Third-order differential subordination and superordination involving a fractional operator

Rabha W. Ibrahim; Muhammad Zaini Ahmad; Hiba F. Al-Janaby

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

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The third-order differential subordination and the corresponding differential superordination problems for a new linear operator convoluted the fractional integral operator with the Carlson-Shaffer operator, are investigated in this study. The new operator satisfies the required first-order differential recurrence (identity) relation. This property employs the subordination and superordination methodology. Some classes of admissible functions are determined, and these significant classes are exploited to obtain fractional differential subordination and superordination results. The new third-order differential sandwich-type outcomes are investigated in subsequent research.

How to cite

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Rabha W. Ibrahim, Muhammad Zaini Ahmad, and Hiba F. Al-Janaby. "Third-order differential subordination and superordination involving a fractional operator." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/275994>.

@article{RabhaW2015,
abstract = {The third-order differential subordination and the corresponding differential superordination problems for a new linear operator convoluted the fractional integral operator with the Carlson-Shaffer operator, are investigated in this study. The new operator satisfies the required first-order differential recurrence (identity) relation. This property employs the subordination and superordination methodology. Some classes of admissible functions are determined, and these significant classes are exploited to obtain fractional differential subordination and superordination results. The new third-order differential sandwich-type outcomes are investigated in subsequent research.},
author = {Rabha W. Ibrahim, Muhammad Zaini Ahmad, Hiba F. Al-Janaby},
journal = {Open Mathematics},
keywords = {Fractional calculus; Fractional integral operator; Subordination; Superordination; Unit disk; Analytic function},
language = {eng},
number = {1},
pages = {null},
title = {Third-order differential subordination and superordination involving a fractional operator},
url = {http://eudml.org/doc/275994},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Rabha W. Ibrahim
AU - Muhammad Zaini Ahmad
AU - Hiba F. Al-Janaby
TI - Third-order differential subordination and superordination involving a fractional operator
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - The third-order differential subordination and the corresponding differential superordination problems for a new linear operator convoluted the fractional integral operator with the Carlson-Shaffer operator, are investigated in this study. The new operator satisfies the required first-order differential recurrence (identity) relation. This property employs the subordination and superordination methodology. Some classes of admissible functions are determined, and these significant classes are exploited to obtain fractional differential subordination and superordination results. The new third-order differential sandwich-type outcomes are investigated in subsequent research.
LA - eng
KW - Fractional calculus; Fractional integral operator; Subordination; Superordination; Unit disk; Analytic function
UR - http://eudml.org/doc/275994
ER -

