Harmonic univalent functions defined by post quantum calculus operators

Om P. Ahuja; Asena Çetinkaya; V. Ravichandran

doi:10.2478/ausm-2019-0001

Abstract

We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

References

[1] O. P. Ahuja, Planar harmonic univalent and related mappings, J. Inequal. Pure Appl. Math., 6 (2005) (4), Art. 122, 18 pp.
Search in Google Scholar Back to article
[2] O. P. Ahuja, Recent advances in the theory of harmonic univalent mappings in the plane, Math. Student, 83 (1–4) (2014), 125–154.
Search in Google Scholar Back to article
[3] O. P. Ahuja, A. Çetinkaya, Y. Polatõglu,Bieberbach-de Branges and Fekete-Szegö inequalities for certain families of q-convex and q-close-to-convex functions, J. Comput. Anal. Appl., 26 (4) (2019), 639–649.
Search in Google Scholar Back to article
[4] O. P. Ahuja, A. Çetinkaya, Connecting quantum calculus and harmonic starlike functions, preprint, 2018.
Search in Google Scholar Back to article
[5] J. Clunie, T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 9 (1984), 3–25.10.5186/aasfm.1984.0905
Search in Google Scholar Back to article
[6] R. Jagannathan, K. Srivanasava, Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, arXiv:math/0602613v1 [math.NT] 27 Feb 2006.
Search in Google Scholar Back to article
[7] P. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156, Cambridge University Press, Cambridge, 2004.10.1017/CBO9780511546600
Search in Google Scholar Back to article
[8] M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Var. Theory Appl., 14 (1–4) (1990), 77–84.10.1080/17476939008814407
Search in Google Scholar Back to article
[9] F. H. Jackson, On q-functions and a certain difference operator, Trans. R. Soc. Edinburg, 46 (part II) (11) (1909), 253–281.10.1017/S0080456800002751
Search in Google Scholar Back to article
[10] F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203.
Search in Google Scholar Back to article
[11] F. H. Jackson, q-Difference equations, Amer. J. Math., 32 (4) (1910), 305–314.10.2307/2370183
Search in Google Scholar Back to article
[12] J. M. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. Appl., 235 (3) (1999), 470–477.10.1006/jmaa.1999.6377
Search in Google Scholar Back to article
[13] J. M. Jahangiri, Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 52 (2) (1998), 57–66.
Search in Google Scholar Back to article
[14] J. M. Jahangiri, Harmonic univalent functions defined by q-calculus operators, Int. J. Math. Anal. Appl., 5 (2) (2018), 39–43.
Search in Google Scholar Back to article
[15] S. Nagpal and V. Ravichandran, A comprehensive class of harmonic functions defined by convolution and its connection with integral transforms and hypergeometric functions, Stud. Univ. Babeş-Bolyai Math., 59 (1) (2014), 41–55.
Search in Google Scholar Back to article
[16] N. Ravikumar, Certain classes of analytic functions defined by fractional q-calculus operator, Acta Univ. Sapientiae Math., 10 (1) (2018), 178–188.10.2478/ausm-2018-0015
Search in Google Scholar Back to article
[17] G. S. Sălăgean, Subclasses of univalent functions, in Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), 362–372, Lecture Notes in Math., 1013, Springer, Berlin.10.1007/BFb0066543
Search in Google Scholar Back to article
[18] S. Yalçin, A new class of Salagean-type harmonic univalent functions, Appl. Math. Lett., 18 (2) (2005), 191–198.10.1016/j.aml.2004.05.003
Search in Google Scholar Back to article

Harmonic univalent functions defined by post quantum calculus operators

Abstract

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