Some properties of a class of generalized Janowski-type $q$-starlike functions associated with Opoola $q$-differential operator and $q$-differential subordination

Abstract

Without qualms, studies show that quantum calculus has received great attention in recent times. This can be attributed to its wide range of applications in many science areas. In this exploration, we study a new qdifferential operator that generalized many known differential operators. The new q-operator and the concept of subordination were afterwards, used to define a new subclass of analytic-univalent functions that invariably consists of several known and new generalizations of starlike functions. Consequently, some geometric properties of the new class were investigated. The properties include coefficient inequality, growth, distortion and covering properties. In fact, we solved some radii problems for the class and also established its subordinating factor sequence property. Indeed, varying some of the involving parameters in our results led to some existing results.

Keywords

• Analytic function
• univalent function
• Schwarz function
• Janowski $q$-starlike function
• subordination
• radius problem
• distortion theorem
• growth theorem
• covering theorem
• coefficient inequality
• subordinating factor sequence
• Opoola $q$-differential operator

References

1. Alexander, J. W., Functions which map the interior of the unit circle upon simple regions, Ann. Math. Sec. Ser., 17(1) (1915), 12–22. https://doi.org/10.2307/2007212
2. Ali-Shah, S. G., Khan, S. Hussain, S., Darus, M., q-Noor integral operator associated with starlike functions and q-conic domains, AIMS Math., 7(6) (2022), 10842-10859. https://doi.org/10.3934/math.2022606
3. Al-Oboudi, F. M., On univalent functions defined by a generalised Salagean operator, Internat. J. Math. Math. Sci., 2004(27) (2004), 1429–1436. https://doi.org/10.1155/S0161171204108090
4. Al-Shbeil, I., Shaba, T. G., Cˇat¸as, A., Second Hankel determinant for the subclass of biunivalent functions using q-Chebyshev polynomial and Hohlov operator, Fractal Fract., 6(186) (2022), 19 pages. https://doi.org/10.3390/ fractalfract6040186
5. Annaby, M. H., Mansour, Z. S., q-Fractional Calculus and Equations, Springer Science+Business Media, New York, 2012. https://doi.org/10.1007/978-3-642-30898-7
6. Aral, A., Gupta, V., Agarwal, R. P., Applications of q-Calculus in Operator Theory, Springer Science+Business Media, New York, 2013. https://doi.org/10.1007/978-1-4614-6946-9 1
7. Arif, M., Barukab, O. M., Khan, A. S., Abbas, M., The sharp bounds of Hankel determinants for the families of three-leaf-type analytic functions, Fractal Fract. 6(291) (2022), 35 pages. https://doi.org/10.3390/fractalfract6060291
8. Cho, N. E., Kumar, V., Ravichandran, V., A survey on coefficient estimates for Caratheodory functions, Appl. Math. E-Notes, 19 (2019), 370–396. https://www.emis.de/journals/AMEN/2019/AMEN-180518.pdf