References

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  1. [1] Alexander J. W., Functions which map the interior of the unit circle upon simple regions, Ann. of Math., 1915, 17, 12–22. [Crossref] Zbl45.0672.02
  2. [2] Libera R. J., Some classes of regular univalent functions, Proc. Amer. Math. Soc., 1965, 16, 755–758. [Crossref] Zbl0158.07702
  3. [3] Bernardi S. D., Convex and starlike univalent functions, Trans. Amer. Math. Soc., 1969, 135, 429–446. Zbl0172.09703
  4. [4] Miller S. S., Mocanu P. T., Reade M. O., Starlike integral operators, Pacific J. Math., 1978, 79, 157–168. Zbl0398.30007
  5. [5] Miller S. S., Mocanu P. T., Classes of univalent integral operators, J. Math. Anal. Appl., 1991, 157, 147–165. [Crossref] Zbl0729.30011
  6. [6] Singh R., On Bazilevic functions, Proc. Amer. Math. Soc., 1973, 18 261–271. Zbl0262.30014
  7. [7] Pascu N. N., Pescar V., On integral operators of Kim-Merkes and Pfaltz-graff, Mathematica (Cluj), 1990, 2, 185–192. Zbl0761.30011
  8. [8] Pescar V., Breaz D., Some integral operators and their univalence, Acta Univ. Apulensis Math., Inform. 2008, 15, 147–152. Zbl1199.30094
  9. [9] Breaz D., Breaz N., Srivastava H. M., An extension of the univalent condition for a family of integral operators, Appl. Math. Lett., (2009), 22, 41–44. [WoS][Crossref] Zbl1163.30304
  10. [10] Breaz D., Darus M., Breaz N., Recent Studies on Univalent Integral Operators, Alba Iulia: Aeternitas, 2010. Zbl1207.41012
  11. [11] Darus M., Ibrahim R. W., On subclasses of uniformly Bazilevic type functions involving generalized differential and integral operators, FJMS, 2009, 33, 401–411. Zbl1168.30306
  12. [12] Darus M., Ibrahim R. W., On inclusion properties of generalized integral operator involving Noor integral, FJMS, 2009, 33, 309–321. Zbl1168.30305
  13. [13] Hernandez R., Prescribing the preschwarzian in several complex variables, Annales Academiae Scientiarum Fennicae Mathematica, 2011, 36, 331–340. [Crossref][WoS] Zbl1227.32021
  14. [14] Ong K. W., Tan S. L., Tu Y. E., Integral operators and univalent functions, Tamkang Journal of Mathematics, 2012, 43(2), 215–221. Zbl1255.30029
  15. [15] Goluzin G. M., On the majorization principle in function theory (Russian). Dokl. Akad. Nauk. SSSR, 1953, 42, 647–650. 
  16. [16] Suffridge T. J., Some remarks on convex maps of the unit disk. Duke Math. J., 1970, 37, 775–777. Zbl0206.36202
  17. [17] Robinson R. M., Univalent majorants, Trans. Amer. Math. Soc., 1947, 61, 1–35. [Crossref] 
  18. [18] Hallenbeck D. J., Ruscheweyh S., Subordination by convex functions, Proc. Amer. Math. Soc., 1975, 52, 191–195. [Crossref] Zbl0311.30010
  19. [19] Miller S.S., Mocanu P.T., Differential subordinations and univalent function, Michig. Math. J., 1981, 28, 157–171. [Crossref] Zbl0439.30015
  20. [20] Miller S.S., Mocanu P.T., Differential subordinations and inequalities in the complex plane, J. Diff. Eqn., 1987, 67, 199–211. Zbl0633.34005
  21. [21] Miller S.S., Mocanu P.T., The theory and applicatins of second-order differential subordinations, Studia Univ. Babes-Bolyai, math., 1989, 34, 3–33. 
  22. [22] Miller S. S., Mocanu P. T., Differential Subordinations, Theory and applications, Monographs and Textbooks in Pure and Applied Mathematics, 225, Dekker, New York, 2000. 
  23. [23] Miller S. S., Mocanu P. T., Subordinants of differetial superordinations, Complex Var. Theory Appl., 2003, 48, 815–826. Zbl1039.30011
  24. [24] Bulboac Ma T., Differential subordinations and superordinations, Recent Results, House of Scientific Book Publ., Cluj-Napoca, 2005. 
  25. [25] Baricz A., Deniz E., Caglar M., Orhan H., Differential subordinations involving the generalized Bessel functions, Bull. Malays. Math. Sci. Soc., DOI: 10.1007/s40840-014-0079-8. [WoS][Crossref] Zbl1316.30010
  26. [26] Cho N. E., Bilboaca T., Srivastava H. M., A general family of integral operators and associated subordination and superordination properties of some special analytic function classes, Appl. Math. Comput., 2012, 219, 2278–2288. [WoS] Zbl1293.30027
  27. [27] Kuroki K., Srivastava H. M., Owa S., Some applications of the principle of differential, Electron. J. Math. Anal. Appl., 2013, 1 (50), 40–46. 
  28. [28] Xu Q.-H., Xiao H.-G., Srivastava H. M., Some applications of differential subordination and the Dziok-Srivastava convolution operator, Appl. Math. Comput., 2014, 230, 496–508. 
  29. [29] Ali R. M., Ravichandran V., Seenivasagan N., Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava operator, J. Franklin Inst., 2010, 347, 1762–1781. [WoS] Zbl1204.30008
  30. [30] Ali R. M., Ravichandran V., Seenivasagan N., On Subordination and superordination of the multiplier transformation for meromorphic functions, Bull. Malays. Math. Sci. Soc., 2010, 33, 311–324. Zbl1189.30009
  31. [31] Ponnusamy S., Juneja O. P., Third-order differential inequalities in the complex plane, Current Topics in Analytic Function Theory, World Scientific, Singapore, London, 1992. Zbl0991.30012
  32. [32] Antonion J. A., Miller S. S., Third-order differential inequalities and subordinations in the complex plane, Complex Var. Theory Appl., 2011, 56, 439–454. Zbl1220.30035
  33. [33] Jeyaraman M. P., Suresh T. K., Third-order differential subordination of analysis functions, Acta Universitatis Apulensis, 2013, 35, 187–202. Zbl1340.30103
  34. [34] Tang H., Srivastiva H. M., Li S., Ma L., Third-order differential subordinations and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava Operator, Abstract and Applied Analysis, 2014, 1–11. [WoS][Crossref] 
  35. [35] Tang H., Deniz E., Third-order differential subordinations results for analytic functions involving the generalized Bessel functions, Acta Math. Sci., 2014, 6, 1707–1719. [WoS][Crossref] Zbl1340.30080
  36. [36] Tang H., Srivastiva H. M., Deniz E., Li S., Third-order differential superordination involving the generalized Bessel functions, Bull. Malays. Math. Sci. Soc., 2014, 1–22. [WoS] 
  37. [37] Farzana H. A., Stephen B. A., Jeyaraman M. P., Third-order differential subordination of analytic function defined by functional derivative operator, Annals of the Alexandru Ioan Cuza University - Mathematics, 2014, 1–16. 
  38. [38] B. C. Carlson and D. B. Shaffer,Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 1984, 15, 737–745. Zbl0567.30009
  39. [39] Machado J. T., Discrete-time fractional-order controllers, Fractional Calculus and Applied Analysis, 2001, 4, 47–66. Zbl1111.93307
  40. [40] Pu Y.-F., Zhou J.-L., Yuan X., Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement, Image Processing, IEEE Transactions on, 2010, 19, 491–511. [WoS] 
  41. [41] Jalab H. A., Ibrahim R. W., Fractional Conway polynomials for image denoising with regularized fractional power parameters, J. Math. Imaging Vis., 2015, 51, 442–450. [Crossref][WoS] Zbl1331.94021
  42. [42] Jalab H A, Ibrahim R. W., Fractional Alexander polynomials for image denoising, Signal Processing, 2015, 107, 340–354. 
  43. [43] Wu G.C., Baleanu D., Zeng S.D., Deng Z.G., Discrete fractional diffusion equation, Nonlinear Dynamics, 2015, 80, 1–6. Zbl06496111

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