Inclusion relations of $ q $-Bessel functions associated with generalized conic domain

Shahid Khan; Saqib Hussain; Maslina Darus; Shahid Khan; Saqib Hussain; Maslina Darus

doi:10.3934/math.2021216

AIMS Mathematics

2021, Volume 6, Issue 4: 3624-3640. doi: 10.3934/math.2021216

Previous Article Next Article

Research article Special Issues

Inclusion relations of $ q $-Bessel functions associated with generalized conic domain

1.
Department of Mathematics, Riphah International University Islamabad 44000, Pakistan
2.
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus 22060, Pakistan
3.
Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia

Received: 01 December 2020 Accepted: 22 December 2020 Published: 22 January 2021
MSC : 05A30, 30C45, 11B65, 47B38

In this paper, we investigate the geometric properties of Jackson and Hahn-Exton $ q $-Bessel functions and perform their normalization for the analyticity in open unit disk $ E $. By applying normalized Jackson and Hahn-Exton $ q $-Bessel functions and idea of convolution we introduce a new operator and define new family of subclasses of analytic functions related with generalized conic domain. For these subclasses of analytic functions, we investigate inclusion relations and integral preserving properties. Also we will use $ q $-Bernardi integral operator to discuss some applications of our main results.
- analytic functions,
- $ q $-calculus,
- $ q $-derivative,
- $ q $-starlike functions,
- $ q $-convex functions,
- $ q $-Bessel functions,
- generalized conic domain $ \Omega _{k, q} $
Citation: Shahid Khan, Saqib Hussain, Maslina Darus. Inclusion relations of $ q $-Bessel functions associated with generalized conic domain[J]. AIMS Mathematics, 2021, 6(4): 3624-3640. doi: 10.3934/math.2021216

Related Papers:

Abstract

In this paper, we investigate the geometric properties of Jackson and Hahn-Exton $ q $-Bessel functions and perform their normalization for the analyticity in open unit disk $ E $. By applying normalized Jackson and Hahn-Exton $ q $-Bessel functions and idea of convolution we introduce a new operator and define new family of subclasses of analytic functions related with generalized conic domain. For these subclasses of analytic functions, we investigate inclusion relations and integral preserving properties. Also we will use $ q $-Bernardi integral operator to discuss some applications of our main results.

