Quasi-subordination for a subclass of non-Bazilevic functions connected with a q-derivative operator

Authors

  • Risikat Ayodeji Bello
    Department of Mathematics and Statistics, Kwara State University, PMB 1530, Malete, Kwara State, Nigeria
  • Ayotunde Olajide Lasode
    Department of Mathematics, Federal College of Education, PMB 11, Ilawe-Ekiti, Ekiti State, Nigeria
  • Rasheed Olawale Ayinla
    Department of Mathematics and Statistics, Kwara State University, PMB 1530, Malete, Kwara State, Nigeria
  • Adediran Dauda Adeshola
    Department of Mathematics and Statistics, Kwara State University, PMB 1530, Malete, Kwara State, Nigeria

Keywords:

Non-Bazilevic functions, Modified Opoola q-differential operator, q-calculus, Quasi-subordination, Fekete-Szego inequality

Abstract

This study examines a broad subclass of non-Bazileviˇc functions that includes several subclasses of q-bounded turning functions and q-Sakaguchi functions. We link the definition of the class with a modified Opoola q-derivative operator, quasi-subordination, and a few number of mathematical concepts such as q-calculus and infinite series formations. Among the achievements in this work are the estimates for the early upper coefficient bounds and the Fekete-Szego inequalities having complex parameters. In general, this unique class reduces to various recognized classes of non-Bazilevic functions when some of the parameters take values within their interval of definition.

Dimensions

[1] M. S. Robertson, “Quasi-subordination and coefficients conjectures”, Bulletin of American Mathematical Society 76 (1970) 1. https://doi.org/10.1090/S0002-9904-1970-12356-4. DOI: https://doi.org/10.1090/S0002-9904-1970-12356-4

[2] T. H. MacGregor, “Majorization by univalent functions”, Duke Mathematical Journal 34 (1967) 1. https://doi.10.1215/S0012-7094-67-03411-4. DOI: https://doi.org/10.1215/S0012-7094-67-03411-4

[3] R. O. Ayinla & A. O. Lasode, “Estimates for a class of analytic and univalent functions connected with quasi-subordination”, Proceedings of the Nigerian Society of Physical Sciences Conferences 1 (2024) 82. https://doi.org/10.61298/pnspsc.2024.1.82. DOI: https://doi.org/10.61298/pnspsc.2024.1.82

[4] I. E. Bazileviˇc, “On a case of integrability by quadratures of the equation of Loewner Kufarev”, Matematicheskii Sbornik 37 (1955) 79. http://mi.mathnet.ru/eng/sm5227.

[5] M. Obradovic, “A class of univalent functions”, Hokkaido Mathematical Journal 27 (1998) 2. https://doi.org/10.14492/HOKMJ/1351001289. DOI: https://doi.org/10.14492/hokmj/1351001289

[6] S. G. Shah, S. Al-Sa’di, S. Hussain, A. Tasleem, A. Rasheed, I. Z. Cheema & M. Darus, “Fekete-Szeg¨o functional for a class of non-Bazileviˇc functions related to quasi-subordination”, Demonstratio Mathematica 56 (2023) 1. https://doi.org/10.1515/dema-2022-0232. DOI: https://doi.org/10.1515/dema-2022-0232

[7] S. Umar, M. Arif, M. Raza & S. K. Lee, “On a subclass related to Bazileviˇc functions”, AIMS Mathematics 5 (2020) 3. https://doi.org/10.3934/math.2020135. DOI: https://doi.org/10.3934/math.2020135

[8] W. Janowski, “Some extremal problems for certain families of analytic functions I”, Annales Polonici Mathematici 28 (1973) 3. https://doi.org/10.4064/ap-28-3-297-326. DOI: https://doi.org/10.4064/ap-28-3-297-326

[9] Y. Polatoˇglu, S. M. Bolcal, A. S¸en & E. Yavuz, “A study on the generalization of Janowski functions in the unit disc”, Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 22 (2006) 27. https://www.emis.de/journals/AMAPN/vol22_1/amapn22 04.pdf.

