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On quantum hybrid fractional conformable differential and integral operators in a complex domain

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Newly, the hybrid fractional differential operator (HFDO) is presented and studied in Baleanu et al. (Mathematics 8.3:360, 2020). This work deals with the extension of HFDO to the complex domain and its generalization by using the quantum calculus. The outcome of the above conclusion is a q-HFDO, which will employ to introduce some classes of normalized analytic functions containing the well-known starlike and convex classes. Moreover, we utilize the quantum calculus to formulate the q-integral operator corresponding to q-HFDO. As a result, the upper solution is exemplified by utilizing the notion of subordination inequality.

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References

  1. Kac, V., Cheung, P.: Quantum Calculus. Springer Science & Business Media; (2001)

  2. Johnson, W.P.: An Introduction to q-analysis, American Mathematical Society, (2020), Print ISBN: 978-1-4704-5623-8

  3. Jackson, F.H.: On q-functions and a certain difference operator. Earth Environ. Sci. Trans. R. Soc. Edinburgh 46(2), 253–281 (1909)

    Google Scholar 

  4. Jackson, F.H.: On q-definite integrals. Quart. J. Pure Appl. Math. 41, 193–203 (1910)

    MATH  Google Scholar 

  5. Aouf, M.K., Seoudy, T.M.: “Convolution properties for classes of bounded analytic functions with complex order defined by q-derivative operator.” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 113.2 (2019): 1279–1288

  6. Arif, Muhammad, Srivastava, H.M., Umar, Sadaf: “Some applications of a q-analogue of the Ruscheweyh type operator for multivalent functions.” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 113.2 (2019): 1211–1221

  7. Anderson, Douglas R., Onitsuka, Masakazu: Hyers-Ulam Stability for Quantum Equations of Euler Type. Discrete Dyn. Nat. Soc. 2020, (2020)

  8. Usman, M., Ibrahim, M.S., Ahmed, J., Hussain, S.S., Moinuddin, M.: Quantum calculus-based volterra LMS for nonlinear channel estimation. In 2019 Second International Conference on Latest trends in Electrical Engineering and Computing Technologies (INTELLECT) (pp. 1-4). IEEE (2019, November)

  9. Srivastava, H.M., Bansal, D.E.: Close-to-convexity of a certain family of q-Mittag–Leffler functions. J. Nonlinear Var. Anal 1(1), 61–69 (2017)

    MATH  Google Scholar 

  10. Srivastava, Hari M., Ahmad, Qazi Zahoor, Khan, Nasir, Khan, Nazar, Khan, Bilal: Hankel and toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics 7, no. 2 (2019): 181

  11. Mahmood, Shahid, Srivastava, Hari M., Khan, Nazar, Ahmad, Qazi Zahoor, Khan, Bilal, Ali: Upper bound of the third Hankel determinant for a subclass of q-starlike functions. Symmetry 11, no. 3 (2019): 347

  12. Shi, Lei, Khan, Qaiser, Srivastava, Gautam, Liu, Jin-Lin, Arif, Muhammad: A study of multivalent q-starlike functions connected with circular domain. Mathematics 7(8), 670 (2019)

    Article  Google Scholar 

  13. Ibrahim, Rabha W., Darus, Maslina: On a class of analytic functions associated to a complex domain concerning q-differential-difference operator. Adv. Differ. Equ. 2019(1), 515 (2019)

    Article  MathSciNet  Google Scholar 

  14. Srivastava, H.M.: Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran J. Sci. Technol. Trans. A Sci. (2020): 1-18

  15. Ul-Haq, Miraj, Raza, Mohsan, Arif, Muhammad, Khan, Qaiser, Tang, Huo: q-analogue of differential subordinations. Mathematics 7(8), 724 (2019)

    Article  Google Scholar 

  16. Ibrahim, R. W., Hadid, S. B., Momani, S.: Generalized Briot–Bouquet differential equation by a quantum difference operator in a complex domain. Int. J. Dyn. Control 1–10 (2020)

