Harmonic close-to-convex mappings associated with Sălăgean q-differential operator
DOI:
https://doi.org/10.24193/subbmath.2025.1.03Keywords:
Sălăgean q-differential operator, analytic functions, harmonic functionsAbstract
In this paper, we define a new subclass W(n; _; q) of analytic functions and a new subclass W0H (n; _; q) of harmonic functions f = h+g 2 H0 associated with Sălăgean q-differential operator. We prove that a harmonic function f = h+_g belongs to the class W0H (n; _; q) if and only if the analytic functions h+_g belong to W(n; _; q) for each _ (j_j = 1), and using a method by Clunie and Sheil-Small, we determine a sufficient condition for the class W0H (n; _; q) to be close-to-convex. We provide sharp coefficient estimates, sufficient coefficient condition, and convolution properties for such functions classes. We also determine several conditions of partial sums of f 2 W0H (n; _; q).
Mathematics Subject Classification (2010): 31A05, 30C45, 30C55.
Received 05 November 2023; Accepted 28 December 2024.
References
[1] Beig, S., Ravichandran, V., Convolution and convex combination of harmonic mappings, Bull. Iranian Math. Soc., 45(2019), no. 5, 1467-1486.
[2] Chichra, P.N., New subclass of the class of close-to-convex function, Proc. Amer. Math. Soc., 62(1977), 37–43.
[3] Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I., 9(1984), 3–25.
[4] Dorff, M., Rolf, J., Anamorphosis, mapping problems, and harmonic univalent functions, Explorations in Complex Analysis, Math. Assoc. of America, Inc., Washington, DC, (2012), 197–269.
[5] Gasper, G., Rahman, M., Basic Hypergeometric Series, Cambridge University Press, Cambridge, 2004.
[6] Ghosh, N., Vasudevarao, A., On a subclass of harmonic close-to-convex mappings, Monatsh. Math., 188(2019), 247–267.
[7] Govindaraj, M., Sivasubramanian, S., On a class of analytic functions related to conic domains involving q-calculus, Analysis Math., 43(2017), 475–487.
[8] Hussain, S., Khan, S., Zaighum, M.A., Darus, M., Certain subclass of analytic functions related with conic domains and associated with Sălăgean q-differential operator, AIMS Mathematics, 2(2017), 622–634.
[9] Ismail, M.E.H., Merkes, E., Styer, D., A generalizations of starlike functions, Complex Variables Theory Appl., 14(1990), 77–84.
[10] Jackson, F.H., On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46(1909), 253–281.
[11] Jackson, F.H., On q-definite integrals, Quart. J. Pure Appl. Math., 41(1910), 193–203.
[12] Jahangiri, J.M., Harmonic univalent functions defined by q-calculus operator, Int. J. Math. Anal. Appl., 5(2018), 39–43.
[13] Jahangiri, J.M., Murugusundaramoorthy, G., Vijaya, K., Sălăgean-type harmonic univalent functions, South J. Pure. Appl. Math., 2(2002), 77–82.
[14] Kac, V., Cheung, P., Quantum Calculus, Springer-Verlag, New York, 2002.
[15] Lewy, H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc., 42(1936), 689–692.
[16] Li, L., Ponnusamy, S., Disk of convexity of sections of univalent harmonic functions, J. Math. Anal. Appl., 408(2013), 589–596.
[17] Li, L., Ponnusamy, S., Solution to an open problem on convolutions of harmonic mappings, Complex Var. Elliptic Equ., 58(2013), no. 12, 1647–1653.
[18] Liu, Z., Ponnusamy, S., Univalency of convolutions of univalent harmonic right half-plane mappings, Comput. Methods Funct. Theory, 17(2017), no. 2, 289–302.
[19] Liu, M.S., Yang, L.M., Geometric properties and sections for certain subclasses of harmonic mappings, Monatsh. Math., 190(2019), 353–387.
[20] MacGregor, T.H., Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104(1962), 532–537.
[21] Mahmood, S., Sokół, J., New subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator, Results Math., 71(2017), 1345–1357.
[22] Mishra, O., Sokol, J., Generalized q-starlike harmonic functions, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 115(2021).
[23] Murugusundaramoorthy, G., Jahangiri, J.M., Ruscheweyh-type harmonic functions defined by q-differential operators, Khayyam J. Math., 5(2019), 79–88.
[24] Nagpal, S., Ravichandran, V., Construction of subclasses of univalent harmonic mappings, J. Korean Math. Soc., 53(2014), 567–592.
[25] Ponnusamy, S., Yamamoto, H., Yanagihara, H., Variability regions for certain families of harmonic univalent mappings, Complex Variables and Elliptic Equations, 58(2013), no. 1, 23–34.
[26] Ruscheweyh, S., Sheil-Small, T., Hadamard products of Schlicht functions and the Pólya- Schoenberg conjecture, Comment. Math. Helv., 48(1973), 119–135.
[27] Sălăgean, G.S., Subclasses of univalent functions, Lecture notes in Math., Springer-Verlage, 1013(1983), 362–372.
[28] Sharma, P., Mishra, O., A class of harmonic functions associated with a q−Sălăgean operator, U.P.B. Sci. Bull., Series A, 82(2020), no. 3, 10 pages.
[29] Silverman, H., Partial sums of starlike and convex functions, J. Math. Anal. Appl.,
209(1997), 221–227.
[30] Silvia, E.M., On partial sums of convex functions of order alpha, Houston J. Math.,
11(1985), 397–404.
[31] Singh, R., Singh, S., Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., 106(1989), 145–152.
[32] Szegö, G., Zur Theorie der schlichten Abbildungen, Math. Ann., 100 (1928), 188–211.
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