Applications of $ q- $Ultraspherical polynomials to bi-univalent functions defined by $ q- $Saigo's fractional integral operators

Tariq Al-Hawary; Ala Amourah; Abdullah Alsoboh; Osama Ogilat; Irianto Harny; Maslina Darus; Tariq Al-Hawary; Ala Amourah; Abdullah Alsoboh; Osama Ogilat; Irianto Harny; Maslina Darus

doi:10.3934/math.2024828

AIMS Mathematics

2024, Volume 9, Issue 7: 17063-17075. doi: 10.3934/math.2024828

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Applications of $ q- $Ultraspherical polynomials to bi-univalent functions defined by $ q- $Saigo's fractional integral operators

1.
Department of Applied Science, Ajloun College, Al Balqa Applied University, Ajloun 26816, Jordan
2.
Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 3111, Oman
3.
Department of Mathematics, Faculty of Science, Philadelphia University, 19392 Amman, Jordan
4.
Department of Basic Sciences, Faculty of Arts and Science, Al-Ahliyya Amman University, Amman 19328, Jordan
5.
Department General Education, Faculty of Resilience, Rabdan Academy, Abu Dhabi, United Arab Emirates
6.
Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received: 06 December 2023 Revised: 04 April 2024 Accepted: 10 April 2024 Published: 16 May 2024
MSC : 30C45

This study established upper bounds for the second and third coefficients of analytical and bi-univalent functions belonging to a family of particular classes of analytic functions utilizing $ q- $Ultraspherical polynomials under $ q- $Saigo's fractional integral operator. We also discussed the Fekete-Szegö family function problem. As a result of the specialization of the parameters used in our main results, numerous novel outcomes were demonstrated.
- $ q- $Ultraspherical polynomials,
- univalent functions,
- bi-univalent functions,
- $ q- $calculus,
- analytic functions
Citation: Tariq Al-Hawary, Ala Amourah, Abdullah Alsoboh, Osama Ogilat, Irianto Harny, Maslina Darus. Applications of $ q- $Ultraspherical polynomials to bi-univalent functions defined by $ q- $Saigo's fractional integral operators[J]. AIMS Mathematics, 2024, 9(7): 17063-17075. doi: 10.3934/math.2024828

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Abstract

This study established upper bounds for the second and third coefficients of analytical and bi-univalent functions belonging to a family of particular classes of analytic functions utilizing $ q- $Ultraspherical polynomials under $ q- $Saigo's fractional integral operator. We also discussed the Fekete-Szegö family function problem. As a result of the specialization of the parameters used in our main results, numerous novel outcomes were demonstrated.