9. Friedland, S., Schiffer, M., Global results in control theory with applications to univalent functions, Bull. Amer. Math. Soc., 82(6) (1976), 913–915. https://doi.org/10.1090/S0002-9904-1976-14211-5
10. Goodman, A. W., Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8 (1957), 598–601. https://doi.org/10.1090/S0002-9939-1957-0086879-9
11. Govindaraj, M., Sivasubramanian, S., On a class of analytic functions related to conic domains involving q-calculus, Anal. Math., 43(3) (2017), 475–487. https://doi.org/10.1007/s10476-017-0206-5
12. Hu, Q., Srivastava, H. M., Ahmad, B., Khan, N., Khan, M. G., Mashwani, W. K., Khan, B., A subclass of multivalent Janowski type q-starlike functions and its consequences, Symmetry, 13(1275) (2021), 14 pages. https://doi.org/10.3390/ sym13071275
13. Jackson, F. H., On q-functions and a certain difference operator, Trans. Roy. Soc. Edinb., 46(2) (1908), 253–281. https://doi.org/10.1017/S0080456800002751
14. Jackson, F. H., On q-difference equation, Amer. J. Math., 32(4) (1910), 305–314. https://doi.org/10.2307/2370183
15. James, A. A., Lasode, A. O., Moses, B. O., Geometric conditions for starlikeness and convexity of univalent functions, IOSR J. Math., 3 (2012), 15–23. https://doi.org/10.9790/5728-0361523
16. Janowski, W., Some extremal problems for certain families of analytic functions I, Ann. Polon. Math., 28(3) (1973), 297–326. https://doi.org/10.4064/ap-28-3-297-326
17. Kac, V., Cheung, P., Quantum Calculus, Springer Science+Business Media, New York, 2002. https://doi.org/10.1007/978-1-4613-0071-7
18. Khan, B., Liu, Z. -G., Shaba, T. G., Araci, S., Khan, N., Khan, M. G., Applications of q-derivative operator to the subclass of bi-univalent functions involving q-Chebyshev polynomials, J. Math., 2022 (2022), Artice ID 8162182, 7 pages. https://doi.org/10.1155/2022/8162182
19. Khan, B., Liu, Z. -G., Srivastava, H. M., Araci, S., Khan, N., Ahmad, Q. Z., Higher-order q-derivatives and their applications to subclasses of multivalent Janowski type q-starlike functions, Adv. Diff. Equ., 2021(440) (2021), 15 pages. https://doi.org/10.1186/s13662-021-03611-6
20. Khan, B., Srivastava, H. M., Arjika, S., Khan, S., Khan, N., Ahmad, Q. Z., A certain q-Ruscheweyh type derivative operator and its applications involving multivalent functions, Adv. Differ. Equ., 2021(279) (2021), https://doi.org/10.1186/s13662-021-03441-6
21. Lasode, A. O., Opoola, T. O., On a generalized class of bi-univalent functions defined by subordination and q-derivative operator, Open J. Math. Anal., 5(2) (2021), 46–52. https://doi.org/10.30538/psrp-oma2021.0092
22. Lasode, A. O., Opoola, T. O., Fekete-Szegö estimates and second Hankel determinant for a generalized subfamily of analytic functions defined by q-differential operator, Gulf J. Math., 11(2) (2021), 36–43. https://doi.org/10.56947/gjom.v11i2.583
23. Lasode, A. O., Opoola, T. O., Some investigations on a class of analytic and univalent functions involving q-differentiation, Eur. J. Math. Anal., 2(12) (2022), 1–9. https://doi.org/10.28924/ada/ma.2.12
24. Lasode, A. O., Opoola, T. O., Coefficient problems of a class of q-starlike functions associated with q-analogue of Al-Oboudi-Al-Qahtani integral operator and nephroid domain, J. Class. Anal., 20(1) (2022), 35–47. https://doi.org/10.7153/jca-2022-20-04
25. Lasode, A. O., Opoola, T. O., Al-Shbeil, I., Shaba, T. G., Alsaud, H., Concerning a novel integral operator and a specific category of starlike functions, Mathematics 11(4519) (2023), 17 pages. https://doi.org/10.3390/math11214519
26. Opoola, T. O., On a subclass of univalent functions defined by a generalised differential operator, Internat. J. Math. Anal., 11(18) (2017), 869–876. https://doi.org/10.12988/ijma.2017.7232
27. Oyekan, E. A., Lasode, A. O., Estimates for some classes of analytic functions associated with Pascal distribution series, error function, Bell numbers and q-differential operator, Nigerian J. Math. Appl., 32 (2022), 163–173. http://www.njmaman.com/articles/2022/PAPER14.pdf
28. Oyekan, E. A., Olatunji, T. A., Lasode, A. O., Applications of (p, q)-Gegenbauer polynomials on a family of bi-univalent functions, Earthline J. Math. Sci., 12(2) (2023), 271–284. https://doi.org/10.34198/ejms.12223.271284
29. Oyekan, E. A., Swamy, S. R., Adepoju, P. O. , Olatunji, T. A., Quasi-convolution properties of a new family of close-to-convex functions involving q-p-Opoola differential operator, Intern. J. Math. Trends Technol., 69(5) (2023), 70-77. https://doi.org/10.14445/22315373/IJMTTV69I5P506
30. Polatoğlu, Y., Bolcal, S. M., Şen, A., Yavuz, E., A study on the generalization of Janowski functions in the unit disc, Acta Math. Acad. Paedagog. Nyh´azi., 22 (2006), 27–31.
31. Raza, M., Srivastava, H. M., Arif, M., Ahmad, K., Coefficient estimates for a certain family of analytic functions involving a q-derivative operator, Ramanujan J., 55 (2021), 53–71. https://doi.org/10.1007/s11139-020-00338-y
32. Rensaa, R. J., Univalent functions and frequency analysis, Rocky J. Math., 33(2) (2003), 743–758. https://doi.org/10.1216/rmjm/1181069976
33. Salagean, G. S., Subclasses of univalent functions, Lect. Notes Math., 1013 (1983), 362–372. https://doi.org/10.1007/BFb0066543
34. Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51(1) (1975), 109–116. https://doi.org/10.1090/S0002-9939-1975-0369678-0
35. Shaba, T. G., Wanas, A. K., Coefficient bounds for a certain families of m-fold symmetric bi-univalent functions associated with q-analogue of Wanas operator, Acta Univ. Apulensis Math. Inform., 68 (2021), 25–37. https://doi.org/10.17114/j.aua.2021.68.03
36. Srivastava, H. M., 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., Review Paper, (2020), 18 pages. https://doi.org/10.1007/s40995-019-00815-0
37. Srivastava, H. M., Attiya, A. A., Some subordination results associated with certain subclasses of analytic functions, J. Inequal. Pure Appl. Math., 5(4) (2004), 1–14. https://www.emis.de/journals/JIPAM/images/113 04 JIPAM/113 04.pdf
38. Srivastava, H. M., Khan, B., Khan, N., Hussain, A., Khan, N., Tahir, M., Applications of certian basic (or q-) derivatives to subclasses of multlivalemt Janowski type q-starlike functions involving conic domain, J. Nonlinear Var. Anal., 5(4) (2021), 531–547. https://doi.org/10.23952/jnva.5.2021.4.03
39. Thomas, D. K., Tuneski, N., Vasudevarao, A., Univalent Functions: A Primer, Walter de Gruyter Inc., Berlin, 2018. https://doi.org/10.1515/9783110560961-001
40. Ul-Haq, M., Raza, M., Arif, M., Khan, Q., Tang, H., q-Analogue of differential subordinations, Math. 7(8) (2019), pp. 16. https://doi.org/10.3390/math7080724
41. Vasilev, A., Univalent functions in the dynamics of viscous flow, Comput. Method Funct. Theory, 1(2) (2001), 311–337. https://doi.org/10.1007/BF03320993
42. Wanas, A. K., Mahdi, A. M., Applications of the q-Wanas operator for a certain family of bi-univalent functions defined by subordination, Asian-Eur. J. Math., 16(6) (2023), 1–16. https://doi.org/ 10.1142/S179355712350095X
43. Wilf, H. S., Subordinating factor sequences for convex maps of the unit circle, Proc. Amer. Math. Soc., 12 (1961), 689–693. https://doi.org/10.1090/S0002-9939-1961-0125214-5