References

[1]	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
[2]	S. D. Bernardi, Convex and starlike univalent functions, Trans. Am. Math. Soc., 135 (1969), 429–446. doi: 10.1090/S0002-9947-1969-0232920-2
[3]	P. Eeinigenburg, S. S. Miller, P. T. Mocanu, M. D. Reade, General inequalities, Birkhäuser, Basel, 64 (1983), 339–348.
[4]	S. Hussain, S. Khan, M. A. Zaighum, M. Darus, Applications of a $q$-Salagean type operator on multivalent function, J. Inequalities Appl., 1 (2018), 301–312.
[5]	M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Variables, Theory Appl., 14 (1990), 77–84. doi: 10.1080/17476939008814407
[6]	M. E. H. Ismail, M. E. Muldoon, On the variation with respect to a parameter of zeros of Bessel and $q$-Bessel functions, J. Math. Anal. Appl., 135 (1988), 187–207. doi: 10.1016/0022-247X(88)90148-5
[7]	F. H. Jackson, On $q$-functions and a certain difference operator, Trans. R. Soc. Edinburgh, 46 (1908), 253–281.
[8]	F. H. Jackson, On $q$-definite integrals, Q. J. Pure Appl. Math., 41 (1910), 193–203.
[9]	S. Kanas, S. R. Mondal, A. D. Mohammed, Relations between the generalized Bessel functions and the Janowski class, Math. Inequalities Appl., 21 (2018), 165–178.
[10]	S. Kanas, D. Raducanu, Some class of analytic functions related to conic domains, Math. Slovaca, 64 (2014), 1183–1196.
[11]	S. Kanas, H. M. Srivastava, Linear operators associated with $k$ -uniform convex functions, Integr. Transforms Spec. Funct., 9 (2000), 121–132. doi: 10.1080/10652460008819249
[12]	S. Kanas, A. Wisniowska, Conic regions and $k$-uniform convexity Ⅱ, Zeszyty Nauk. Politech. Rzeszowskiej Mat., 22 (1998), 65–78.
[13]	S. Kanas, A. Wisniowska, Conic regions and $k$-uniform convexity, J. Comput. Appl. Math., 105 (1999), 327–336. doi: 10.1016/S0377-0427(99)00018-7
[14]	S. Kanas, A. Wisniowska, Conic domains and $k$-starlike functions, Rev. Roum. Math. Pure Appl., 45 (2000), 647–657.
[15]	N. Khan, B. Khan, Q. Z. Ahmad, S. Ahmad, Some convolution properties of multivalent analytic functions, AIMS Math., 2 (2017), 260–268. doi: 10.3934/Math.2017.2.260
[16]	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), 1–12.
[17]	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.
[18]	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), 1–15.
[19]	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.
[20]	H. T. Koelink, R. F. Swarttouw, On the zeros of the Hahn-Exton $ q$-Bessel function and associated $q$-Lommel polynomials, J. Math. Anal. Appl., 186 (1994), 690–710. doi: 10.1006/jmaa.1994.1327
[21]	T. H. Koornwinde, R. F. Swarttouw, On $q$-analogues of the hankel and fourier transforms, Trans. Amer. Math. Soc., 333 (1992), 445–461.
[22]	S. Mahmood, Q. Z. Ahmad, H. M. Srivastava, N. Khan, B. Khan, M. Tahir, A certain subclass of meromorphically $q$-starlike functions associated with the Janowski functions, J. Inequalities Appl., 2019 (2019), 1–11. doi: 10.1186/s13660-019-1955-4
[23]	A. Mahammed, M. Darus, A generalized operator involving the $q$-hypergeometric function, Matematichki Vesnik, 65 (2013), 454–465.
[24]	S. Mahmood, N. Raza, E. S. A. Abujarad, G. Srivastava, H. M. Srivastava, S. N. Malik, Geometric properties of certain classes of analytic functions associated with a $q$-integral operator, Symmetry, 11 (2019), 1–14.
[25]	S. Mahmood, H. M. Srivastava, N. Khan, Q. Z. Ahmad, B. Khan, I. Ali, Upper bound of the third Hankel determinant for a subclass of $q$-Starlike functions, Symmetry, 11 (2019), 1–13.
[26]	S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Mich. Math. J., 28 (1981), 157–172. doi: 10.1307/mmj/1029002507
[27]	S. S. Miller, P. T. Mocanu, Differential subordinations and inequalities in complex plane, J. Differ. Equations, 67 (1987), 199–211. doi: 10.1016/0022-0396(87)90146-X
[28]	K. I. Noor, S. Riaz, M. A. Noor, On $q$-Bernardi integral operator, TWMS J. Pure Appl. Math., 8 (2017), 3–11.
[29]	C. Ramachandran, S. Annamalai, S. Sivasubramanian, Inclusion relations for Bessel functions for domains bounded by conical domains, Adv. Differ. Equations, 2014 (2014), 288. doi: 10.1186/1687-1847-2014-288
[30]	M. Raza, M. D. Din, S. N. Malik, Certain geometric properties of normalized wright functions, J. Funct. Spaces, 2016 (2016), 1896154.
[31]	M. S. Rehman, Q. Z. Ahmad, B. Khan, M. Tahir, N. Khan, Generalisation of certain subclasses of analytic and univalent functions, Maejo Internat. J. Sci. Technol., 13 (2019), 1–9.
[32]	M. S. Rehman, Q. Z. Ahmad, H. M. Srivastava, B. Khan, N. Khan, Partial sums of generalized $q$-Mittag-Leffler functions, AIMS Math., 5 (2019), 408–420.
[33]	M. S. Rehman, Q. Z. Ahmad, H. M. Srivastava, N. Khan, M. Darus, B. Khan, Applications of higher-order $q$-derivatives to the subclass of $q$-starlike functions associated with the Janowski functions, AIMS Math., 6 (2020), 1110–1125.
[34]	M. S. Robertson, On the theory of univalent functions, Ann. Math., 37 (1936), 374–408. doi: 10.2307/1968451
[35]	M. Shafiq, N. Khan, H. M. Srivastava, B. Khan, Q. Z. Ahmad, M. Tahir, Generalisation of close-to-convex functions associated with Janowski functions, Maejo Internat. J. Sci. Technol., 14 (2020), 141–155.
[36]	L. Shi, Q. Khan, G. Srivastava, J. L. Liu, M. Arif, A study of multivalent $q$-starlike functions connected with circular domain, Mathematics, 7 (2019), 670. doi: 10.3390/math7080670
[37]	H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: H. M. Srivastava, S. Owa, Univalent functions, Fractional Calculus, and Their Applications: Srivastava, New York: John Wiley & Sons, (1989), 329–354.
[38]	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), 327–344. doi: 10.1007/s40995-019-00815-0
[39]	H. M. Srivastava, Q. Z. Ahmad, N. Khan, N. Khan, B. Khan, Hankel and toeplitz determinants for a subclass of $q$-starlike functions associated with a general conic domain, Mathematics, 7 (2019), 1–15.
[40]	H. M. Srivastava, S. Khan, Q. Z. Ahmad, N. Khan, S. Hussain, The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain $q$-integral operator, Stud. Univ. Babes-Bolyai Math., 63 (2018), 419–436. doi: 10.24193/subbmath.2018.4.01
[41]	H. M. Srivastava, B. Khan, N. Khan, Q. Z. Ahmad, Coefficient inequalities for $q$-starlike functions associated with the Janowski functions, Hokkaido Math. J., 48 (2019), 407–425. doi: 10.14492/hokmj/1562810517
[42]	H. M. Srivastava, B. Khan, N. Khan, Q. Z. Ahmad, M. Tahir, A generalized conic domain and its applications to certain subclasses of analytic functions, Rocky Mountain J. Math., 49 (2019), 2325–2346. doi: 10.1216/RMJ-2019-49-7-2325
[43]	H. M. Srivastava, S. Owa, Current Topics in Analytic Function Theory, River Edge: World Scientific, 1992.
[44]	H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad, N. Khan, Some general classes of $q$-starlike functions associated with the Janowski functions, Symmetry, 11 (2019), 1–14.
[45]	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
[46]	H. E. O. Uçar, Coefficient inequality for $q$-starlike functions, Appl. Math. Comput., 76 (2016), 122–126.

Reader Comments

Your name:*

Email:*
© 2021 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)