International Journal of Nonlinear Analysis and Applications
Maclaurin coefficient estimates of te-univalent functions connected with the (p,q)-derivative
Document Type : Research Paper
Authors
1 Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt
2 Department of Mathematics, College of Science, University of Al-Qadisiyah, Al Diwaniyah, Al-Qadisiyah, Iraq
Abstract
In this paper, we introduce a new subclass of analytic and te-univalent functions in the open unit disc associated with the operator $\mathcal{T}_{\zeta }^{\lambda ,p,q}$, which is defined by using the (p,q)-derivative. We obtain the coefficient estimates and Fekete-Szeg\H{o} inequalities for the functions belonging to this class.The various results presented in this paper would generalize and improve those in related works of several earlier authors.
Keywords
[1] A.M. Abd-Eltawab, Coefficient estimates of te-univalent functions associated with the Dziok-Srivastava operator,
Int. J. Open Prob. Complex Anal. 13 (2021), no. 2, 14–28.
[2] T. Acar, A. Aral and S.A. Mohiuddine, On Kantorovich modification of (p,q)-Baskakov operators, J. Inequal.
Appl. 2016 (2016), 1–14.[3] R.M. Ali, S. K. Lee, V. Ravichandran and S. Supramanian, Coefficient estimates for bi-univalent Ma-Minda
starlike and convex functions, Appl. Math. Lett. 25 (2012), no. 3, 344–351.
[4] S¸. Altınkaya, Inclusion properties of Lucas polynomials for bi-univalent functions introduced through the q-analogue
of the Noor integral operator, Turk. J. Math. 43 (2019), 620–629.
[5] M. Arif, M.U. Haq and J.L. Liu, A subfamily of univalent functions associated with q-analogue of Noor integral
operator, J. Funct. Spaces 2018 (2018), ID 3818915, 1–5.
[6] D.A. Brannan and J. Clunie, Aspects of Contemporary Complex Analysis, Academic Press, New-York and London,
1980.
[7] D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math. 31
(1986), 70-77.
[8] J.D. Bukweli-Kyemba and M.N. Hounkonnou, Quantum deformed algebras: coherent states and special functions,
arXiv preprint arXiv:1301.0116 (2013).
[9] T. Bulboaca, Differential Subordinations and Superordinations, Recent Results, House of Scientific Book Publ.,
Cluj-Napoca, 2005.
[10] R. Chakrabarti and R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum, J. Phys. Math.
Gen. 24 (1991), no. 13, L711–L718.
[11] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal. 2 (2013),
no. 1, 49–60.
[12] P.L. Duren, A Univalent functions, Springer-Verlag, Berlin-New York, 1983.
[13] S.M. El-Deeb and T. Bulboac˘a, Differential sandwich-type results for symmetric functions connected with a qanalog integral operator, Math. 7 (2019), no. 12, 1185.
[14] S.M. El-Deeb and T. Bulboac˘a, Fekete-Szeg˝o inequalities for certain class of analytic functions connected with
q-anlogue of Bessel function, J. Egypt. Math. Soc. 27 (2019), 1–11.
[15] S. M. El-Deeb, T. Bulboac˘a and B. M. El-Matary, Maclaurin coefficient estimates of bi-univalent functions
connected with the q-derivative, Math. 8 (2020), no. 3, 418.
[16] M. Fekete and G. Szeg˝o, Eine Bemerkung ¨uber ungerade schlichte Funktionen, J. Lond. Math. Soc. 8 (1933),
85–89.
[17] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
[18] F.H. Jackson, q-difference equations, Am. J. Math. 32 (1910), no. 4, 305–314.
[19] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
[20] W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proc. Conf. Complex
Anal. Z. Li, F. Ren, L. Yang, and S. Zhang (Eds), 1992, pp. 157–169.
[21] S.S. Miller and P.T. Mocanu, Differential Subordination: Theory and Applications, Series on Monographs and
Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker Inc., New York and Basel, 2000.
[22] E. Netanyahu, The minimal distance of the image boundary from origin and the second coefficient of a univalent
function in |z| < 1, Arch. Rational Mech. Anal. 32 (1969), 100–112.
[23] H. Orhan and H. Arikan, (P, Q)–Lucas polynomial coefficient inequalities of bi-univalent functions defined by the
combination of both operators of Al-Aboudi and Ruscheweyh, Afr. Mat. 32 (2021), no. 3, 589–598.
[24] Ch. Pommerenke, Univalent Functions, Vandenhoeck and Rupercht, Gttingen, 1975.
[25] S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal. 2014
(2014), 984135.
[26] J.K. Prajapat, Subordination and superordination preserving properties for generalized multiplier transformation
operator, Math. Comput. Model. 55 (2012), 1456–1465.
[27] P.N. Sadjang, On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas, arXiv:1309.3934[math.QA] (2013).
[28] H.M. Srivastava, A.K. Mishra and M.K. Das, The fekete-szeg˝o problem for a subclass of close-to-convex functions,
Complex Var. Elliptic Equ. 44 (2001), 145–163.
[29] H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl.
Math. Lett. 23 (2010), 1188–1192.
[30] H.M. Srivastava, A.K. Wanas and R. Srivastava, Applications of the q-Srivastava-Attiya operator involving a
certain family of bi-univalent functions associated with the Horadam polynomials, Symmetry 13 (2021), Article
ID 1230, 1–14.
[31] H.M. Srivastava, N. Raza, E.S.A. AbuJarad and M.H. AbuJarad, Fekete-Szeg¨o inequality for classes of (p, q)-
starlike and (p, q)-convex functions, RACSAM 113 (2019), 3563–3584.
