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Accessible Unlicensed Requires Authentication Published by De Gruyter July 19, 2019

A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator

Shahid Khan, Saqib Hussain, Muhammad A. Zaighum and Maslina Darus
From the journal Mathematica Slovaca


Making use of Ruscheweyh q-differential operator, we define a new subclass of uniformly convex functions and corresponding subclass of starlike functions with negative coefficients. The main object of this paper is to obtain, coefficient estimates, closure theorems and extreme point for the functions belonging to this new class. The results are generalized to families with fixed finitely many coefficients.

MSC 2010: Primary 30C45; 30C50

  1. (Communicated by Stanisława Kanas)


The authors would like to thank the Editor and referees for their valuable comments which helped to improve the manuscript. The 4th author is supported by UKM grant: GUP-2017-064.


[1] Aldweby, H.—Darus, M.: Some subordination results onq-analogue of Ruscheweyh differential operator, Abstr. Appl. Anal. 2014 (2014), Art. ID 958563.Search in Google Scholar

[2] Anastassiou, G. A.—Gal, S. G.: Geometric and approximation properties of generalized singular integrals in the unit disk, J. Korean Math. Soc. 23(2) (2006), 425–443.Search in Google Scholar

[3] Aral, A.: On the generalized Picard and Gauss Weierstrass singular integrals, J. Comput. Anal. Appl. 8(3), (2006) 249–261.Search in Google Scholar

[4] Aral, A.—Gupta, V.: Generalizedq-Baskakov operators, Math. Slovaca 61(4) (2011), 619–634.Search in Google Scholar

[5] Aral, A.—Gupta, V.: Onq-Baskakov type operators, Demonstratio Math. 42(1) (2009), 109–122.Search in Google Scholar

[6] Goodman, A. W: On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87–92.Search in Google Scholar

[7] Jackson, F. H.: Onq-functions and a certain difference operator, Earth and Environmental Science Transactions of the Royal Society of Edinburgh 46(2) (1909), 253–281.Search in Google Scholar

[8] Jackson, F. H.: Onq-definite integrals, Quart. J. Mech. Appl. Math. 41 (1910), 193–203.Search in Google Scholar

[9] Kanas, S.—Raducanu, D.: Some class of analytic functions related to conic domains, Math. Slovaca 64(5) (2014), 1183–1196.Search in Google Scholar

[10] Kanas, S.—Sugawa, T.: On conformal representation of the interior of an ellipse, Ann. Acad. Sci. Fenn. Math. 31 (2006), 329–348.Search in Google Scholar

[11] Kanas, S.—Wisniowska, A.: Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl. 45(3) (2000), 647–657.Search in Google Scholar

[12] Mahmood, S.—Sokol, J.: New subclass of analytic functions in conical domain associated with Ruscheweyhq-differential operator, Results Math. 71 (2017), 1–13.Search in Google Scholar

[13] Ma, W.—Minda, D.: Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165–175.Search in Google Scholar

[14] Rønning, F.: Uniformly convex functions and corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189–196.Search in Google Scholar

[15] Ruscheweyh, S. T.: New criteria for univalent functions, Proc. Amer. Math. Soc 49 (1975), 109–115.Search in Google Scholar

[16] Silverman, H.: Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109–116.Search in Google Scholar

[17] Silverman, H.—Silvia, E. M.: Fixed coefficients for subclasses of starlike functions, Houston J. Math. 7 (1997), 129–136.Search in Google Scholar

Received: 2018-08-25
Accepted: 2018-12-19
Published Online: 2019-07-19
Published in Print: 2019-08-27

© 2019 Mathematical Institute Slovak Academy of Sciences