[10] F. H. Jackson, “On q-functions and a certain difference operator”, Transactions of the Royal Society of Edinburgh 46 (1908) 2. https://doi.org/10.1017/S0080456800002751. DOI: https://doi.org/10.1017/S0080456800002751

[11] A. A. Alatawi & M. Darus, “On a certain subclass of analytic functions involving the modified q-Opoola derivative operator”, International Journal of Nonlinear Analysis and Applications 14 (2023) 5. http://dx.doi.org/10.22075/ijnaa.2023.29137.4072.

[12] A. Aral, V. Gupta & R. P. Agarwal, Applications of q-calculus in operator theory, Springer Science+Business Media, New York, 2013, pp. 3–5. https://doi.org/10.1007/978-1-4614-6946-9_1. DOI: https://doi.org/10.1007/978-1-4614-6946-9

[13] A. O. Lasode & T. O. Opoola, “Some properties of a family of univalent functions defined by a generalized Opoola differential operator”, General Mathematics 30 (2022) 1. https://doi.org/10.2478/gm-2022-0001. DOI: https://doi.org/10.2478/gm-2022-0001

[14] A. O. Lasode & T. O. Opoola, “Some properties of a class of generalized Janowski-type q-starlike functions associated with Opoola q-differential operator and q-differential subordination”, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics 73 (2024) 2. https://doi.org/10.31801/cfsuasmas.1281348. DOI: https://doi.org/10.31801/cfsuasmas.1281348

[15] A. Saliu & S. O. Oladejo, “On lemniscate of Bernoulli of q-Janowski type”, Journal of the Nigerian Society of Physical Sciences 4 (2022) 961. https://doi.org/10.46481/jnsps.2022.961. DOI: https://doi.org/10.46481/jnsps.2022.961

[16] M. Govindaraj & S. Sivasubramanian, “On a class of analytic functions related to conic domains involving q-calculus”, Analysis Mathematica 43 (2017) 3. https://doi.org/10.1007/s10476-017-0206-5. DOI: https://doi.org/10.1007/s10476-017-0206-5

[17] R. A. Bello, “Coefficient estimate for a subclass of analytic functions defined by a generalized differential operator”, International Journal of Advances in Scientific Research and Engineering 9 (2023) 7. https://doi.org/10.31695/IJASRE.2023.9.7.8. DOI: https://doi.org/10.31695/IJASRE.2023.9.7.8

[18] R. A. Bello & T. O. Opoola, “On some properties of a class of analytic Functions defined by Sˇalˇagean differential operator”, Journal of Progressive Research in Mathematics 18 (2021) 3. https://scitecresearch.com/ojs-3.2.1-5/index.php/jprm/article/view/1985.

[19] L. M. Fatunsin, “On a new subclass of univalent functions defined by Sˇalˇagean differential operator”, Numerical Analysis and Applicable Mathematics 3 (2022) 25. https://doi.org/10.36686/Ariviyal.NAAM.2022.03.06.014.

[20] G. S. Sˇalˇagean, “Subclasses of univalent functions”, Lecture Notes in Mathematics 1013 (1983) 362. https://doi.org/10.1007/BFb0066543. DOI: https://doi.org/10.1007/BFb0066543

[21] R. O. Ayinla & R. A. Bello, “Toeplitz determinants for a subclass of analytic functions.”, Journal of Progressive Research Mathematics 18 (2021) 1. http://scitecresearch.com/journals/index.php/jprm/article/view/2034/1426.

[22] F. M. Al-Oboudi, “On univalent functions defined by a generalised Sˇalˇagean operator”, International Journal of Mathematics and Mathematical Sciences 2004 (2004) 27. http://dx.doi.org/10.1155/S0161171204108090. DOI: https://doi.org/10.1155/S0161171204108090

[23] M. K. Aouf & F. Z. El-Emam, “Fekete-Szeg¨o problems for certain classes of meromorphic functions involving q-Al-Oboudi differential operator”, Journal of Mathematics 2022 (2022) 1. https://doi.org/10.1155/2022/4731417. DOI: https://doi.org/10.1155/2022/4731417

[24] T. O. Opoola, “On a subclass of univalent functions defined by a generalised differential operator”, International Journal of Mathematical Analysis 11 (2017) 18. https://doi.org/10.12988/ijma.2017.7232. DOI: https://doi.org/10.12988/ijma.2017.7232