[32] A.K. Wanas and L.-I. Cotˆırlˇa, Initial coefficient estimates and Fekete–Szeg¨o inequalities for new families of biunivalent functions governed by (p − q)-Wanas operator, Symmetry 13 (2021), Article ID 2118, 1–17.
Int. J. Open Prob. Complex Anal. 13 (2021), no. 2, 14–28.
[2] T. Acar, A. Aral and S.A. Mohiuddine, On Kantorovich modification of (p,q)-Baskakov operators, J. Inequal.
Appl. 2016 (2016), 1–14.[3] R.M. Ali, S. K. Lee, V. Ravichandran and S. Supramanian, Coefficient estimates for bi-univalent Ma-Minda
starlike and convex functions, Appl. Math. Lett. 25 (2012), no. 3, 344–351.
[4] S¸. Altınkaya, Inclusion properties of Lucas polynomials for bi-univalent functions introduced through the q-analogue
of the Noor integral operator, Turk. J. Math. 43 (2019), 620–629.
[5] M. Arif, M.U. Haq and J.L. Liu, A subfamily of univalent functions associated with q-analogue of Noor integral
operator, J. Funct. Spaces 2018 (2018), ID 3818915, 1–5.
[6] D.A. Brannan and J. Clunie, Aspects of Contemporary Complex Analysis, Academic Press, New-York and London,
1980.
[7] D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math. 31
(1986), 70-77.
[8] J.D. Bukweli-Kyemba and M.N. Hounkonnou, Quantum deformed algebras: coherent states and special functions,
arXiv preprint arXiv:1301.0116 (2013).
[9] T. Bulboaca, Differential Subordinations and Superordinations, Recent Results, House of Scientific Book Publ.,
Cluj-Napoca, 2005.
[10] R. Chakrabarti and R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum, J. Phys. Math.
Gen. 24 (1991), no. 13, L711–L718.
[11] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal. 2 (2013),
no. 1, 49–60.
[12] P.L. Duren, A Univalent functions, Springer-Verlag, Berlin-New York, 1983.
[13] S.M. El-Deeb and T. Bulboac˘a, Differential sandwich-type results for symmetric functions connected with a qanalog integral operator, Math. 7 (2019), no. 12, 1185.
[14] S.M. El-Deeb and T. Bulboac˘a, Fekete-Szeg˝o inequalities for certain class of analytic functions connected with
q-anlogue of Bessel function, J. Egypt. Math. Soc. 27 (2019), 1–11.
[15] S. M. El-Deeb, T. Bulboac˘a and B. M. El-Matary, Maclaurin coefficient estimates of bi-univalent functions
connected with the q-derivative, Math. 8 (2020), no. 3, 418.
[16] M. Fekete and G. Szeg˝o, Eine Bemerkung ¨uber ungerade schlichte Funktionen, J. Lond. Math. Soc. 8 (1933),
85–89.
[17] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
[18] F.H. Jackson, q-difference equations, Am. J. Math. 32 (1910), no. 4, 305–314.
[19] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
[20] W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proc. Conf. Complex
Anal. Z. Li, F. Ren, L. Yang, and S. Zhang (Eds), 1992, pp. 157–169.
[21] S.S. Miller and P.T. Mocanu, Differential Subordination: Theory and Applications, Series on Monographs and
Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker Inc., New York and Basel, 2000.
[22] E. Netanyahu, The minimal distance of the image boundary from origin and the second coefficient of a univalent
function in |z| < 1, Arch. Rational Mech. Anal. 32 (1969), 100–112.
[23] H. Orhan and H. Arikan, (P, Q)–Lucas polynomial coefficient inequalities of bi-univalent functions defined by the
combination of both operators of Al-Aboudi and Ruscheweyh, Afr. Mat. 32 (2021), no. 3, 589–598.
[24] Ch. Pommerenke, Univalent Functions, Vandenhoeck and Rupercht, Gttingen, 1975.
[25] S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal. 2014
(2014), 984135.
[26] J.K. Prajapat, Subordination and superordination preserving properties for generalized multiplier transformation
operator, Math. Comput. Model. 55 (2012), 1456–1465.
[27] P.N. Sadjang, On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas, arXiv:1309.3934[math.QA] (2013).
[28] H.M. Srivastava, A.K. Mishra and M.K. Das, The fekete-szeg˝o problem for a subclass of close-to-convex functions,
Complex Var. Elliptic Equ. 44 (2001), 145–163.
[29] H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl.
Math. Lett. 23 (2010), 1188–1192.
[30] H.M. Srivastava, A.K. Wanas and R. Srivastava, Applications of the q-Srivastava-Attiya operator involving a
certain family of bi-univalent functions associated with the Horadam polynomials, Symmetry 13 (2021), Article
ID 1230, 1–14.
[31] H.M. Srivastava, N. Raza, E.S.A. AbuJarad and M.H. AbuJarad, Fekete-Szeg¨o inequality for classes of (p, q)-
starlike and (p, q)-convex functions, RACSAM 113 (2019), 3563–3584.
[32] A.K. Wanas and L.-I. Cotˆırlˇa, Initial coefficient estimates and Fekete–Szeg¨o inequalities for new families of biunivalent functions governed by (p − q)-Wanas operator, Symmetry 13 (2021), Article ID 2118, 1–17.
)
Volume 13, Issue 2
July 2022Pages 2751-2762
- Receive Date: 10 April 2022
- Accept Date: 15 July 2022