Throughout the paper, we introduce a new subclass $ \mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}\ f(z)$ by using the Bazilevič functions with the idea of bounded boundary rotation and $ q $-analogue Cătaş operator. Also we find the estimate of the coefficients for functions in this class. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward $ (p, q) $-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter $ p $ is obviously redundant.
Citation: S. M. Madian. Some properties for certain class of bi-univalent functions defined by $ q $-Cătaş operator with bounded boundary rotation[J]. AIMS Mathematics, 2022, 7(1): 903-914. doi: 10.3934/math.2022053
Throughout the paper, we introduce a new subclass $ \mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}\ f(z)$ by using the Bazilevič functions with the idea of bounded boundary rotation and $ q $-analogue Cătaş operator. Also we find the estimate of the coefficients for functions in this class. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward $ (p, q) $-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter $ p $ is obviously redundant.
| [1] |
F. Al-Aboudi, On univalent functions defined by a generalized Sălăgean operator, Int. J. Math. Math. Sci., 27 (2004), 481–494. doi: 10.1155/S0161171204108090. doi: 10.1155/S0161171204108090
|
| [2] |
B. Ahmad, M. G. Khan, B. A. Frasin, M. K. Aouf, T. Abdeljawad, W. K. Mashwani, et al., On $q$-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain, AIMS Math., 6 (2021), 3037–3052. doi: 10.3934/math.2021185. doi: 10.3934/math.2021185
|
| [3] |
Q. Z. Ahmad, N. Khan, M. Raza, M. Tahir, B. Khan, Certain $q$-difference operators and their applications to the subclass of meromorphic $q$-starlike functions, Filomat, 33 (2019), 3385–3397. doi: 10.2298/FIL1911385A. doi: 10.2298/FIL1911385A
|
| [4] | M. K. Aouf, A generalization of functions with real part bounded in the mean on the unit disc, Math. Japonica., 33 (1988), 175–182. |
| [5] |
M. K. Aouf, Generalization of certain subclasses of multivalent functions with negative coefficients defined by using a differential operator, Math. Comput. Model., 50 (2009), 1367–1378. doi: 10.1016/j.mcm.2008.08.026. doi: 10.1016/j.mcm.2008.08.026
|
| [6] | M. K. Aouf, On certain multivalent functions with negative coefficients defined by using a differential operator, Mat. Vesn., 62 (2010), 23–35. doi: 0025-51651001023A. |
| [7] |
M. K. Aouf, S. M. Madian, Inclusion and properties neighborhood for certain $p$-valent functions associated with complex order and $q$-$p$-valent Cătaş operator, J. Taibah Univ. Sci., 14 (2020), 1226–1232. doi: 10.1080/16583655.2020.1812923. doi: 10.1080/16583655.2020.1812923
|
| [8] |
M. K. Aouf, S. M. Madian, Coefficient bounds for bi-univalent classes defined by Bazilevič functions and convolution, Bol. Soc. Mat. Mex., 26 (2020), 1045–1062. doi: 10.1007/s40590-020-00304-0. doi: 10.1007/s40590-020-00304-0
|
| [9] |
M. K. Aouf, S. M. Madian, Certain classes of analytic functions associated with $q$-analogue of $p$-valent Cătaş operator, Moroccan J. Pure Appl. Anal., 7 (2021), 430–447. doi: 10.2478/mjpaa-2021-0029. doi: 10.2478/mjpaa-2021-0029
|
| [10] |
M. K. Aouf, S. M. Madian, Subordination factor sequence results for starlike and convex classes defined by $q$-Cătaş operator, Afr. Mat., 2021, 1–13. doi: 10.1007/s13370-021-00896-4. doi: 10.1007/s13370-021-00896-4
|
| [11] |
M. K. Aouf, S. M. Madian, A. O. Mostafa, Bi-univalent properties for certain class of Bazilevič functions defined by convolution and with bounded boundary rotation, J. Egypt. Math. Soc., 27 (2019), 11. doi: 10.1186/s42787-019-0012-2. doi: 10.1186/s42787-019-0012-2
|
| [12] | I. E. Bazilevič, On a case of integrability in quadratures of the Lowner-Kufarev equation, Mat. Sb., 37 (1955), 471–476. |
| [13] |
M. Çaǧlar, E. Deniz, H. M. Srivastava, Second Hankel determinant for certain subclasses of bi-univalent functions, Turk. J. Math., 41 (2017), 694–706. doi: 10.3906/mat-1602-25
|
| [14] | A. Cătaş, On certain classes of $p$-valent functions defined by multiplier transformations, Proceedings of the international symposium on geometric function theory and applications: GFTA, 2007,241–250. |
