Certain differential subordination results for univalent functions associated with $ q $-Salagean operators

Ebrahim Amini; Mojtaba Fardi; Shrideh Al-Omari; Rania Saadeh; Ebrahim Amini; Mojtaba Fardi; Shrideh Al-Omari; Rania Saadeh

doi:10.3934/math.2023811

AIMS Mathematics

2023, Volume 8, Issue 7: 15892-15906. doi: 10.3934/math.2023811

Previous Article Next Article

Research article

Certain differential subordination results for univalent functions associated with $ q $-Salagean operators

1.
Department of Mathematics, Payme Noor University, P. O. Box 19395-4697, Tehran, Iran
2.
Department of Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord, P. O. Box 115, Iran
3.
Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Amman 11134, Jordan
4.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan

Received: 29 January 2023 Revised: 03 April 2023 Accepted: 06 April 2023 Published: 04 May 2023
MSC : 05A30, 30C45, 39A13

In this paper, we employ the concept of the $ q $-derivative to derive certain differential and integral operators, $ D_{q, \lambda}^{n} $ and $ I_{q, \lambda}^{n} $, resp., to generalize the class of Salagean operators over the set of univalent functions. By means of the new operators, we establish the subclasses $ M^n_{q, \lambda} $ and $ D^n_{q, \lambda} $ of analytic functions on an open unit disc. Further, we study coefficient inequalities for each function in the given classes. Over and above, we derive some properties and characteristics of the set of differential subordinations by following specific techniques. In addition, we study the general properties of $ D_{q, \lambda}^{n} $ and $ I_{q, \lambda}^{n} $ and obtain some interesting differential subordination results. Several results are also derived in some details.
- $q$-derivative operator,
- univalent function,
- Salagean operator,
- geometric function theory
Citation: Ebrahim Amini, Mojtaba Fardi, Shrideh Al-Omari, Rania Saadeh. Certain differential subordination results for univalent functions associated with $ q $-Salagean operators[J]. AIMS Mathematics, 2023, 8(7): 15892-15906. doi: 10.3934/math.2023811

Related Papers:

Abstract

In this paper, we employ the concept of the $ q $-derivative to derive certain differential and integral operators, $ D_{q, \lambda}^{n} $ and $ I_{q, \lambda}^{n} $, resp., to generalize the class of Salagean operators over the set of univalent functions. By means of the new operators, we establish the subclasses $ M^n_{q, \lambda} $ and $ D^n_{q, \lambda} $ of analytic functions on an open unit disc. Further, we study coefficient inequalities for each function in the given classes. Over and above, we derive some properties and characteristics of the set of differential subordinations by following specific techniques. In addition, we study the general properties of $ D_{q, \lambda}^{n} $ and $ I_{q, \lambda}^{n} $ and obtain some interesting differential subordination results. Several results are also derived in some details.

