Properties of a Subclass of Harmonic Univalent Functions Using the Al-Oboudi $q-$Differential Operator

Year 2025, Volume: 8 Issue: 2, 104 - 114, 30.06.2025

Serkan Çakmak

https://doi.org/10.33401/fujma.1549452

Abstract

In this paper, we introduce the Al-Oboudi $q-$differential operator, a generalized S\u{a}l\u{a}gean operator, for harmonic functions and define a new subclass of harmonic univalent functions using this operator. We investigate several fundamental properties of this subclass, including coefficient conditions, extreme points, distortion bounds, convex combination, and radii of convexity.

Keywords

Modified generalized Sălăgean operator , Al-Oboudi $q-$differential operator , Convolution , $q-$Harmonic univalent functions , Subordination

References

• [1] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn., Ser. A. I., 9(1) (1984), 3–25. $ \href{https://doi.org/10.5186/aasfm.1984.0905}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1984AAH6600001}{\mbox{[Web of Science]}} $
• [2] S. Owa, M. Nunokawa, H. Saitoh, and H.M. Srivastava, Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett., 15(1) (2002), 63–69. $ \href{https://doi.org/10.1016/S0893-9659(01)00094-5}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-31244433724&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Close-to-convexity+starlikeness+and+convexity+of+certain+analytic+functions%22%29&sessionSearchId=0ba82c3e12b3d827062759371e1d7172}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000172499400012}{\mbox{[Web of Science]}} $
• [3] F.H. Jackson, On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh, 46(2) (1909), 253–281. $\href{http://dx.doi.org/10.1017/S0080456800002751}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84974497034&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28On+q-functions+and+a+certain+difference+operator%29&sessionSearchId=a09225bb9f200802a23900162bf129bd&relpos=2}{\mbox{[Scopus]}} $
• [4] F. Jackson, T. Fukuda, and O. Dunn, On q-definite integrals, Quart. J. Pure Appl. Math., 41(15) (1910), 193–203. $ \href{https://www.scirp.org/reference/referencespapers?referenceid=1489517}{\mbox{[Web]}} $
• [5] J.M. Jahangiri, Harmonic univalent functions defined by q-calculus operators, Int. J. Math. Anal. Appl., 5(2) (2018), 39–43. $\href{https://doi.org/10.48550/arXiv.1806.08407}{\mbox{[CrossRef]}} $
• [6] E. Yaşar and S. Yalçın, Generalized Salagean-type harmonic univalent functions, Stud. Univ. Babes-Bolyai Math., 57(3) (2012), 395–403. $ \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000453574500007}{\mbox{[Web of Science]}} $
• [7] J.M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, Salagean-type harmonic univalent functions, South J. Pure Appl. Math., 2 (2002), 77–82. $ \href{https://eudml.org/doc/225293}{\mbox{[Web]}} $
• [8] M.K. Aouf, A.O. Mostafa, and R.E. Elmorsy, Certain subclasses of analytic functions with varying arguments associated with q-difference operator, Afr. Matematika, 32(3-4) (2021), 621–630. $ \href{https://doi.org/10.1007/s13370-020-00849-3}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85092221197}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000577373400001}{\mbox{[Web of Science]}} $
• [9] F.M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci., 27 (2004), 1429–1436. $ \href{https://doi.org/10.1155/S0161171204108090}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-17844378171&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+univalent+functions+defined+by+a+generalized+Salagean+operator%22%29&sessionSearchId=041cbf211db4d12abe0163abf3e147bc&relpos=1}{\mbox{[Scopus]}} $
• [10] G.S. Salagean, Subclasses of Univalent Functions, Lecture Notes in Math., Springer-Verlag, Heidelberg, 1013 (1983), 362–372. $ \href{https://doi.org/10.1007/BFb0066543}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1983RX23800030}{\mbox{[Web of Science]}} $
• [11] O.P. Ahuja and A. Çetinkaya, Use of quantum calculus approach in mathematical sciences and its role in geometric function theory, AIP Conf. Proc., 2095(1) (2019), 020001–14. $ \href{https://doi.org/10.1063/1.5097511}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85064819305&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28Use+of+quantum+calculus+approach+in+mathematical+sciences+and+its+role+in+geometric+function+theory%29}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000472953700001}{\mbox{[Web of Science]}} $
