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AIMS Mathematics

2023, Volume 8, Issue 8: 18474-18490. doi: 10.3934/math.2023939
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Research article

Subclasses of spiral-like functions associated with the modified Caputo's derivative operator

  • 1.
    Department of Basic Science, College of Health and Applied Sciences, A'Sharqiyah University, Ibra Oman
  • 2.
    Center for Language and Foundation Studies, A'Sharqiyah University, Ibra Oman
  • Received: 06 March 2023 Revised: 11 May 2023 Accepted: 17 May 2023 Published: 31 May 2023

  • MSC : 30C45, 30C50

  • In this paper, the authors apply the modified Caputo's derivative operator, to introduce two new subclasses of spiral-like functions, namely the spiral-starlike functions and spiral-convex functions. In addition to this we, elaborate on the inclusion properties of these subclasses by considering the generalization of the Mittag-Leffler function and its integral transformation. Consequently, we obtain the subordination result for the functions in the class of spiral-like functions.

    Citation: Jamal Salah, Hameed Ur Rehman, Iman Al Buwaiqi, Ahmad Al Azab, Maryam Al Hashmi. Subclasses of spiral-like functions associated with the modified Caputo's derivative operator[J]. AIMS Mathematics, 2023, 8(8): 18474-18490. doi: 10.3934/math.2023939

    Related Papers:

  • In this paper, the authors apply the modified Caputo's derivative operator, to introduce two new subclasses of spiral-like functions, namely the spiral-starlike functions and spiral-convex functions. In addition to this we, elaborate on the inclusion properties of these subclasses by considering the generalization of the Mittag-Leffler function and its integral transformation. Consequently, we obtain the subordination result for the functions in the class of spiral-like functions.


