Subordination problems for a new class of Bazilevič functions associated with $ k $-symmetric points and fractional $ q $-calculus operators

Haiyan Zhou; K. A. Selvakumaran; S. Sivasubramanian; S. D. Purohit; Huo Tang; Haiyan Zhou; K. A. Selvakumaran; S. Sivasubramanian; S. D. Purohit; Huo Tang

doi:10.3934/math.2021502

AIMS Mathematics

2021, Volume 6, Issue 8: 8642-8653. doi: 10.3934/math.2021502

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Subordination problems for a new class of Bazilevič functions associated with $ k $-symmetric points and fractional $ q $-calculus operators

1.
School of Mathematics and Computer Sciences, Chifeng University, Chifeng 024000, China
2.
Department of Mathematics, R. M. K College of Engineering and Technology, Puduvoyal 601206, Tamil Nadu, India
3.
Department of Mathematics, University College of Engineering Tindivanam, Anna University (Chennai), Tindivanam 604001, Tamil Nadu, India
4.
Department of HEAS (Mathematics), Rajasthan Technical University, Kota 324010, Rajasthan, India

Received: 22 February 2021 Accepted: 26 May 2021 Published: 07 June 2021
MSC : 30C45, 30C80

In this article, we introduce and investigate a new class of Bazilevič functions with respect to $ k $-symmetric points defined by using fractional $ q $-calculus operators that are analytic in the open unit disk $ \mathbb{D} $. Several interesting subordination problems are also derived for the functions belonging to this new class.
- univalent functions,
- starlike with respect to symmetric points,
- Bazilevič functions,
- fractional $ q $-calculus operators,
- subordination
Citation: Haiyan Zhou, K. A. Selvakumaran, S. Sivasubramanian, S. D. Purohit, Huo Tang. Subordination problems for a new class of Bazilevič functions associated with $ k $-symmetric points and fractional $ q $-calculus operators[J]. AIMS Mathematics, 2021, 6(8): 8642-8653. doi: 10.3934/math.2021502

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Abstract

In this article, we introduce and investigate a new class of Bazilevič functions with respect to $ k $-symmetric points defined by using fractional $ q $-calculus operators that are analytic in the open unit disk $ \mathbb{D} $. Several interesting subordination problems are also derived for the functions belonging to this new class.

References

[1]	M. H. Abu Risha, M. H. Annaby, M. E. H. Ismail, Z. S. Mansour, Linear $q$-difference equations, Z. Anal. Anwend., 26 (2007), 481-494.
[2]	D. Albayrak, S. D. Purohit, F. Uçar, On $q$-analogues of Sumudu transforms, An. Şt. Univ. Ovidius Constanţa, 21 (2013), 239-260.
[3]	F. M. Al-Oboudi, On univalent functions defined by a generalized Sălăgean operator, Int. J. Math. Math. Sci., 27 (2004), 1429-1436.
[4]	G. Bangerezako, Variational calculus on $q$-nonuniform lattices, J. Math. Anal. Appl., 306 (2005), 161-179. doi: 10.1016/j.jmaa.2004.12.029
[5]	T. Ernst, A Comprehensive Treatment of $q$-Calculus, Basel: Birkhäuser, 2012.
[6]	G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge University Press, 1990.
[7]	D. J. Hallenbeck, S. Ruscheweyh, Subordination by convex functions, Proc. Am. Math. Soc., 52 (1975), 191-195. doi: 10.1090/S0002-9939-1975-0374403-3
[8]	V. Kac, P. Cheung, Quantum Calculus, New York: Springer, 2002.
[9]	Z. S. I. Mansour, Linear sequential $q$-difference equations of fractional order, Fractional Calculus Appl. Anal., 12 (2009), 159-178.
[10]	S. S. Miller, P. T. Mocanu, Differential Subordinations: Theory and Applications, Series in Pure and Applied Mathematics, New York, 2000.
[11]	S. D. Purohit, R. K. Raina, Certain subclasses of analytic functions associated with fractional $q$-calculus operators, Math. Scand., 109 (2011), 55-70. doi: 10.7146/math.scand.a-15177
[12]	S. D. Purohit, A new class of multivalently analytic functions associated with fractional $q$-calculus operators, Fractional Differ. Calculus, 2 (2012), 129-138.
[13]	S. D. Purohit, K. A. Selvakumaran, On certain generalized $q$-integral operators of analytic functions, Bull. Korean Math. Soc., 52 (2015), 1805-1818. doi: 10.4134/BKMS.2015.52.6.1805
[14]	P. M. Rajković, S. D. Marinković, M. S. Stanković, Fractional integrals and derivatives in $q$-calculus, Appl. Anal. Discrete Math., 1 (2007), 311-323. doi: 10.2298/AADM0701072C
[15]	K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11 (1959), 72-75.
[16]	G. Ş. 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, Berlin: Springer, 1983.
[17]	K. A. Selvakumaran, S. D. Purohit, A. Secer, M. Bayram, Convexity of certain $q$-integral operators of $p$-valent functions, Abstr. Appl. Anal., 2014 (2014), 925902.
[18]	K. A. Selvakumaran, S. D. Purohit, A. Secer, Majorization for a class of analytic functions defined by $q$-differentiation, Math. Probl. Eng., 2014 (2014), 653917.
[19]	T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J., 37 (1970), 775-777.

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