References

[1]	A. Alsoboh, A. Amourah, M. Darus, R. I. Al Sharefeen, Applications of Neutrosophic $q–$Poisson Distribution Series for subclass of Analytic Functions and bi-univalent functions, Mathematics, 11 (2023), 868. http://doi.org/10.3390/math11040868 doi: 10.3390/math11040868
[2]	A. Amourah, A. Alsoboh, O. Ogilat, G. M. Gharib, R. Saadeh, M. A. Al Soudi, Generalization of Gegenbauer Polynomials and Bi-Univalent Functions, Axioms, 12, (2023), 128. http://doi.org/10.3390/axioms12020128 doi: 10.3390/axioms12020128
[3]	A. Amourah, B. A. Frasin, G. Murugusundaramoorthy, T. Al-Hawary, Bi-Bazilevič functions of order $\vartheta+i\delta$ associated with $(p, q)-$Lucas polynomials, AIMS Mathematics, 6 (2021), 4296–4305. https://doi.org/10.3934/math.2021254 doi: 10.3934/math.2021254
[4]	A. Amourah, T. Al-Hawary, B. A. Frasin, Application of Chebyshev polynomials to certain class of bi-Bazilevič functions of order $\vartheta +i\beta $, Afr. Mat., 32 (2021), 1–8.
[5]	A. Amourah, M. Alomari, F. Yousef, A. Alsoboh, Consolidation of a Certain Discrete Probability Distribution with a Subclass of Bi-Univalent Functions Involving Gegenbauer Polynomials, Math. Probl. Eng., 2022 (2022), 6354994. http://doi.org/10.1155/2022/6354994 doi: 10.1155/2022/6354994
[6]	R. Askey, M. E. H. Ismail, A generalization of ultraspherical polynomials, In: Studies of Pure Mathematics, Basel: Birkhauser, 1983. http://doi.org/10.1007/978-3-0348-5438-2_6
[7]	R. Askey, J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society, 1985. https://doi.org/10.1090/memo/0319
[8]	D. A. Brannan, J. G. Clunie, Aspects of contemporary complex analysis, New York, London: Academic Press, 1980.
[9]	L. Carlitz, Some polynomials related to the Hermite polynomials, Duke Math. J., 26 (1959), 429–444.
[10]	R. Chakrabarti, R. Jagannathan, S. S. Naina Mohammed, New connection formulae for the $q$-orthogonal polynomials via a series expansion of the $q$–exponential, J. Phys. A: Math. Gen., 39 (2006), 12371.
[11]	G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge: Cambridge university press, 2004.
[12]	F. H. Jackson, On $q$-functions and a certain difference operator, Earth Environ. Sci. Trans., 46 (1909), 253–281. https://doi.org/10.1017/S0080456800002751 doi: 10.1017/S0080456800002751
[13]	M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63–68. https://doi.org/10.1090/S0002-9939-1967-0206255-1 doi: 10.1090/S0002-9939-1967-0206255-1
[14]	S. S. Miller, P. T. Mocanu, Second Order Differential Inequalities in the Complex Plane. J. Math. Anal. Appl., 65 (1978), 289–305.
[15]	S. S. Miller, P. T. Mocanu, Differential Subordinations and Univalent Functions. Mich. Math. J., 28 (1981), 157–172.
[16]	S. S. Miller, P. T. Mocanu, Differential Subordinations. Theory and Applications, New York: Marcel Dekker, 2000.
[17]	E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $ \|z\|<1$, Arch. Ration. Mech. An., 32 (1969), 100–112.
[18]	S. D. Purohit, R. K. Raina, Fractional q-calculus and certain subclass of univalent analytic functions, Mathematica, 55 (2013), 62–74.
[19]	S. D. Purohit, R. K. Raina, Some classes of analytic and multivalent functions associated with q-derivative operators, Acta Univ. Sapientiae, Math., 6 (2014), 5–23. https://doi.org/10.2478/ausm-2014-0015 doi: 10.2478/ausm-2014-0015
[20]	S. D. Purohit, R. K. Raina, On a subclass of p-valent analytic functions involving fractional q-calculus operators, KJS, 42 (2015), 1.
[21]	S. D. Purohit, R.K. Raina, Certain subclasses of analytic functions associated with fractional q-calculus operators, Math. Scand., 109 (2011), 55–70. https://doi.org/10.7146/math.scand.a-15177 doi: 10.7146/math.scand.a-15177
[22]	A. B. Patil, T. G. Shaba, Sharp initial coefficient bounds and the Fekete-Szegö problem for some certain subclasses of analytic and bi-univalent functions, Ukrains'kyi Matematychnyi Zhurnal, 75 (2023), 198–206. https://doi.org/10.37863/umzh.v75i2.6602 doi: 10.37863/umzh.v75i2.6602
[23]	C. Quesne, Disentangling $q$-Exponentials: A General Approach, Int. J. Theor. Phys., 43 (2004), 545–559. https://doi.org/10.1023/B:IJTP.0000028885.42890.f5 doi: 10.1023/B:IJTP.0000028885.42890.f5
[24]	N. Ravikumar, Certain classes of analytic functions defined by fractional $q$-calculus operator, Acta Univ. Sapientiae, Math., 10 (2018), 178–188.
[25]	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. https://doi.org/10.1007/s40995-019-00815-0 doi: 10.1007/s40995-019-00815-0
[26]	M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Kyushu University, 11 (1978), 135–143.
[27]	H. Srivastava, S. Owa, Univalent Functions, Fractional Calculus and Their Applications, New Jersey: John Wiley and Sons, 1989.
[28]	F. Yousef, B. A. Frasin, T. Al-Hawary, Fekete-Szegö inequality for analytic and bi-univalent functions subordinate to Chebyshev polynomials, Filomat, 32 (2018), 3229–3236. https://doi.org/10.2298/FIL1809229Y doi: 10.2298/FIL1809229Y
[29]	P. Zaprawa, On the Fekete-Szegö problem for classes of bi-univalent functions, Bull. Belg. Math. Soc.-Simon Stevin 21 (2014), 169–178. https://doi.org/10.36045/bbms/1394544302 doi: 10.36045/bbms/1394544302

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