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    [1] M. I. S. Robertson, On the theory of univalent functions, Ann. Math., 37 (1936), 374–408. https://doi.org/10.2307/1968451 doi: 10.2307/1968451
    [2] B. A. Frasin, T. Al-Hawary, F. Yousef, Necessary and sufficient conditions for hypergeometric functions to be in a subclass of analytic functions, Afr. Mat., 30 (2019), 223–230. https://doi.org/10.1007/s13370-018-0638-5. doi: 10.1007/s13370-018-0638-5
    [3] H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math., 2011 (2011), 1–51. https://doi.org/10.1155/2011/298628 doi: 10.1155/2011/298628
    [4] Y. C. Kim, T. Sugawa, Correspondence between spirallike functions and starlike functions, Math. Nachr., 285 (2012), 322–331. https://doi.org/10.1002/mana.201010020. doi: 10.1002/mana.201010020
    [5] D. Aharonov, M. Elin, D. Shoikhet, Spiral-like functions with respect to a boundary point, J. Math. Anal. Appl., 280 (2003), 17–29. https://core.ac.uk/download/pdf/82536676.pdf
    [6] B. A. Frasin, I. Aldawish, On subclasses of uniformly spiral-like functions associated with generalized Bessel functions, J. Funct. Space., 2019 (2019), 1–6. https://doi.org/10.1155/2019/1329462 doi: 10.1155/2019/1329462
    [7] H. M. Srivastava, N. Khan, M. Darus, M. T. Rahim, Q. Z. Ahmad, Y. Zeb, Properties of spiral-like close-to-convex functions associated with conic domains, Mathematics, 7 (2019), 706. https://doi.org/10.3390/math7080706 doi: 10.3390/math7080706
    [8] S. A. Shah, L. I. Cotirla, A. Catas, C. Dubau, G. Cheregi, A study of spiral-like harmonic functions associated with quantum calculus, J. Funct. Space., 2022 (2022), 1–7. https://doi.org/10.1155/2022/5495011 doi: 10.1155/2022/5495011
    [9] T. M. Seoudy, A. E. Shammaky, Certain subclasses of spiral-like functions associated with q-analogue of Carlson-Shaffer operator, AIMS Math., 6 (2021), 2525–2538. https://doi.org/10.3934/math.2021153 doi: 10.3934/math.2021153
    [10] L. I. Cotîrlǎ, K. R. Karthikeyan, Classes of multivalent spiral-like functions associated with symmetric regions, Symmetry, 14 (2022), 1598. https://doi.org/10.3390/sym14081598 doi: 10.3390/sym14081598
    [11] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1–15. https://doi.org/10.48550/arXiv.1106.0965 doi: 10.48550/arXiv.1106.0965
    [12] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and Applications, Switzerland: Gordon and Breach Science Publishers, 1993. Available from: https://www.gbv.de/dms/hebis-darmstadt/toc/32759916.pdf.
    [13] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, In: North Holland Mathematics Studies, Vol. 204, Elsevier Science, Publishers BV, Amsterdam, 2006. Available from: http://www.math-frac.org/TAFDE_Table_of_Contents.pdf.
    [14] I. Podlubny, Fractional differential equations, Vol. 198, Academic Press. San Diego, California, USA, 1999. Available from: https://documents.pub/download/igor-podlubny-fractional-differential-equations.html.
    [15] H. U. Rehman, M. Darus, J. Salah, A note on Caputo's derivative operator interpretation in economy, J. Appl. Math., 2018 (2018), 1–7. https://doi.org/10.1155/2018/1260240 doi: 10.1155/2018/1260240
    [16] W. G. Atshan, Fractional calculus on a subclass of spiral-like functions defined by Komatu operator, Int. Math. Forum, 3 (2008), 1587–1594. Available from: http://www.m-hikari.com/imf-password2008/29-32-2008/index.html.
    [17] W. G. Atshan, R. N. Abdul-Hussien, On a subclass of spiral-like Functions by applying fractional calculus, Int. J. Sci. Res., 4 (2015), 2424–2428. Available from: https://www.ijsr.net/archive/v4i5/SUB153569.pdf.
    [18] J. Salah, M. Darus, A subclass of uniformly convex functions associated with a fractional calculus operator involving Caputo's fractional differentiation, Acta Univ. Apulensis. Math.-Inform., 24 (2010), 295–306. https://eudml.org/doc/222861
    [19] G. S. Salagean, Subclasses of univalent functions, In: C. A. Cazacu, N. Boboc, M. Jurchescu, I. Suciu, eds., Springer, Berlin, Heidelberg, Complex Analysis—Fifth Romanian-Finnish Seminar, Lecture Notes in Mathematics, 1013 (1983), 362–372. https://doi.org/10.1007/BFb0066543
    [20] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc., 16 (1969), 755–758. Available from: https://www.ams.org/journals/proc/1965-016-04/S0002-9939-1965-0178131-2/.
    [21] S. Owa, Some properties of fractional calculus operators for certain analytic functions, RIMS Kôkyûroku, 1626 (2009), 86–92. Available from: https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1626-12.pdf.
    [22] J. Salah, Certain subclass of analytic functions associated with fractional calculus operator, Trans. J. Math. Mech., 3 (2011), 35–42. Available from: http://tjmm.edyropress.ro/journal/11030106.pdf.
    [23] J. Salah, A note on Starlike functions of order α associated with a fractional calculus operator involving Caputo's fractional, J. Appl. Comput. Sci. Math., 5 (2011), 97–101. Available from: https://www.jacsm.ro/view/?pid=10_16.