References

[1]	M. Arif, H. M. Srivastava, S. Umar, Some application of a $q$-analogue of the Ruscheweyh type operator for multivalent functions, RACSAM, 113 (2019), 1211–1221. https://doi.org/10.1007/s13398-018-0539-3 doi: 10.1007/s13398-018-0539-3
[2]	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. Sci., 44 (2020), 327–344. https://doi.org/10.1007/s40995-019-00815-0 doi: 10.1007/s40995-019-00815-0
[3]	E. Amini, M. Fardi, S. K. Al-Omari, K. Nonlaopon, Results on univalent functions defined by $q$-analogues of Salagean and Ruscheweh operators, Symmetry, 14 (2022), 1725. https://doi.org/10.3390/sym14081725 doi: 10.3390/sym14081725
[4]	E. Amini, S. K. Al-Omari, K. Nonlaopon, D. Baleanu, Estimates for coefficients of bi-univalent functions associated with a fractional q-difference operator, Symmetry, 14 (2022), 879. https://doi.org/10.3390/sym14050879 doi: 10.3390/sym14050879
[5]	R. W. Ibrahim, R. M. Elobaid, S. Obaiys, Geometric inequalities via a symmetric differential operator defined by quantum calculus in the open unit disk, J. Funct. Space., 2020 (2020), 6932739. https://doi.org/10.1155/2020/6932739 doi: 10.1155/2020/6932739
[6]	M. Arif, B. Ahmad, New subfamily of meromorphic multivalent starlike functions in circular domain involving $q$-differential operator, Math. Slovaca, 68 (2018), 1049–1056. https://doi.org/10.1515/ms-2017-0166 doi: 10.1515/ms-2017-0166
[7]	M. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Variables, Theory and Application: An International Journal, 14 (1990), 77–84. https://doi.org/10.1080/17476939008814407 doi: 10.1080/17476939008814407
[8]	H. M. Srivastava, A. Motamednezhad, 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), 172. https://doi.org/10.3390/math8020172 doi: 10.3390/math8020172
[9]	K. Vijaya, G. Murugusundaramoorthy, M. Kasthuri, Starlike functions of complex order involving $q$-hypergeometric functions with fixed point, Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, 13 (2014), 51–63.
[10]	E. Amini, S. K. Al-Omari, K. Nonlaopon, D. Baleanu, Estimates for coefficients of Bi-univalent functions associated with a fractional $q$-difference Operator, Symmetry, 14 (2022), 879. https://doi.org/10.3390/sym14050879 doi: 10.3390/sym14050879
[11]	S. Al-Omari, Estimates and properties of certain $q$-Mellin transform on generalized $q$-calculus theory, Adv. Differ. Equ., 2021 (2021), 233. https://doi.org/10.1186/s13662-021-03391-z doi: 10.1186/s13662-021-03391-z
[12]	M. Arif, H. M. Srivastava, S. Umar, Some applications of a $q$-analogue of the Ruscheweyh type operator for multivalent functions, RACSAM, 113 (2019), 1211–1221. https://doi.org/10.1007/s13398-018-0539-3 doi: 10.1007/s13398-018-0539-3
[13]	S. Al-Omari, On a $q$-Laplace-type integral operator and certain class of series expansion, Math. Method. Appl. Sci., 44 (2021), 8322–8332. https://doi.org/10.1002/mma.6002 doi: 10.1002/mma.6002
[14]	A. Mohammed, M. Darus, A generalized operator involving the $q$-hypergeometric function, Matematiqki Vesnikmat, 65 (2013), 454–465.
[15]	S. Al-Omari, On a $q$-Laplace-type integral operator and certain class of series expansion, Math. Method. Appl. Sci., 44 (2021), 8322–8332. https://doi.org/10.1002/mma.6002 doi: 10.1002/mma.6002
[16]	S. Al-Omari, D. Baleanu, S. Purohit, Some results for Laplace-type integral operator in quantum calculus, Adv. Differ. Equ., 2018 (2018), 124. https://doi.org/10.1186/s13662-018-1567-1 doi: 10.1186/s13662-018-1567-1
[17]	S. Al-Omari, D. Suthar, S. Araci, A fractional $q$-integral operator associated with certain class of $q$-Bessel functions and $q$-generating series, Adv. Differ. Equ., 2021 (2021), 441. https://doi.org/10.1186/s13662-021-03594-4 doi: 10.1186/s13662-021-03594-4
[18]	S. Al-Omari, On $q$-analogues of Mangontarum transform of some polynomials and certain class of H-functions, Nonlinear Studies, 23 (2016), 51–61.
[19]	G. Gharib, R. Saadeh, Reduction of the self-dual Yang-Mills equations to Sinh-Poisson equation and exact solutions, WSEAS Transactions on Mathematics, 20 (2021), 540–546. https://doi.org/10.37394/23206.2021.20.57 doi: 10.37394/23206.2021.20.57
[20]	X. Zhang, S. Khan, S. Hussain, H. Tang, Z. Shareef, New subclass of $q$-starlike functions associated with generalized conic domain, AIMS Mathematics, 5 (2020), 4830–4848. https://doi.org/10.3934/math.2020308 doi: 10.3934/math.2020308