| [15] | P. L. Duren, Univalent functions, New York: Springer Verlag, 1983. |
| [16] | P. Goswami, B. S. Alkahtani, T. Bulboaca, Estimate for initial Maclaurin coefficients of certain subclasses of bi-univalent functions, arXiv. Available from: https://arXiv.org/abs/1503.04644. |
| [17] |
Q. Hu, H. M. Srivastava, B. Ahmad, N. Khan, M. G. Khan, W. K. Mashwani, et al., A subclass of multivalent Janowski type $q$-starlike functions and its consequences, Symmetry, 13 (2021), 1275. doi: 10.3390/sym13071275. doi: 10.3390/sym13071275
|
| [18] |
Q. H. Xu, H. G. Xiao, H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput., 218 (2012), 11461–11465. doi: 10.1016/j.amc.2012.05.034. doi: 10.1016/j.amc.2012.05.034
|
| [19] |
Q. Khan, M. Arif, M. Raza, G. Srivastava, H. Tang, S. ur Rehman, Some applications of a new integral operator in $q$-analog for multivalent functions, Mathematics, 7 (2019), 1178. doi: 10.3390/math7121178. doi: 10.3390/math7121178
|
| [20] |
B. Khan, Z. G. Liu, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, A study of some families of multivalent $q$-starlike functions involving higher-order $q$-derivatives, Mathematics, 8 (2020), 1470. doi: 10.3390/math8091470. doi: 10.3390/math8091470
|
| [21] | B. Khan, Z. G. Liu, H. M. Srivastava, N. Khan, M. Tahir, Applications of higher-order derivatives to subclasses of multivalent $q$-starlike functions, Maejo Int. J. Sci. Technol., 15 (2021), 61–72. |
| [22] |
N. Khan, M. Shafiq, M. Darus, B. Khan, Q. Z. Ahmad, Upper bound of the third Hankel determinant for a subclass of $q$-starlike functions associated with Lemniscate of Bernoulli, J. Math. Inequal., 14 (2020), 51–63. doi: 10.7153/jmi-2020-14-05. doi: 10.7153/jmi-2020-14-05
|
| [23] |
B. Khan, H. M. Srivastava, N. Khan, M. Darus, Q. Z. Ahmad, M. Tahir, Applications of certain conic domains to a subclass of $ q $-starlike functions associated with the Janowski functions, Symmetry, 13 (2021), 574. doi: 10.3390/sym13040574. doi: 10.3390/sym13040574
|
| [24] |
B. Khan, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, Q. Z. Ahmad, Coefficient estimates for a subclass of analytic functions associated with a certain leaf-like domain, Mathematics, 8 (2020), 1334. doi: 10.3390/math8081334. doi: 10.3390/math8081334
|
| [25] |
B. Khan, H. M. Srivastava, M. Tahir, M. Darus, Q. Z. Ahmad, N. Khan, Applications of a certain $q$-integral operator to the subclasses of analytic and bi-univalent functions, AIMS Math., 6 (2020), 1024–1039. doi: 10.3934/math.2021061. doi: 10.3934/math.2021061
|
| [26] | W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, In: Z. Li, F. Ren, L. Lang, S. Zhang, Proceedings of the conference on complex analysis, International Press Inc., 1 (1994), 157–169. |
| [27] |
N. Magesh, T. Rosy, S. Varma, Coefficient estimate problem for a new subclass of bi-univalent functions, J. Compl. Anal., 2013 (2013), 474231. doi: 10.1155/2013/474231. doi: 10.1155/2013/474231
|
| [28] | H. Orhan, M. Kamali, Neighborhoods of a class of analytic functions with negative coefficients, Acta Math. Acad. Paedagog. Nyhazi., 21 (2005), 55–61. |
| [29] |
H. Orhan, N. Magesh, V. K. Balaji, Certain classes of bi-univalent functions with bounded boundary variation, Tbilisi Math. J., 10 (2017), 17–27. doi: 10.1515/tmj-2017-0042. doi: 10.1515/tmj-2017-0042
|
| [30] |
H. Orhan, N. Magesh, V. K. Balaji, Initial coefficient bounds for a general class of bi-univalent functions, Filomat, 29 (2015), 1259–1267. doi: 10.2298/FIL1506259O. doi: 10.2298/FIL1506259O
|
| [31] |
K. S. Padmanabhan, R. Paravatham, Properties of a class of functions with bounded boundary rotation, Ann. Pol. Math., 31 (1976), 311–323. doi: 10.4064/ap-31-3-311-323
|
| [32] |
B. Pinchuk, Functions of bounded boundary rotation, Israel J. Math., 10 (1971), 6–16. doi: 10.1007/BF02771515. doi: 10.1007/BF02771515
|
| [33] | S. Prema, B. S. Keerthi, Coefficient bounds for certain subclasses of analytic functions, J. Math. Anal., 4 (2013), 22–27. |
| [34] |
