Korean J. Math. Vol. 22 No. 4 (2014) pp.699-709
DOI: https://doi.org/10.11568/kjm.2014.22.4.699

Starlikeness of q−differential operator involving quantum calculus

Main Article Content

Ibtisam Aldawish
Maslina Darus


In the present paper, we investigate starlikeness condi- tions for $q$−differential operator by using the concept of quantum calculus in the unit disk.

Article Details

Supporting Agencies

The work here is supported by FRGSTOPDOWN/2013/ST06/UKM/01/1 and the authors would like to thank the referee for the comments to improve the manu- script.


[1] F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math and Math Sci 25-28 (2004), 1429–1436. Google Scholar

[2] H. Aldweby and M. Darus, Some subordination results on q−Analogue of Ruscheweyh differential operator, Abstr. Appl. Anal. (2014) Article ID958563, 6 pages. Google Scholar

[3] I. Aldawish and M. Darus, New subclass of analytic function associated with the generlalized hypergeometric functions, Electronic J. Math. Anal. Appl. 2 (2) (2014), 163–171. Google Scholar

[4] A. Aral, V. Gupta Ravi and P. Agarwal, Applications of q Calculus in operator Theory, New york, NY: Springer, 2013. Google Scholar

[5] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric func- tions, SIAM J. Math. Anal. 15 (4) (1984), 737–745. Google Scholar

[6] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999), 1–13. Google Scholar

[7] H. Exton, q-hypergeometric Functions and Applications, Ellis Horwood Limited, Chichester, 1983. Google Scholar

[8] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Math- ematics and its Application, Vol. 35, Cambridge University Press, Cambridge, 1990. Google Scholar

[9] H. A. Ghany, q-derivative of basic hypergeometric series with respect to parameters, Int. J. Math. Anal. (Ruse) 3 (33-36) (2009), 1617–1632. Google Scholar

[10] F. H. Jackson, On q−functions and a certain difference operator. Tranc. Roy. Soc. Edin 46 (1908), 253–281. Google Scholar

[11] K. Kuroki and S. Owa, Double integral operators concerning starlike of order β, Int. J. Differ. Equ. (2009), 1–13. Google Scholar

[12] S. S. Miller and P. T. Mocanu, Double integral starlike operators, Integral Trans- forms Spec. Funct. 19 (7-8) (2008), 591–597 . Google Scholar

[13] A. Mohammed and M. Darus, A generalized operator involving the q−hypergeometric function, Mat. Vesnik , 65 (4) (2013), 454–465. Google Scholar

[14] M. Obradovi c, Simple sufficient conditions for univalence, Mat. Vesnik, 49 (3-4) (1997), 241-244. Google Scholar

[15] D. Purohit and R. K. Raina, Generalized q−Taylor’s series and applications, Gen. Math. 18 (3) (2010), 19–28. Google Scholar

[16] S. Rusheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115. Google Scholar

[17] G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Math. (Springer Verlag), 1013 (1983), 362–372. Google Scholar

[18] W.S. Chung, T. Kim, H. I. Kwon, On the q-analog of the Laplace transform, Russ. J. Math. Phys. 21 (2) (2014), 156–168. Google Scholar

[19] J. Yang, U(n+1) extensions of some basic hypergeometric series identities. Adv. Stud. Contemp. Math. (Kyungshang) 18 (2) (2009), 201–218. Google Scholar

[20] Y. S. Kim, C. H. Lee, Exton’s triple hypergeometric series associated with the Kamp De Friet function, Proc. Jangjeon Math. Soc. 14 (4) (2011), 447–453. Google Scholar

[21] K. R. Vasuki, A. A. Kahtan, G. Sharath, On certain continued fractions related to 3ψ3 basic bilateral hypergeometric functions, Adv. Stud. Contemp. Math. (Kyungshang) 20 (3) (2010) 343–357. Google Scholar