    [24] J. Salah, M. Darus, On convexity of general integral operators, J. An. Univ. Vest. Timisoara. Ser. Mat.–Inform. XLIX, 1 (2011), 117–124. Available from: https://ukmsarjana.ukm.my/main/penerbitanterkini/SzAwNjM5Mg==.
    [25] G. Mittag-Leffler, Sur la représentation analytique d'une branche uniforme d'une fonction monogène, Acta Math., 29 (1905), 101–181. https://doi.org/10.1007/BF02403200 doi: 10.1007/BF02403200
    [26] G. Mittag-Leffler, Sur la nouvelle fonction Eα(x), C. R. Acad. Sci. Paris, 137 (1903), 554–558. Available from: https://archive.org/details/comptesrendusheb137acad/page/554/mode/2up.
    [27] D. Bansal, S. Maharana, Sufficient condition for strongly starlikeness of normalized Mittag-Leffler function, J. Complex Anal., 2017 (2017), 1–4. https://doi.org/10.1155/2017/6564705. doi: 10.1155/2017/6564705
    [28] S. Elhaddad, M. Darus, Some properties of certain subclass of analytic functions associated with generalized differential operator involving Mittag-Leffler function, Trans. J. Math. Mech., 10 (2018), 1–7. Available from: http://tjmm.edyropress.ro/journal/18100101.pdf.
    [29] A. A. Attiya, Some applications of Mittag-Leffler function in the unit disc, Filomat, 30 (2016), 2075–2081. https://doi.org/10.2298/FIL1607075A doi: 10.2298/FIL1607075A
    [30] D. Răducanu, On partial sums of normalized Mittag-Leffler functions, An. Univ. Ovid. Co.-Mat., 25 (2017), 123–133. Available from: https://sciendo.com/article/10.1515/auom-2017-0024.
    [31] Y. Kao, Y. Li, J. H. Park, X. Chen, Mittag-Leffler synchronization of delayed fractional memristor neural networks via adaptive control, IEEE T. Neur. Net. Lear., 32 (2021), 2279–2284. https://doi.org/10.1109/TNNLS.2020.2995718 doi: 10.1109/TNNLS.2020.2995718
    [32] Y. Cao, Y. Kao, J. H. Park, H. Bao, Global Mittag-Leffler stability of the delayed fractional-coupled reaction-diffusion system on networks without strong connectedness, IEEE T. Neur. Net. Lear., 33 (2022), 6473–6483. https://doi.org/10.1109/TNNLS.2021.3080830 doi: 10.1109/TNNLS.2021.3080830
    [33] Y. Kao, Y. Cao, X. Chen, Global Mittag-Leffler synchronization of coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks via sliding mode control, Chaos, 32 (2022), 113123. https://doi.org/10.1063/5.0102787 doi: 10.1063/5.0102787
    [34] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460–481.
    [35] J. Salah, M. A. Hashmi, H. U. Rehman, K. A. Mashrafi, Modified mathematical models in biology by the means of Caputo derivative of a function with respect to another exponential function, Math. Stat., 10 (2022), 1194–1205. https://doi.org/10.13189/ms.2022.100605 doi: 10.13189/ms.2022.100605
    [36] O. Jawabreh, A. A. Qader, J. Salah, K. A. Mashrafi, E. A. D. A. L. Fahmawee, B. J. A. Ali, Fractional calculus analysis of tourism mathematical model, Progr. Fract. Differ., 9 (2023), 1–11. https://doi.org/10.18576/pfda/09S101 doi: 10.18576/pfda/09S101
    [37] H. U. Rehman, M. Darus, J. Salah, Coefficient properties involving the generalized k-Mittag-Leffler functions, Transylv. J. Math. Mech., 9 (2017), 155–164. Available from: http://tjmm.edyropress.ro/journal/17090206.pdf.
    [38] K. S Nisar, S. D. Purohit, M. S. Abouzaid, M. A. Qurashi, Generalized k-Mittag-Leffler function and its composition with pathway integral operators, J. Nonlinear Sci. Appl., 9 (2016), 3519–3526. http://dx.doi.org/10.22436/jnsa.009.06.07 doi: 10.22436/jnsa.009.06.07
    [39] A. Wiman, Über den Fundamentalsatz in der Teorie der Funktionen Eα(x), Acta Math., 29 (1905), 191–201. https://doi.org/10.1007/BF02403202 doi: 10.1007/BF02403202
    [40] A. K. Shukla, J. C. Prajapti, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 797–811. https://doi.org/10.1016/j.jmaa.2007.03.018. doi: 10.1016/j.jmaa.2007.03.018
    [41] J. Salah, M. Darus, A note on generalized Mittag-Leffler function and Application, Far East. J. Math. Sci., 48 (2011), 33–46. Available from: http://www.pphmj.com/abstract/5478.html.
    [42] E. Lindelöf, Mémoire sur certaines inégalités dans la théorie des fonctions monogènes et sur quelques propriétés nouvelles de ces fonctions dans le voisinage d'un point singulier essentiel, Acta Societas Scientiarium Fennicae, 1909. Available from: https://ia801001.us.archive.org/8/items/actasocietatissc35suom/actasocietatissc35suom.pdf.
    [43] W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, Proceeding of the Conference on Complex Analysis, New York: International Press, 1994,157–169. Available from: https://cir.nii.ac.jp/crid/1570572700543766144.
    [44] H. S. Wilf, Subordinating factor sequences for convex maps of the unit circle, Proc. Amer. Math. Soc., 12 (1961), 689–693. Available from: https://www.ams.org/journals/proc/1961-012-05/S0002-9939-1961-0125214-5/home.html.
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