The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on the properties of the differential subordination and superordination one of the newest techniques used in this field, we obtain some differential subordination and superordination results for multivalent functions defined by differintegral operator with $ j $-derivatives $ \Im _{p}(\nu, \rho; \ell)f(z) $ for $ \ell > 0, \ \nu, \rho \in \mathbb{R}, \ $such that $ (\rho -j)\geq 0, \nu > -\ell p\, (p\in \mathbb{N}) $ in the open unit disk $ U $. Differential sandwich result is also obtained. Also, the results are followed by some special cases and counter examples.
Citation: Ekram E. Ali, Rabha M. El-Ashwah, R. Sidaoui. Application of subordination and superordination for multivalent analytic functions associated with differintegral operator[J]. AIMS Mathematics, 2023, 8(5): 11440-11459. doi: 10.3934/math.2023579
The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on the properties of the differential subordination and superordination one of the newest techniques used in this field, we obtain some differential subordination and superordination results for multivalent functions defined by differintegral operator with $ j $-derivatives $ \Im _{p}(\nu, \rho; \ell)f(z) $ for $ \ell > 0, \ \nu, \rho \in \mathbb{R}, \ $such that $ (\rho -j)\geq 0, \nu > -\ell p\, (p\in \mathbb{N}) $ in the open unit disk $ U $. Differential sandwich result is also obtained. Also, the results are followed by some special cases and counter examples.
| [1] |
R. M. Ali, A. O. Badghaish, V. Ravichandran, Subordination for higher-order derivatives of multivalent functions, J. Inequal. Appl., 2008 (2008), 1–12. https://doi.org/10.1155/2008/830138 doi: 10.1155/2008/830138
|
| [2] |
E. E. Ali, H. M. Srivastava, R. M. El-Ashwah, A. M. Albalahi, Differential subordination and differential superordination for classes of admissible multivalent functions associated with a linear operator, Mathematics, 10 (2022), 4690. https://doi.org/10.3390/math10244690 doi: 10.3390/math10244690
|
| [3] | M. K. Aouf, R. M. El-Ashwah, E. E. Ali, On Sandwich theorems for higher-order derivatives of $p$-valent analytic functions, Se. Asian B. Math., 37 (2013), 7–14. |
| [4] |
M. K. Aouf, R. M. El-Ashwah, A. M. Abd-Eltawab, Some inclusion relationships of certain subclasses of $p$-valent functions associated with a family of integral operators, ISRN Math. Anal., 2013 (2013), 1–8. https://doi.org/10.1155/2013/384170 doi: 10.1155/2013/384170
|
| [5] | S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135 (1969), 429–446. |
| [6] | N. Breaz, R. M. El-Ashwah, Quasi-Hadamard product of some uniformly analytic and $p$-valent functions with negative coefficients, Carpathian J. Math., 30 (2014), 39–45. Available from: https://www.jstor.org/stable/43999556. |
| [7] | T. Bulboaca, Differential subordinations and superordinations, recent results, Hous of Scientific Book Publ., Cluj-Napoca, 2005. |
| [8] |
B. C. Carlson, D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737–745. https://doi.org/10.1137/0515057 doi: 10.1137/0515057
|
| [9] | M. P. Chen, H. Irmak, H. M. Srivastava, Some multivalent functions with negative coefficients defined by using a differential operator, Pan Amer. Math. J., 6 (1996), 55–64. Available from: http://hdl.handle.net/1828/1655. |
| [10] |
J. H. Choi, M. Saigo, H. M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl., 276 (2002), 432–445. https://doi.org/10.1016/S0022-247X(02)00500-0 doi: 10.1016/S0022-247X(02)00500-0
|
| [11] | R. M. El-Ashwah, M. E. Drbuk, Subordination properties of $ p$-valent functions defined by linear operators, Biritish J. Math. Comput. Sci., 4 (2014), 3000–3013. |