[25] L. M. Fatunsin & T. O. Opoola, “New results on subclasses of analytic functions defined by Opoola differential operator”, Journal of Mathematics and System Science 7 (2017) 295. https://doi.org/10.17265/2159-5291/2017.10.003. DOI: https://doi.org/10.17265/2159-5291/2017.10.003

[26] R. A. Bello, “On some properties of a class of analytic functions defined by Opoola differential operator”, International Journal of Mathematics and Statistics Studies 12 (2023) 4. https://doi.org/10.37745/ijmss.13/vol12n45261. DOI: https://doi.org/10.37745/ijmss.13/vol12n45261

[27] H. Shiraishi & T. Hayami, “Coefficient estimates for Schwarz functions”, (2013) 1. https://doi.org/10.48550/arXiv.1302.6916.

[28] R. M. Ali, V. Ravichandran & N. Seenivasagan, “Coefficient bounds for p-valent functions”, Applied Mathematics and Computation 187 (2007) 11. https://doi.org/10.1016/j.amc.2006.08.100. DOI: https://doi.org/10.1016/j.amc.2006.08.100

[29] M. Nunokawa, M. Obradoviˇc & S. Owa, “One criterion for univalency”, Proceedings of the American Mathematical Society 106 (1989) 4. https://doi.org/10.1090/S0002-9939-1989-0975653-5. DOI: https://doi.org/10.2307/2047290

[30] P. Sharma & R. K. Raina, “On a Sakaguchi-type class of univalent functions associated with quasi-subordination”, Commentarii Mathematici Universitatis Sancti Pauli 64 (2015) 59. https://core.ac.uk/download/pdf/293152375.pdf.

[31] S. Owa, T. Sekine & R. Yamakawa, “On Sakaguchi-type functions”, Applied Mathematics and Computation 187 (2007) 133. https://doi.org/10.1016/j.amc.2006.08.133. DOI: https://doi.org/10.1016/j.amc.2006.08.133

[32] H. M. Srivastava, S. Hussain, A. Raziq & M. Raza, “The Fekete-Szeg¨o functional for a subclass of analytic functions associated with quasi-subordination”, Carpathian Journal of Mathematics 34 (2018) 1. http://www.jstor.org/stable/90021608. DOI: https://doi.org/10.37193/CJM.2018.01.11

[33] K. Sakaguchi, “On a certain univalent mapping”, Journal of the Mathematical Society of Japan 11 (1959) 1. https://doi.org/10.2969/jmsj/01110072. DOI: https://doi.org/10.2969/jmsj/01110072

[34] M. Fekete & G. Szeg¨o, “Eine bemerkung ¨uber ungerade schlichte funktionen”, Journal of the London Mathematical Society 8 (1933) 85. https://doi.org/10.1112/jlms/s1-8.2.85. DOI: https://doi.org/10.1112/jlms/s1-8.2.85

[35] A. Pfluger, “The Fekete-Szeg¨o inequality for complex parameters”, Complex Variables, Thoery and Application: An International Journal 7 (1986) 149. https://doi.org/10.1080/17476938608814195. DOI: https://doi.org/10.1080/17476938608814195

[36] A. O. Lasode, R. O. Ayinla, R. A. Bello, A. A. Amao, L. M. Fatunsin, B. Sambo & O. Awoyale, “Hankel determinants with Fekete-Szeg¨o parameter for a subset of Bazileviˇc functions linked with Ma-Minda function”, Open Journal of Mathematical Analysis 9 (2025) 1. https://doi.org/10.30538/psrp-oma2025.0150. DOI: https://doi.org/10.30538/psrp-oma2025.0150

Published

2025-09-22

How to Cite

Quasi-subordination for a subclass of non-Bazilevic functions connected with a q-derivative operator. (2025). African Scientific Reports, 4(3), 326. https://doi.org/10.46481/asr.2025.4.3.326

Issue

Volume 4, Issue 3, December 2025 (In Progress)

Section

MATHEMATICAL SCIENCES SECTION
Copyright & Licensing

Copyright (c) 2025 Risikat Ayodeji Bello, Ayotunde Olajide Lasode, Rasheed Olawale Ayinla, Adediran Dauda Adeshola

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Quasi-subordination for a subclass of non-Bazilevic functions connected with a q-derivative operator. (2025). African Scientific Reports, 4(3), 326. https://doi.org/10.46481/asr.2025.4.3.326