[21]	S. Al-Omari, On $q$-analogues of the Mangontarum transform for certain $q$-Bessel functions and some application, J. King Saud Univ. Sci., 28 (2016), 375–379. http://doi.org/10.1016/j.jksus.2015.04.008 doi: 10.1016/j.jksus.2015.04.008
[22]	M. Caglar, E. Deniz, Initial coefficients for a subclass of bi-univalent functions defined by Salagean differential operator, Commun. Fac. Sci. Univ., 66 (2017), 85–91. https://doi.org/10.1501/Commua1_0000000777 doi: 10.1501/Commua1_0000000777
[23]	M. Caglar, L. Cotirla, A. Catas, A new family of harmonic functions defined by an integral operator, Acta Universitatis Apulensis, 72 (2022), 1–13.
[24]	A. R. S. Juma, L. Cotirla, On harmonic univalent function defined by generalized salagean derivatives, Acta Universitatis Apulensis, 23 (2010), 179–188.
[25]	R. Saadeh, A. Qazza, A. Burqan, On the double ARA-Sumudu transform and its applications, Mathematics, 10 (2022), 2581. https://doi.org/10.3390/math10152581 doi: 10.3390/math10152581
[26]	R. Saadeh, A. Burqan, A. El-Ajou, Reliable solutions to fractional Lane-Emden equations via Laplace transform and residual error function, Alex. Eng. J., 61 (2022), 10551–10562. https://doi.org/10.1016/j.aej.2022.04.004 doi: 10.1016/j.aej.2022.04.004
[27]	R. Saadeh, Applications of double ARA integral transform, Computation, 10 (2022), 216. https://doi.org/10.3390/computation10120216 doi: 10.3390/computation10120216
[28]	F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203.
[29]	I. Graham, G. Kohr, Geometric function theory in one and higher dimensions, New York: Marcel Dekker, 2003.
[30]	M. Aouf, A. Mostafa, H. Zayed, Certain family of integral operators associated with multivalent functions preserving subordination and superordination, Filomat, 32 (2018), 2395–2401. https://doi.org/10.2298/FIL1807395A doi: 10.2298/FIL1807395A
[31]	E. Amini, S. Al-Omari, H. Rahmatan, On geometric properties of certain subclasses of univalent functions defined by Noor integral operator, Analysis, 42 (2022), 251–259. https://doi.org/10.1515/anly-2022-1043 doi: 10.1515/anly-2022-1043
[32]	H. Tang, G. Deng, Subordination and superordination preserving properties for a family of integral operators involving the Noor integral operator, Journal of the Egyptian Mathematical Society, 22 (2014), 352–361. https://doi.org/10.1016/j.joems.2013.09.003 doi: 10.1016/j.joems.2013.09.003
[33]	S. Ruscheweyh, Convolutions in geometric function theory, Presses de l'Université de Montréal, 1982.
[34]	S. Miller, P. Mocanu, Differential subordinations: theory and aApplications, CRC Press, 2000.
[35]	I. G. Oros, Geometrical theory of analytic functions, Mathematics, 10 (2022), 3267. https://doi.org/10.3390/math10183267 doi: 10.3390/math10183267
[36]	H. S. Wilf, Subordinating factor sequences for convex maps of the unit circle, Proc. Amer. Math. Soc., 12 (1961), 689–693. https://doi.org/10.1090/s0002-9939-1961-0125214-5 doi: 10.1090/s0002-9939-1961-0125214-5
[37]	S. Miller, P. Mocanu, Subordinations of differential superordinations, Complex Variables, Theory and Application: An International Journal, 48 (2003), 815–826. https://doi.org/10.1080/02781070310001599322 doi: 10.1080/02781070310001599322
[38]	G. S. Salagean, Subclass of univalent functios, In: Complex analysis—Fifth Romanian-Finnish seminar, Berlin: Springer, 1983,362–372. https://doi.org/10.1007/BFb0066543
[39]	C. Ramachandran, D. Kavitha, Coefficient estimates for a subclass of bi-univalent functions defined by Salagean operator using quasi subordination, Applied Mathematical Sciences, 11 (2017), 1725–1732. https://doi.org/10.12988/ams.2017.75165 doi: 10.12988/ams.2017.75165
[40]	B. Şeker, On a new subclass of bi-univalent functions defined by using Salagean operator, Turk. J. Math., 42 (2018), 2891–2896. https://doi.org/10.3906/mat-1507-100 doi: 10.3906/mat-1507-100
[41]	G. I. Oros, R. Sendrutiu, A. O. Taut, On a class of univalent functions defined by Salagean differential operator, Banach J. Math. Anal., 3 (2009), 61–67. https://doi.org/ 10.15352/bjma/1240336424 doi: 10.15352/bjma/1240336424
[42]	F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, International Journal of Mathematics and Mathematical Sciences, 2004 (2004), 172525. https://doi.org/10.1155/S0161171204108090 doi: 10.1155/S0161171204108090

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)