| [12] | D. Z. Hallenbeck, S. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc., 52 (1975), 191–195. |
| [13] |
I. B. Jung, Y. C. Kim, H. M. Srivastava, The hardy space of analytic functions associated with certain parameter families of integral operators, J. Math. Anal. Appl., 176 (1993), 138–147. https://doi.org/10.1006/jmaa.1993.1204 doi: 10.1006/jmaa.1993.1204
|
| [14] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 204 (2006). |
| [15] | V. Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics, Longman Scientic and Technical, Harlow (Essex), 301 (1993). |
| [16] |
A. Y. Lashin, F. Z. El-Emam, On certain classes of multivalent analytic functions defined with higher-order derivatives, Mathematics, 11 (2023), 83. https://doi.org/10.3390/math11010083 doi: 10.3390/math11010083
|
| [17] |
R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc., 16 (1965), 755–758. http://dx.doi.org/10.1090/S0002-9939-1965-0178131-2 doi: 10.1090/S0002-9939-1965-0178131-2
|
| [18] | J. L. Liu, S. Owa, Properties of certain integral operator, Int. J. Math. Sci., 3 (2004), 45–51. |
| [19] | S. S. Miller, P. T. Mocanu, Differential subordinations: Theory and applications, CRC Press, Boca Raton, 2000. |
| [20] | S. S. Miller, P. T. Mocanu, Subordinations of differential superordinations, Complex Var., 48 (2003), 815–826. |
| [21] | K. I. Noor, On new classes of integral operators, J. Nat. Geom., 16 (1999), 71–80. |
| [22] |
R. K. Raina, P. Sharma, Subordination preserving properties associated with a class of operators, Le Mat., 68 (2013), 217–228. http://dx.doi.org/10.4418/2013.68.1.16 doi: 10.4418/2013.68.1.16
|
| [23] | S. Ruscheweyh, New criteria for univalent functions, P. Am. Math. Soc., 49 (1975), 109–115. |
| [24] |
H. Saitoh, S. Owa, T. Sekine, M. Nunokawa, R. Yamakawa, An application of a certain integral operator, Appl. Math. Lett., 5 (1992), 21–24. http://dx.doi.org/10.1016/0893-9659(92)90104-H doi: 10.1016/0893-9659(92)90104-H
|
| [25] | T. N. Shanmugam, S. Sivasubramanian, M. Darus, C. Ramachandran, Subordination and superordination results for subclasses of analytic functions, Int. J. Math. Forum, 2007, 1039–1052. |
| [26] | T. N. Shanmugam, V. Ravichandran, S. Sivasubramanian, Differential sandwich theorems for subclasses of analytic functions, Aust. J. Math. Anal. Appl., 3 (2006), 1–11. |
| [27] | H. Silverman, Higher order derivatives, Chinese J. Math., 23 (1995), 189–191. Available from: https://www.jstor.org/stable/43836593. |
| [28] | H. M. Srivastava, R. M. El-Ashwah, N. Breaz, A certain subclass of multivalent functions involving higher-order derivatives, Filomat, 30 (2016), 113–124. Available from: https://www.jstor.org/stable/24898417. |
| [29] |
H. M. Srivastava, An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions, J. Adv. Eng. Comput., 5 (2021), 135–166. http://dx.doi.org/10.55579/jaec.202153.340 doi: 10.55579/jaec.202153.340
|
| [30] | E. T. Whittaker, G. N. Watson, A course on modern analysis: An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, 4 Eds., Cambridge University Press, Cambridge, 1927. https://doi.org/10.1002/zamm.19630430916 |
| [31] | T. Yaguchi, The radii of starlikeness and convexity for certain multivalent functions, Current Topics in Analytic Function Theory, World Scientific, River Edge, NJ, USA, 1992,375–386. |
Ekram E. Ali, Rabha M. El-Ashwah, R. Sidaoui. Application of subordination and superordination for multivalent analytic functions associated with differintegral operator[J]. AIMS Mathematics, 2023, 8(5): 11440-11459. doi: 10.3934/math.2023579
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