  17. Ibrahim, R.W., Elobaid, R.M., Obaiys, S.J.: A class of quantum Briot–Bouquet differential equations with complex coefficients. Mathematics 8(5), 794 (2020)

    Article  Google Scholar 

  18. Ibrahim, R.W., Elobaid, R.M., Obaiys, S.J.: On sub-classes of analytic functions based on a quantum symmetric conformable differential operator with the application. Adv. Differ. Equ. 2020, 1–14 (2020)

    Article  Google Scholar 

  19. Baleanu, D., Fernandez, A., Akgul, A.: On a fractional operator combining proportional and classical differintegrals. Mathematics 8.3 (2020): 360

  20. Anderson, D.R., Ulness, D.J.: Newly defined conformable derivatives. Adv. Dyn. Syst. Appl 10(2), 109–137 (2015)

    MathSciNet  Google Scholar 

  21. Duren, P.: Univalent Functions, Grundlehren der mathematischen Wissenschaften; 259 Springer-Verlag New York Inc. 1983. ISBN 0-387-90795-5. MR0708494

  22. Srivastava, H.M., Owa, S.: Univalent Functions, Fractional calculus, and their applications, Halsted Press, John Wiley and Sons, New York. Brisbane, and Toronto, Chichester (1989)

  23. Ibrahim, R. W., Jahangiri, Jay M.: Cloud computing center. “Conformable differential operator generalizes the Briot-Bouquet differential equation in a complex domain.” AIMS Math 4, 6: 1582-1595 (2019)

  24. Miller, S.S., Mocanu, P.T.: Differential subordinations: theory and applications. CRC Press, Boca Raton (2000)

    Book  Google Scholar 

  25. Ma W.C., Minda D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, China, (1992): 19-23

  26. Khatter, K., Ravichandran, V., Sivaprasad Kumar, S.: Starlike functions associated with exponential function and the lemniscate of Bernoulli. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 113, 1 (2019): 233–253

  27. Kumar, V., Nak Eun C., Ravichandran, V., Srivastava, H. M.: Sharp coefficient bounds for starlike functions associated with the Bell numbers. Mathematica Slovaca 69, 5 (2019): 1053–1064

  28. Singh, R., Singh, S.: Some sufficient conditions for univalence and starlikeness. In Colloq. Math. 2(47), 309–314 (1982)

    Article  MathSciNet  Google Scholar 

  29. Ruscheweyh, S.: Convolutions in geometric function theory. Presses University, Montreal (1982)

    MATH  Google Scholar 

  30. Jack, I.S.: Functions starlike and convex of order \(\alpha \). J. Lond. Math. Soc. 3, 469–474 (1971)

    Article  MathSciNet  Google Scholar 

  31. Saitoh, H.: Properties of certain analytic functions. Proc. Jpn. Acad. Ser. A Math. Sci. 65(5), 131–134 (1989)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank both anonymous reviewers and the editor for their helpful advice and comments.

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Authors and Affiliations

  1. Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Rabha W. Ibrahim

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Rabha W. Ibrahim

  3. Department of Mathematics, Cankaya University, Balgat, 06530, Ankara, Turkey

    Dumitru Baleanu

  4. Institute of Space Sciences, Magurele, 76900, Bucharest, Romania

    Dumitru Baleanu

  5. Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

    Dumitru Baleanu

Authors
  1. Rabha W. Ibrahim
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  2. Dumitru Baleanu
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All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Correspondence to Rabha W. Ibrahim.

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Ibrahim, R.W., Baleanu, D. On quantum hybrid fractional conformable differential and integral operators in a complex domain. RACSAM 115, 31 (2021). https://doi.org/10.1007/s13398-020-00982-5

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  • DOI: https://doi.org/10.1007/s13398-020-00982-5

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