematical Notes, 21, (2020), 1031-1040. ˇ
  • [31] A.k. Wanas, Applications of (M,N)-Lucas polynomials for holomorphic and bi-univalent functions, Filomat, 34, (2020), 3361-3368.
  • [32] A. K. Wanas, A. L. Alina, Applications of Horadam polynomials on Bazilevic bi-univalent function satisfying subordinate conditions, J. Phys. : Conf. ˇ Ser., 1294 (2019), 1-6.
  • [33] A. K. Wanas and A. H. Majeed, Certain new subclasses of analytic and m-fold symmetric bi-univalent functions, Applied Mathematics E-Notes, 18, (2018), 178-188.
  • [34] A. k. Wanas, S. Yalc¸in, Horadam polynomials and their applications to new family of bi-univalent functions with respect to symmetric conjugate points, Proyecciones, 40, (2021), 107-116.
  • [35] A. K. Wanas, H. Tang, Initial coefficient estimates for a classes of m-fold symmetric bi-univalent functions involving Mittag-Leffler function. Mathematica Moravica, 24 (2), (2020), 51-61.

Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function

Year 2022, Volume: 10 Issue: 1, 59 - 68, 15.04.2022

Timilehin Shaba , Abbas Kareem

Abstract

In the present investigation, we introduce and study a certain subclasses $\mathcal{H}_{\Sigma_{m}}(\eta,\gamma,\lambda,\delta,\tau,\phi,\upsilon;\alpha)$ and $\mathcal{H}_{\Sigma_{m}}^{*}(\eta,\gamma,\lambda,\delta,\tau,\phi,\upsilon;\beta)$ of analytic and m-fold symmetric bi-univalent functions involving $\phi$-pseudo-starlike functions associated with Mittag-Leffler Function. We establish upper bounds for the second and third Taylor-Maclaurin coefficients for functions in each of these subclasses. Furthermore, we indicate several certain special cases for our results.

Keywords

Analytic function m-fold symmetric bi-univalent function Coefficient bounds Mittag-Leffler function pseudo-starlike function

