Differential subordination and superordination of a $q$-derivative operator connected with the $q$-exponential function

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq

2 Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia

Abstract

In this article, we define a $q$-derivative operator of univalent functions associated with the $q$-exponential function. Moreover, we introduce differential subordination and differential superordination for the subordination class defined by this operator. Sandwich-type theorems of several known results also are derived by applying these results.

Keywords

[1] RM. Ali, V. Ravichandran, MH. Khan and K.G. Subramanian, Differential sandwich theorems for certain analytic
functions, Far East J. Math. Sci. 15 (2004), 87–94.[2] AA. Attiya and MF. Yassen, Some subordination and superordination results associated with generalized
Srivastava-Attiya operator, Filomat 31 (2017), 53—60.
[3] A.A. Attiya, M.K. Aouf, E.E. Ali and M.F. Yassen, Differential Subordination and Superordination Results Associated with Mittag–Leffler Function, Math. 9 (2021), 226.
[4] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean operator, J. Math. Math. Sci. 27 (2004),
1429–1436.
[5] M.K. Aouf, A.O. Mostafa and R.E. Elmorsy, Certain subclasses of analytic functions with varying arguments
associated with q-difference operator, Afr. Mat. 32 (2021), 621-–630.
[6] S.S. Billing, A subordination theorem with applications to analytic functions, Bull. Math. Ana. App. 3 (2011),
1–8.
[7] T. Bulboac˘a, A class of superordination-preserving integral operators, Indag. Math., N.S. 13 (2002), 301-–311.
[8] T. Bulboac˘a, Classes of first order differential superordinations, Demonst. Math. 35 (2002), 287-–292.
[9] W.S. Chung and H.J. Kang, The q-gamma, q-beta functions, and q-multiplication formula, J. Math. Phys. 35
(1994), 4268.
[10] J.L. Cie´sli´nski, Improved q-exponential and q-trigonometric functions, Appl. Math. Lett. 24 (2011), 2110–2114.
[11] M. Darus and R.W. Ibrahim, On applications of differential subordination and differential operator, J. Math.
Statist. 8 (2012), 165–168.
[12] R.M. El-Ashwah, M.K. Aouf and T. Bulboaca, Differential subordinations for classes of meromorphic p-valent
functions defined by multiplier transformations, Bull. Aust. Math. Soc. 83 (2011), 353-–368.
[13] S. Elhaddad, H. Aldweby and M. Darus, On certain subclasses of analytic functions involving differential operator,
Jnanabha. 48 (2018), 55-–64.
[14] S.M. El-Deeb and T. Bulboac˘a, Differential sandwich-type results for symmetric functions connected with a qanalog integral operator, Math. 7 (2019), 1185.
[15] B.A. Frasin and G. Murugusundaramoorthy, A subordination results for a class of analytic functions defined by
q-differential operator, Ann. Univ. Paedagog. Crac. Stud. Math. 19 (2020), 53–64.
[16] M. Govindaraj and S. Sivasubramanian, On a class of analytic functions related to conic domains involving
q-calculus, Anal. Math. 43 (2017), 475-–487.
[17] S.P. Goyal, P. Goswami and H. Silverman, Subordination and superordination results for a class of analytic
multivalent functions, Int. J. Math. Math. Sci. 2008 (2008), Article ID 561638, 1–12.
[18] S.H. Hadi, M. Darus, C. Park, J.R. Lee., Some geometric properties of multivalent functions associated with a
new generalized q-Mittag-Leffler function, AIMS Math. 7 (2022), 11772–11783.
[19] F.H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh. 46 (1908), 253—281.
[20] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193—203.
[21] S. Kanas and D. Raducanu, Some subclass of analytic functions related to conic domains, Math. Slovaca 64
(2014), 1183—1196.
[22] Q. Khan, M. Arif, M. Raza, G. Srivastava, H. Tang and S.U. Rehman, Some applications of a new integral
operator in q-analog for multivalent functions, Math. 7 (2019), 1178.
[23] B. Khan, H.M. Srivastava, M. Tahir, M. Darus, Q.Z. Ahmad and N. Khan, Applications of a certain q-integral
operator to the subclasses of analytic and bi-univalent functions, AIMS Math. 6 (2012), 1024—1039.
[24] A.A. Lupas and G.I. Oros, Differential subordination and superordination results using fractional integral of
confluent hypergeometric function, Symmetry 13 (2021), 327.
[25] M.S. McAnally, q-exponential and q-gamma functions, II. q-gamma functions, J. Math. Phys. 36 (1995), 574.
[26] S.S. Miller and P.T. Mocanu, Differential Subordinations: Theory and Applications, Series on Monographs and
Textbooks in Pure and Applied Mathematics 225, Marcel Dekker Inc., New York, Basel, 2000.[27] S.S. Miller and P.T. Mocanu, Subordinants of differential superordinations, Complex Variabl. Ellipt. Equ.. 48
(2003), 815-–826.
[28] A.K. Mishra and M.M. Soren, Sandwich results for subclasses of multivalent meromorphic functions associated
with iterations of the Cho-Kwon-Srivastava transform, Filomat 33 (2019), 255-–266.
[29] G. Murugusundaramoorthy and N. Magesh, Differential subordinations and superordinations for analytic functions
defined by the Dziok-Srivastava linear operator, J. Ineq. Pure App. Math. 7 (2006), no. 4, Article 152.
[30] GS. S˘al˘agean, Subclasses of univalent functions, Complex Analysis—Fifth Romanian-Finnish Seminar, Springer,
Berlin, Heidelberg, 1983, pp. 362–372.
[31] S.A. Shah and K.I. Noor, Study on the q-analogue of a certain family of linear operators, Turk. J. Math. 43
(2019), 2707–2714.
[32] 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 Sci. 44 (2020), 1–18.
[33] T.N. Shanmugam, S. Srikandan, B.A. Frasin and S. Kavitha, On sandwich theorems for certain subclasses of
analytic functions involving Carlson-Shaffer operator, J. Korean Math. Soc. 45 (2008), 611–620.
[34] TN. Shanmugam, S. Sivasubramanian, and M. Darus, Subordination and Superordination Results for Φ-Like
Functions, J. Ineq. Pure and App. Math. 8 (2007), Article ID 20.
[35] A. Oshah and M. Darus, Differential sandwich theorems with new generalized derivative operator, Adv. Math.
Sci. 3 (2014), 117–125.
[36] AK. Wanas and M. Darus, Applications of fractional derivative on a differential subordinations and superordinators for analytic functions associated with differential Operator, Kragujevac J. Math. 45 (2021), 379-–392.
Volume 13, Issue 2
July 2022
Pages 2795-2806
  • Receive Date: 08 May 2022
  • Revise Date: 14 June 2022
  • Accept Date: 15 July 2022