Abstract
Quantum theory has wide applications in special functions and quantum physics. In this paper, we discuss the geometric properties of analytic functions using q-differential operator. We introduce some new subclasses of analytic functions which are obtained from the q-derivative and conic domains. We investigate interesting results involving dual sets and convolution properties of these new subclasses. We also study the inclusion properties of neighborhood of analytic functions. Our results continue to hold for the known and new subclasses of analytic functions which can be obtained as special case.
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References
Agarwal S, Sahoo SK (2014) Geometric properties of basic hypergeometric functions. Difference Equ Appl 20:1502–1522
Dziok J (2011) Properties of new classes of multivalent analytic functions. Mat Vesnik 63:201–217
Gairola AR, Deepmala Mishra LN (2017) On the \(q\)- derivatives of a certain linear positive operators. Iran J Sci Technol Trans A Sci. https://doi.org/10.1007/s40995-017-0227-8
Gairola AR, Mishra LN (2016) Rate of approximation by finite Iterates of \(q\)-Durrmeyer operators. Proc Natl Acad Sci India A Phys Sci 86:229–234
Ismail MEH, Merkes E, Styer D (1990) A generalization of starlike functions. Complex Var Theory Appl 14:77–84
Jackson FH (1909) On \(q\)-functions and a certain difference operator. Trans R Soc Edinburgh 46:253–281
Jackson FH (1910) On \(q\)-definite integrals. Quart J Pure Appl Math 41:193–203
Kanas S (1998) Stability of convolution and dual sets for the class of \(k\)-uniformly convex and \(k\)-starlike functions. Foia Sci Univ Tech Resov 22:51–63
Kanas S, Wisniaskawa A (1999) Conic regions and \(k\)-uniform convexity. J Comput Appl Math 105:327–336
Kanas S (2003) Techniques of the differential subordination for domains bounded by conic sections. Inter J Math Math Sci 38:2389–2400
Mishra VN, Khatri K, Mishra LN (2013) Some approximation properties of \(q\)-Baskakov-Beta-Stancu type operators. J Calc Var 2013:814824. https://doi.org/10.1155/2013/814824
Mishra VN, Khatri K, Mishra LN (2012) On simultaneous approximation for Baskakov-Durrmeyer-Stancu type operators. J Ultra Sci Phy Sci 24:567–577
Mohammed A, Darus M (2013) A generalized operator involving the \(q\)-hypergeometric function. Mat Vesnik 65:454–465
Ruscheweyh S, Sheil-Small T (1973) Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture. Comment Math Helv 48:119–135
Ruscheweyh S (1975) New criteria for univalent functions. Proc Am Math Soc 49:109–115
Ruscheweyh S (1975) Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc. Trans Am Math Soc 210:63–74
Ruscheweyh S (1981) Neighborhoods of univalent functions. Proc Am Math Soc 81:521–527
Sahoo SK, Sharma NL (2015) On a generalization of close-to-convex functions. Ann Polon Math 113:93–108
Seoudy T, Aouf M (2016) Coefficient estimates of new classes of \(q\)-starlike and \(q\)-convex functions of complex order. J Math Inequal 10:135–145
Acknowledgements
The authors are grateful to Dr. S. M. Junaid Zaidi (H.I, S.I), Rector, COMSATS Institute of Information Technology, Pakistan for providing excellent research and academic environment. This research is supported by the HEC NPRU Project No: 20-1966/R&D/11-2553, titled, Research unit of Academic Excellence in Geometric Functions Theory and Applications.
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Noor, K.I., Shahid, H. On Dual Sets and Neighborhood of New Subclasses of Analytic Functions Involving q-Derivative. Iran J Sci Technol Trans Sci 42, 1579–1585 (2018). https://doi.org/10.1007/s40995-018-0525-9
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DOI: https://doi.org/10.1007/s40995-018-0525-9
Keywords
- Convex
- Starlike
- Quantum calculus
- Analytic functions
- Dual sets
- Neighborhood
- Univalent functions
- Convolution
- Inclusion results