• [12] O.P. Ahuja and A. Çetinkaya, Connecting quantum calculus and harmonic starlike functions, Filomat, 34(5) (2020), 1431–1441. $\href{https://doi.org/10.2298/FIL2005431A}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85098556356}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000600791000002}{\mbox{[Web of Science]}} $
• [13] O.P. Ahuja, A. Çetinkaya, and Y. Polatoğlu, Harmonic univalent convex functions using a quantum calculus approach, Acta Univ. Apulensis Math. Inform. Apulensis, 58(1) (2019), 67–81. $ \href{https://doi.org/10.17114/j.aua.2019.58.06}{\mbox{[CrossRef]}} $
• [14] S. Yalçın and H. Bayram, On harmonic univalent functions involving q-Poisson distribution series, Muthanna J. Pure Sci., 8(2) (2021). $\href{http://dx.doi.org/10.52113/2/08.02.2021/105-111}{\mbox{[CrossRef]}} $
• [15] P.L. Duren, Univalent Functions, Springer, New York, Berlin, Heidelberg, Tokyo, (2001).$\href{https://link.springer.com/book/9780387907956#bibliographic-information}{\mbox{[Web]}} $
• [16] S. Çakmak, S. Yalçın, and Ş. Altınkaya, On a subclass of harmonic univalent functions based on subordination, Theory Appl. Math. Comput. Sci., 7(2) (2017), 51–62. $ \href{https://www.researchgate.net/publication/321481365_On_a_subclass_of_harmonic_univalent_functions_based_on_subordination}{\mbox{[Web]}} $
• [17] J. Dziok, J.M. Jahangiri, and H. Silverman, Harmonic functions with varying coefficients, J. Inequal. Appl., 139(1) (2016). $ \href{https://doi.org/10.1186/s13660-016-1079-z}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84971517389}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000376505300001}{\mbox{[Web of Science]}}$
• [18] J. Dziok, Classes of harmonic functions defined by subordination, Abstr. Appl. Anal., 2015 (2015), 756928. $ \href{http://dx.doi.org/10.1155/2015/756928}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84949294735&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Classes+of+harmonic+functions+defined+by+subordination%22%29}{\mbox{[Scopus]}} $
• [19] J.M. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. Appl., 235(2) (1999), 470–477. $ \href{https://doi.org/10.1006/jmaa.1999.6377}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0347776137&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Harmonic+functions+starlike+in+the+unit+disk%22%29&sessionSearchId=a14ed67b88ce5723607a09b68e56daea}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000081612200005}{\mbox{[Web of Science]}}$
• [20] H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl., 220(1) (1998), 283–289. $\href{https://doi.org/10.1006/jmaa.1997.5882}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0037522384&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22harmonic+univalent+functions+with+negative+coefficients%22%29&relpos=2}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000072753200018}{\mbox{[Web of Science]}} $
• [21] H. Silverman and E.M. Silvia, Subclasses of harmonic univalent functions, New Zealand. J. Math., 28 (1999), 275–284. $\href{https://web.archive.org/web/20231014020952/https://www.thebookshelf.auckland.ac.nz/document.php?action=null&wid=2637}{\mbox{[Web]}} $
• [22] H. Bayram and S. Yalçın, A subclass of harmonic univalent functions defined by a linear operator, Palestine J. Math., 6(2) (2017), 320–326. $ \href{https://pjm.ppu.edu/sites/default/files/papers/PJM_June_2017_29.pdf}{\mbox{[Web]}} $
• [23] J.M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, Classes of harmonic starlike functions defined by Salagean-type q-differential operators, Hacettepe J. Math. Stat., 49(1) (2020), 416–424. $ \href{https://www.scopus.com/pages/publications/85079457761}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000514797000036}{\mbox{[Web of Science]}} $
• [24] G. Murugusundaramoorthy and J.M. Jahangiri, Ruscheweyh-type harmonic functions defined by differential operators, Khayyam J. Math., 5(1) (2019), 79–88. $\href{https://doi.org/10.22034/kjm.2019.81212}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85059702302&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28Ruscheweyh-type+harmonic+functions+defined+by+differential+operators%29&relpos=1}{\mbox{[Scopus]}} $
• [25] A. Canbulat, F.M. Sakar, B. Şeker, A study on a harmonic univalent functions in terms of q difference operator, Bull. Inter. Math. Virt. Inst., 14(1) (2024), 1-11. $ \href{https://doi.org/10.7251/BIMVI2401001C}{\mbox{[CrossRef]}} $
• [26] J. Dziok, On Janowski harmonic functions, J. Appl. Anal., 21(2) (2015), 99–107. $ \href{https://doi.org/10.1515/jaa-2015-0010}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84949254010}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000436465400004}{\mbox{[Web of Science]}} $