References

  • [1] R. M. Ali, S. K. Lee, V. Ravichandran, S. Subramanian, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions. Appl. Math. Lett., 2012, 25: 344–351.
  • [2] S. Altinkaya, S. Yalcin, Coefficient Bounds for Certain Subclasses of m-Fold Symmetric Bi-Univalent Functions, Journal of Mathematics, Art. ID 241683, (2015), 1-5.
  • [3] S. Altinkaya, S. Yalcin, On Some Subclasses of m-Fold Symmetric Bi-Univalent Functions, Commun. Fac. Sci. Univ. Ank., Series A1, 67(1), (2018), 29-36.
  • [4] A. A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat, 30 ( 7) (2016), 2075-2081.
  • [5] K. O. Babalola, On λ-pseudo-starlike functions, J. Class. Anal., 3(2), (2013), 137–147.
  • [6] D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Universitatis Babes¸-Bolyai, Mathematica, 31(2) (1986) 70-7.
  • [7] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenchaften, Springer, New York, NY, USA, 1983.
  • [8] S. S. Eker, Coefficient Bounds for Subclasses of m-Fold Symmetric Bi-Univalent Functions, Turk. J. Math., 40, (2016), 641-646.
  • [9] B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24, (2004), 1529-1573.
  • [10] V. B. Girgaonkar, S. B. Joshi, P. P. Yadav, Certain special subclasses of analytic function associated with bi-univalent functions. Palestine Journal of Mathematics, 2017, 6(2): 617-623.
  • [11] D. Guo, H. Tang, E. Ao, L. Xiong, Coeffient estimates for a class of m-fold symmetric bi-univalent function defined by subordination. Communication in Mathematical Research, 2019, 35(1): 57–64.
  • [12] W. Koepf, Coefficients of Symmetric Functions of Bounded Boundary Rotations, Proc. Amer. Math. Soc., 105, (1989), 324-329.
  • [13] M. Lewin, On a coefficients problem of bi-univalent functions, Proc. Am. Math. Soc., 18 (1967), 63-68.
  • [14] G. M. Mittag-Leffler, Sur lanouvelle function, Comptes Rendus de l’Academie des Sciences Paris, 137 (1903), 554-558.
  • [15] G. M. Mittag-Leffler, Sur la representation analytique d’une function monogene (cinquieme note), Acta Mathematica, 29 (1905), 101-181.
  • [16] F. Muge Sakar, M. O. G ¨ uney, Coefficient estimates for certain subclasses of ¨ m-mold symmetric bi-univalent functions defined by the q-derivative operator, Konuralp Journal of Mathematics, 6(2) (2018), 279-285.
  • [17] C. Pommerenke, On the Coefficients of Close-to-Convex Functions, Michigan Math. J., 9, (1962), 259-269.
  • [18] H. M. Srivastava, S. Sivasubramanian, R. Sivakumar, Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions. Tbilisi Math. J., 2014, 7(2): 1–10.
  • [19] H.M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23(10), (2010), 1188-1192.
  • [20] H. M. Srivastava, S. Gaboury, F. Ghanim, Coefficient Estimates for Some Subclasses of m-Fold Symmetric Bi-Univalent Functions, Acta Universitatis Apulensis, 41, (2015), 153-164.
  • [21] H. M. Srivastava, S. Gaboury, F. Ghanim, Initial Coefficient Estimates for Some Subclasses of m-Fold Symmetric Bi-Univalent Functions, Acta Mathematica Scientia, 36B(3), (2016), 863-871.
  • [22] H. M. Srivastava, A. Motamednezhad and E. A. Adegani, Faber polynomial coefficient estimates for bi-univalent functions defined by using differential subordination and a certain fractional derivative operator, Mathematics, 8 (2020), Article ID 172, 1-12.
  • [23] H. M. Srivastava, A. K. Wanas, Initial Maclaurin coefficient bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination, Kyungpook Math. J., 59, (2019), 493-503.
  • [24] H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernal, Applied Mathematics and Computation, 211 (2009), 198-210.
  • [25] T. G. Shaba, A. B. Patil, 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.
  • [26] T. G. Shaba, Subclass of bi-univalent functions satisfying subordinate conditions defined by Frasin differential operator, Turkish Journal of Inequalities, 4 (2) (2020), 50–58.
  • [27] T. G. Shaba, On some subclasses of bi-pseudo-starlike functions defined by Salagean differential operator, Asia Pac. J. Math., 8 (6) (2021), 1–11; Available online at https://doi:10.28924/apjm/8-6
  • [28] T. G. Shaba, On some new subclass of bi-univalent functions associated with the Opoola differential operator, Open J. Math. Anal., 4 (2), (2020), 74–79. [29] H. Tang, H. M. Srivastava, S. Sivasubramanian, P. Gurusamy, The Fekete-Szego Functional Problems for Some Subclasses of m-Fold Symmetric
  • Bi-Univalent Functions, J. Math. Ineq., 10, (2016), 1063-1092. [30] A. k. Wanas, Coefficient Estimates for Bazilevic Functions of Bi-Prestarlike Functions, Miskolc Mathematical Notes, 21, (2020), 1031-1040. ˇ
  • [31] A.k. Wanas, Applications of (M,N)-Lucas polynomials for holomorphic and bi-univalent functions, Filomat, 34, (2020), 3361-3368.
  • [32] A. K. Wanas, A. L. Alina, Applications of Horadam polynomials on Bazilevic bi-univalent function satisfying subordinate conditions, J. Phys. : Conf. ˇ Ser., 1294 (2019), 1-6.
  • [33] A. K. Wanas and A. H. Majeed, Certain new subclasses of analytic and m-fold symmetric bi-univalent functions, Applied Mathematics E-Notes, 18, (2018), 178-188.
  • [34] A. k. Wanas, S. Yalc¸in, Horadam polynomials and their applications to new family of bi-univalent functions with respect to symmetric conjugate points, Proyecciones, 40, (2021), 107-116.
  • [35] A. K. Wanas, H. Tang, Initial coefficient estimates for a classes of m-fold symmetric bi-univalent functions involving Mittag-Leffler function. Mathematica Moravica, 24 (2), (2020), 51-61.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Timilehin Shaba 0000-0001-5881-9260

Abbas Kareem 0000-0001-5838-7365

Submission Date June 6, 2021
Acceptance Date March 23, 2022
Publication Date April 15, 2022
Published in Issue Year 2022 Volume: 10 Issue: 1

Cite

APA Shaba, T., & Kareem, A. (2022). Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function. Konuralp Journal of Mathematics, 10(1), 59-68.
AMA Shaba T, Kareem A. Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function. Konuralp J. Math. April 2022;10(1):59-68.
Chicago Shaba, Timilehin, and Abbas Kareem. “Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function”. Konuralp Journal of Mathematics 10, no. 1 (April 2022): 59-68.
EndNote Shaba T, Kareem A (April 1, 2022) Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function. Konuralp Journal of Mathematics 10 1 59–68.
IEEE T. Shaba and A. Kareem, “Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function”, Konuralp J. Math., vol. 10, no. 1, pp. 59–68, 2022.
ISNAD Shaba, Timilehin - Kareem, Abbas. “Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function”. Konuralp Journal of Mathematics 10/1 (April2022), 59-68.
JAMA Shaba T, Kareem A. Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function. Konuralp J. Math. 2022;10:59–68.
MLA Shaba, Timilehin and Abbas Kareem. “Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function”. Konuralp Journal of Mathematics, vol. 10, no. 1, 2022, pp. 59-68.
Vancouver Shaba T, Kareem A. Initial Coefficient Estimates for a Certain Subclasses of $m$-Fold Symmetric Bi-Univalent Functions Involving $\phi$-Pseudo-Starlike Functions Defined by Mittag-Leffler Function. Konuralp J. Math. 2022;10(1):59-68.
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