Abstract
The main object of the present paper is to define q-analogue of Ruscheweyh operator for multivalent functions. We investigate a number of useful properties including coefficient estimates, sufficiency criteria and the familiar Feke–Szegö type inequality for a newly defined class. Several known consequences of the main results are also pointed out.
Similar content being viewed by others
References
Jackson, F.H.: On \(q\)-functions and a certain difference operator. Earth Environ. Sci. Trans. R. Soc. Edinb. 46(2), 253–281 (1909)
Jackson, F.H.: On \(q\)-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)
Aral, A., Gupta, V.: Generalized \(q\)-Baskakov operators. Math. Slovaca 61(4), 619–634 (2011)
Aral, A., Gupta, V.: On \(q\)-Baskakov type operators. Demonstr. Math. 42(1), 109–122 (2009)
Aral, A., Gupta, V.: On the Durrmeyer type modification of the q-Baskakov type operators. Nonlinear Anal. Theory Methods Appl. 72(3), 1171–1180 (2010)
Anastassiou, G.A., Gal, S.G.: Geometric and approximation properties of generalized singular integrals in the unit disk. J. Korean Math. Soc. 43(2), 425–443 (2006)
Anastassiou, G.A., Gal, S.G.: Geometric and approximation properties of some singular integrals in the unit disk. J. Inequal. Appl. 2006, Article ID 17231, 19 (2006)
Aral, A.: On the generalized Picard and Gauss Weierstrass singular integrals. J. Comput. Anal. Appl. 8(3), 249–261 (2006)
Kanas, S., Răducanu, D.: Some class of analytic functions related to conic domains. Mathematica slovaca 64(5), 1183–1196 (2014)
Aldweby, H., Darus, M.: Some subordination results on q -analogue of Ruscheweyh differential operator. Abstract Appl. Anal. 2014, Article ID 958563, 6 (2014)
Mahmood, S., Sokół, J.: New subclass of analytic functions in conical domain associated with Ruscheweyh \(q\)-differential operator. Results Math. 71(4), 1345–1357 (2017)
Mohammed, A., Darus, M.: A generalized operator involving the \(q\)-hypergeometric function. Matematički Vesnik 65(4), 454–465 (2013)
Goel, R.M., Sohi, N.S.: A new criterion for \(p\)-valent functions. Proc. Am. Math. Soc. 78(3), 353–357 (1980)
Ruscheweyh, S.: New criteria for univalent functions. Proc. Am. Math. Soc. 49, 109–115 (1975)
Noor, K.I., Arif, M.: On some applications of Ruscheweyh derivative. Comput. Math. Appl. 62(12), 4726–4732 (2011)
Aldawish, I., Darus, M.: Starlikeness of \(q\) -differential operator involving quantum calculus. Korean J. Math. 22(4), 699–709 (2014)
Aldweby, H., Darus, M.: A subclass of harmonic univalent functions associated with \(q\)-analogue of Dziok–Srivastava operator. ISRN Math. Anal. 2013, Article ID 382312, 6 (2013)
Kanas, S., Wiśniowska, A.: Conic regions and \(k\)-uniform convexity. J. Comput. Appl. Math. 105(1), 327–336 (1999)
Kanas, S., Wiśniowska, A.: Conic domains and starlike functions. Revue Roumaine de Mathématiques Pures et Appliquées 45(4), 647–658 (2000)
Noor, K.I., Arif, M., Ul-Haq, W.: On \(k\) -uniformly close-to-convex functions of complex order. Appl. Math. Comput. 215(2), 629–635 (2009)
Seoudy, T.M., Aouf, M.K.: Coefficient estimates of new classes of \(q\)-starlike and \(q\)-convex functions of complex order. J. Math. Inequal. 10(1), 135–145 (2016)
Agrawal, S., Sahoo, S.K.: A generalization of starlike functions of order \(\alpha \). Hokkaido Math. J. 46, 15–27 (2017)
Arif, M., Sokół, J., Ayaz, M.: Coefficient inequalities for a subclass of \(p\)-valent analytic functions. Sci. World J. 2014, Article ID 801751, 5 (2014)
Bharati, R., Parvatham, R., Swaminathan, A.: On subclasses of uniformly convex functions and corresponding class of starlike functions. Tamkang J. Math. 28, 17–32 (1997)
Owa, S.: On uniformly convex functions. Math. Jpn. 48(3), 377–384 (1998)
Arif, M., Mahmood, S., Sokol, J., Dziok, J.: New subclass of analytic functions in conical domain associated with a linear operator. Acta Math. Sci. 36(3), 704–716 (2016)
Kanas, S., Srivastava, H.M.: Linear operators associated with \(k\)-uniformly convex functions. Integral Transforms Spec. Funct. 9, 121–132 (2000)
Owa, S., Polatoglu, Y., Yavuz, E.: Coefficient inequalities for classes of uniformly starlike and convex functions. J. Inequal. Pure Appl. Math. 7(5), Article ID 160, 4 (2006)
Srivastava, H.M., Khan, M.R., Arif, M.: Some subclasses of close-to-convex mappings associated with conic regions. Appl. Math. Comput. 285, 94–102 (2016)
Kanas, S.: Coefficient estimates in subclasses of the Carath éodory class related to conical domains. Acta Math. Univ. Comen. LXXIV(2), 149–161 (2005)
Ma, W.C., Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the conference on complex analysis. Tianjin, pp. 157–169 (1992)
Merkes, E.P., Robertson, M.S., Scott, W.T.: On products of starlike functions. Proc. Am. Math. Soc. 13(6), 960–964 (1962)
Silverman, H.: Univalent functions with negative coefficients. Proc. Am. Math. Soc. 51(1), 109–116 (1975)
Robertson, M.S.: On the theory of univalent functions. Ann. Math. 37, 374–408 (1936)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arif, M., Srivastava, H.M. & Umar, S. Some applications of a q-analogue of the Ruscheweyh type operator for multivalent functions. RACSAM 113, 1211–1221 (2019). https://doi.org/10.1007/s13398-018-0539-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-018-0539-3