M. S. Robertson, Variational formulas for several classes of analytic functions, Math. Z., 118 (1976), 311–319. doi: 10.1007/BF01109867. doi: 10.1007/BF01109867
|
| [35] | G. S. Sălăgean, Subclasses of univalent functions, In: C. A. Cazacu, N. Boboc, M. Jurchescu, I. Suciu, Complex analysis-fifth Romanian-finnish seminar, Lecture Notes in Mathematics, Springer-Verlag, 1013 (1983), 362–372. doi: 10.1007/BFb0066543. |
| [36] |
R. Singh, On Bazilevič functions, Proc. Am. Math. Soc., 38 (1973), 261–271. doi: 10.1090/S0002-9939-1973-0311887-9. doi: 10.1090/S0002-9939-1973-0311887-9
|
| [37] |
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. doi: 10.1007/s40995-019-00815-0. doi: 10.1007/s40995-019-00815-0
|
| [38] | H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: Univalent functions, fractional calculus and their applications, Halsted Press, 1989,329–354. |
| [39] |
H. M. Srivastava, M. Arif, M. Raza, Convolution properties of meromorphically harmonic functions defined by a generalized convolution $q$-derivative operator, AIMS Math., 6 (2021), 5869–5885. doi: 10.3934/math.2021347. doi: 10.3934/math.2021347
|
| [40] |
H. M. Srivastava, D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, J. Egypt. Math. Soc., 23 (2015), 242–246. doi: 10.1016/j.joems.2014.04.002. doi: 10.1016/j.joems.2014.04.002
|
| [41] |
H. M. Srivastava, S. S. Eker, R. M. Ali, Coefficient estimates for a certain class of analytic and bi-univalent functions, Filomat, 29 (2015), 1839–1845. doi: 10.2298/FIL1508839S. doi: 10.2298/FIL1508839S
|
| [42] |
H. M. Srivastava, S. Gaboury, F. Ghanim, Initial coefficient estimates for some subclasses of $m$-fold symmetric bi-univalent functions, Acta Math. Sci., 36 (2016), 863–871. doi: 10.1016/S0252-9602(16)30045-5. doi: 10.1016/S0252-9602(16)30045-5
|
| [43] |
H. M. Srivastava, S. Gaboury, F. Ghanim, Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat., 28 (2017), 693–706. doi: 10.1007/s13370-016-0478-0. doi: 10.1007/s13370-016-0478-0
|
| [44] | H. M. Srivastava, S. B. Joshi, S. S. Joshi, H. Pawar, Coefficient estimates for certain subclasses of meromorphically bi-univalent functions, Palest. J. Math., 5 (2016), 250–258. |
| [45] |
H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188–1192. doi: 10.1016/j.aml.2010.05.009. doi: 10.1016/j.aml.2010.05.009
|
| [46] |
H. M. Srivastava, A. O. Mostafa, M. K. Aouf, H. M. Zayed, Basic and fractional $q$-calculus and associated Fekete-Sezgö problem for $p$-valent $q$-starlike functions and $p$-valent $q$-convex functions of complex order, Miskolc Math. Notes, 20 (2019), 489–509. doi: 10.18514/MMN.2019.2405. doi: 10.18514/MMN.2019.2405
|
| [47] |
H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad, N. Khan, Some general families of $q$-starlike functions associated with the Janowski functions, Filomat, 33 (2019), 2613–2626. doi: 10.2298/FIL1909613S. doi: 10.2298/FIL1909613S
|
| [48] |
H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad, N. Khan, Some general classes of $q$-starlike functions Aassociated with the Janowski functions, Symmetry, 11 (2019), 292. doi: 10.3390/sym11020292. doi: 10.3390/sym11020292
|
| [49] |
H. M. Srivastava, B. Khan, N. Khan, M. Tahir, S. Ahmad, N. Khan, Upper bound of the third Hankel determinant for a subclass of $ q $-starlike functions associated with the $q$-exponential function, Bull. Sci. Math., 167 (2021), 102942. doi: 10.1016/j.bulsci.2020.102942. doi: 10.1016/j.bulsci.2020.102942
|
| [50] |
H. Tang, H. M. Srivastava, S. Sivasubramanian, P. Gurusamy, The Fekete-Szegö functional problems for some subclasses of $m$ -fold symmetric bi-univalent functions, J. Math. Ineq., 10 (2016), 1063–1092. doi: 10.7153/jmi-10-85. doi: 10.7153/jmi-10-85
|
| [51] | D. K. Thomas, On Bazilevič functions, Am. Math. Soc., 132 (1968), 353–361. |
| [52] |
D. K. Thomas, On the coefficients of gamma-starlike functions, J. Korean Math. Soc., 55 (2018), 175–184. doi: 10.4134/JKMS.j170105. doi: 10.4134/JKMS.j170105
|
S. M. Madian. Some properties for certain class of bi-univalent functions defined by $ q $-Cătaş operator with bounded boundary rotation[J]. AIMS Mathematics, 2022, 7(1): 903-914. doi: 10.3934/math.2022053
DownLoad: