The study examines third-order sandwich-type outcomes for analytic and univalent functions that are characterized by means of special functions like the Hurwitz–Lerch functions related to the Srivastava–Attiya linear operator. By employing techniques of differential subordination and superordination, we establish several new inclusion relationships and admissibility conditions that extend and unify earlier results in the literature. Further, optimal dominant and subordinant functions are identified, yielding sharp outcomes within the framework of third-order differential inequalities. The derived outcomes not only generalize a number of previously known results but also provide applications to special subclasses of analytic functions associated with the convolution operator $ N_{\alpha}^{\beta} $.
Citation: Mohammad Faisal Khan, Mohammed AbaOud, Muhammed Salih Muhammed. Application on third-order sandwich results for analytic functions defined by using a linear operator[J]. AIMS Mathematics, 2026, 11(9): 28254-28272. doi: 10.3934/math.20261125
The study examines third-order sandwich-type outcomes for analytic and univalent functions that are characterized by means of special functions like the Hurwitz–Lerch functions related to the Srivastava–Attiya linear operator. By employing techniques of differential subordination and superordination, we establish several new inclusion relationships and admissibility conditions that extend and unify earlier results in the literature. Further, optimal dominant and subordinant functions are identified, yielding sharp outcomes within the framework of third-order differential inequalities. The derived outcomes not only generalize a number of previously known results but also provide applications to special subclasses of analytic functions associated with the convolution operator $ N_{\alpha}^{\beta} $.
| [1] |
J. A. Antonino, S. S. Miller, Third-order differential inequalities and subordination in the complex plane, Complex Var. Elliptic, 56 (2011), 439–454. https://doi.org/10.1080/17476931003728404 doi: 10.1080/17476931003728404
|
| [2] |
D. Breaz, S. Khan, F. M. O. Tawfiq, F. Tchier, Applications of fuzzy differential subordination to the subclass of analytic functions involving Riemann–Liouville fractional integral operator, Mathematics, 11 (2023), 4975. https://doi.org/10.3390/math11244975 doi: 10.3390/math11244975
|
| [3] | T. Bulboacă, Differential subordinations and superordinations: Recent results, House of Scientific Book Publishing, Cluj-Napoca, Romania, 2005. |
| [4] | R. M. El-Ashwah, A. H. Hassan, Some third-order differential subordination and superordination results of some meromorphic functions using a Hurwitz–Lerch zeta type operator, Ilir. J. Math., 4 (2015), 1–15. |
| [5] |
H. A. Farzana, B. A. Stephen, M. P. Jeyaraman, Third-order differential subordination of analytic functions defined by functional derivative operator, Ann. Alexandru Ioan Cuza Univ.-Math., 2014 (2014), 1–16. https://doi.org/10.2478/aicu-2014-0028 doi: 10.2478/aicu-2014-0028
|
| [6] | P. Gochhayat, Sandwich-type results for a class of functions defined by a generalized differential operator, Mat. Vesn., 65 (2013), 178–186. |
| [7] | P. Gochhayat, Sandwich-type theorems of some subclasses of multivalent functions involving Dziok–Srivastava operator, Acta Univ. Apulensis Math. Inform., 32 (2012), 31–47. |
| [8] | M. P. Jeyaraman, T. K. Suresh, Third-order differential subordination of analytic functions, Acta Univ. Apulensis Math. Inform., 35 (2013), 187–202. |
| [9] |
B. K. Mihsin, W. G. Atshan, On new sandwich results of univalent function defined by a linear operator, Iraqi J. Sci., 63 (2022), 5467–5475. https://doi.org/10.24996/ijs.2022.63.12.32 doi: 10.24996/ijs.2022.63.12.32
|
| [10] |
S. S. Miller, P. T. Mocanu, Subordinants of differential superordinations, Complex Var. Theory Appl., 48 (2003), 815–826. https://doi.org/10.1080/02781070310001599322 doi: 10.1080/02781070310001599322
|
| [11] |
S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Mich. Math. J., 28 (1981), 157–171. https://doi.org/10.1307/mmj/1029002507 doi: 10.1307/mmj/1029002507
|
| [12] | S. S. Miller, P. T. Mocanu, Differential subordinations: Theory and applications, CRC Press, 2000. |
| [13] |
A. K. Mishra, A. Prajapati, P. Gochhayat, Third-order differential subordination and superordination results for analytic functions involving the Hohlov operator, Tbil. Math. J., 13 (2020), 95–109. https://doi.org/10.32513/tbilisi/1601344901 doi: 10.32513/tbilisi/1601344901
|
| [14] |
J. Morais, H. M. Zayed, Applications of differential subordination and superordination theorems to fluid mechanics involving a fractional higher-order integral operator, Alex. Eng. J., 60 (2021), 3901–3914. https://doi.org/10.1016/j.aej.2021.02.037 doi: 10.1016/j.aej.2021.02.037
|
| [15] |
G. I. Oros, G. Oros, L. F. Preluca, New applications of Gaussian hypergeometric function for developments on third-order differential subordinations, Symmetry, 15 (2023), 1306. https://doi.org/10.3390/sym15071306 doi: 10.3390/sym15071306
|
| [16] | S. Ponnusamy, O. P. Juneja, Third-order differential inequalities in the complex plane, In: Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, London, 1992, 274–290. https://doi.org/10.1142/9789814355896_0023 |
| [17] |
D. Răducanu, Third-order differential subordinations for analytic functions associated with generalized Mittag-Leffler functions, Mediterr. J. Math., 14 (2017), 167. https://doi.org/10.1007/s00009-017-0969-8 doi: 10.1007/s00009-017-0969-8
|
| [18] | Q. A. Shakir, F. M. Sakar, On third-order differential subordination and superordination properties of analytic functions defined by Tayyah–Atshan fractional integral operator, Adv. Nonlinear Var. Inequal., 28 (2025), 176–200. |
| [19] |
H. M. Srivastava, A. A. Attiya, An integral operator associated with the Hurwitz–Lerch zeta function and differential subordination, Integr. Transf. Spec. F., 18 (2007), 207–216. https://doi.org/10.1080/10652460701208577 doi: 10.1080/10652460701208577
|
| [20] | H. M. Srivastava, J. Choi, Series associated with the Zeta and related functions, Springer Science & Business Media, 530 (2001). |
| [21] | S. R. Swamy, Sandwich theorems for $p$-valent functions defined by certain integral operator, Int. J. Math. Arch., 4 (2013), 101–107. |
| [22] |
H. A. Janaby, F. Ghanim, M. Darus, On the third-order complex differential inequalities of $\zeta$-generalized-Hurwitz-Lerch zeta functions, Mathematics, 8 (2020), 845. https://doi.org/10.3390/math8050845 doi: 10.3390/math8050845
|
| [23] |
H. Tang, H. M. Srivastava, S. Li, L. Ma, Third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu–Srivastava operator, Abstr. Appl. Anal., 2014 (2014), 792175. https://doi.org/10.1155/2014/792175 doi: 10.1155/2014/792175
|
| [24] |
A. S. Tayyah, W. G. Atshan, G. I. Oros, Third-order differential subordination results for meromorphic functions associated with the inverse of the Legendre chi function via the Mittag-Leffler identity, Mathematics, 13 (2025), 2089. https://doi.org/10.3390/math13132089 doi: 10.3390/math13132089
|
| [25] |
A. S. Tayyah, W. G. Atshan, Differential subordination and superordination results for $p$-valent analytic functions associated with $(r, k)$-Srivastava fractional integral calculus, MethodsX, 13 (2024), 103079. https://doi.org/10.1016/j.mex.2024.103079 doi: 10.1016/j.mex.2024.103079
|
| [26] |
H. M. Zayed, T. Bulboacă, Applications of differential subordinations involving a generalized fractional differintegral operator, J. Inequal. Appl., 2019 (2019), 242. https://doi.org/10.1186/s13660-019-2198-0 doi: 10.1186/s13660-019-2198-0
|
Figures(1)
Mohammad Faisal Khan, Mohammed AbaOud, Muhammed Salih Muhammed. Application on third-order sandwich results for analytic functions defined by using a linear operator[J]. AIMS Mathematics, 2026, 11(9): 28254-28272. doi: 10.3934/math.20261125
DownLoad: