Hostname: page-component-76d6cb85b7-hqrjx Total loading time: 0 Render date: 2026-07-15T00:54:44.317Z Has data issue: false hasContentIssue false
  • English
  • Français

On a class of difference operator and its applications to a family of analytic functions

Part of: Geometric function theory Special classes of linear operators

Published online by Cambridge University Press:  27 February 2023

Ravinder Krishna Raina ,
Janusz Sokół Open the ORCID record for Janusz Sokół [Opens in a new window]  and
Katarzyna Tra̧bka-Wiȩcław
Ravinder Krishna Raina
Affiliation:
M.P., University of Agriculture and Technology, Udaipur 313001, Rajasthan, India (rkraina_7@hotmail.com)
Janusz Sokół
Affiliation:
University of Rzeszów, College of Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland (jsokol@ur.edu.pl, jsokol@prz.edu.pl)
Katarzyna Tra̧bka-Wiȩcław
Affiliation:
Lublin University of Technology, Mechanical Engineering Faculty, ul. Nadbystrzycka 36, 20-618 Lublin, Poland (k.trabka@pollub.pl)
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper mainly considers the problem of generalizing a certain class of analytic functions by means of a class of difference operators. We consider some relations between starlike or convex functions and functions belonging to such classes. Some other useful properties of these classes are also considered.

Keywords

q-calculusp,q-difference operatorq-derivativeanalytic functionsq-starlike functions

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C50: Coefficient problems for univalent and multivalent functions 47B99: None of the above, but in this section

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 1 , February 2024 , pp. 248 - 258
Check for updates
Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

*

Present address: 10/11 Ganpati Vihar, Opposite Sector 5, Udaipur 313002, Rajasthan, India.

References

Abu-Risha, M. H., Annaby, M. H., Ismail, M. E. H. and Mansour, Z. S.. Linear $q$-difference equations. Z. Anal. Anwend. 26 (2007), 481–494.CrossRefGoogle Scholar
Agrawal, S. and Sahoo, S. K.. A generalization of starlike functions of order alpha. Hokkaido Math. J. 46 (2017), 15–27.CrossRefGoogle Scholar
Annaby, M. H. and Mansour, Z. S.. q-fractional calculus and equations (Berlin, Heidelberg: Springer, 2012).CrossRefGoogle Scholar
Alvarez-Nodarse, R.. On characterizations of classical polynomials. J. Comput. Appl. Math. 196 (2006), 320–337.CrossRefGoogle Scholar
Aouf, M. K. and Seoudy, T. M.. Convolution properties for classes of bounded analytic functions with complex order defined by $q$-derivative operator. Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat. 113 (2019), 1279–1288.CrossRefGoogle Scholar
Araci, S., Duran, U., Acikgoz, M. and Srivastava, H. M.. A certain $(p, q)$-derivative operator and associated divided differences. J. Inequal. Appl. 2016 (2016), 301.CrossRefGoogle Scholar
Corcino, R. B.. On $P, Q$-binomial coefficients. Electron. J. Comb. Number Theory 8 (2008), A29.Google Scholar
Duren, U.. Post quantum calculus. Msc. thesis, 2016.Google Scholar
Gupta, V., Rassis, T. M., Agarwal, P. N. and Acu, A. M.. Basics of post quantum calculus. Recent advances in constructive approximation theory, Springer Optimization and Its Applications, vol. 138 (Cham: Springer, 2018).CrossRefGoogle Scholar
Hahn, W.. Beiträge zur theorie der heineschen reihen. Math. Nachr. 2 (1949), 340–379.CrossRefGoogle Scholar
Hamza, A., Sarhan, A., Shehata, E. and Aldwoah, K.. A general quantum difference calculus. Adv. Differ. Equ. 2015 (2015), 182.CrossRefGoogle Scholar
Ismail, M. E. H., Merkes, E. and Styer, D.. A generalization of starlike functions. Complex Var. 14 (1990), 77–84.Google Scholar
Jackson, F. H.. On $q$-functions and certain difference operator. Trans. R. Soc. Edinburgh 46 (1908), 253–281.CrossRefGoogle Scholar
Jackson, F. H.. On $q$-definite integrals. Q. J. Pure Appl. Math. 41 (1910), 193–203.Google Scholar
Chakrabarti, R. and Jagannathan, R.. A $p, q$-oscillator realization of two-parameter quantum algebras. J. Phys. A, Math. Gen. 24 (1991), 5683.CrossRefGoogle Scholar
Jagannathan, R. and Rao, K. S.. Two-parameter quantum algebras, twin-basic numbers and associated generalized hypergeometric series. ArXiv:math/0602613 [math.NT].Google Scholar
Raghavendar, K. and Swaminathan, A.. Close-to-convexity of basic hypergeometric functions using their Taylor coefficients. J. Math. Appl. 35 (2012), 111–125.Google Scholar
Rønning, F.. A Szegö quadrature formula arising from $q$-starlike functions. In Continued fractions and orthogonal functions, theory and applications (ed. S. Clement Cooper and W. J. Thron), pp. 345–352 (New York: Marcel Dekker Inc., 1994).CrossRefGoogle Scholar
Ruscheweyh, St. and Sheil-Small, T.. Hadamard product of Schlicht functions and the Poyla-Schoenberg conjecture. Comment. Math. Helv. 48 (1973), 119–135.CrossRefGoogle Scholar
Sadjang, P. N.. On the fundamental theorem of $p, q$-calculus and some $p, q$-Taylor formulas. ArXiv:1309.3934 [math.QA].Google Scholar