Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access November 4, 2015

Upper and lower bounds of integral operator defined by the fractional hypergeometric function

  • Rabha W. Ibrahim , Muhammad Zaini Ahmad and Hiba F. Al-Janaby
From the journal Open Mathematics

Abstract

In this article, we impose some studies with applications for generalized integral operators for normalized holomorphic functions. By using the further extension of the extended Gauss hypergeometric functions, new subclasses of analytic functions containing extended Noor integral operator are introduced. Some characteristics of these functions are imposed, involving coefficient bounds and distortion theorems. Further, sufficient conditions for subordination and superordination are illustrated.

References

[1] Mandelbrot B.B., Van Ness J.W., Fractional Brownian motions, fractional noises and applications, SIAM review, 1968, 10, 422-437. 10.1137/1010093Search in Google Scholar

[2] Arovas D., Schrieffer J.R., Wilczek F., Fractional statistics and the quantum Hall effect, Physical review letters, 1984, 53, 722. 10.1103/PhysRevLett.53.722Search in Google Scholar

[3] Wilczek F., Quantum mechanics of fractional-spin particles, Physical review letters, 1982, 49, 957. 10.1103/PhysRevLett.49.957Search in Google Scholar

[4] Baillie R.T., Long memory processes and fractional integration in econometrics, Journal of Econometrics, 1996, 73, 5-59. 10.1016/0304-4076(95)01732-1Search in Google Scholar

[5] He J.-H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 1998, 167 57-68. 10.1016/S0045-7825(98)00108-XSearch in Google Scholar

[6] Hilfer R., et al., Applications of fractional calculus in physics, World Scientific, 2000. 10.1142/3779Search in Google Scholar

[7] Baleanu D.,et al., Fractional Calculus: Models and Numerical Methods, World Scientific, 2012 10.1142/8180Search in Google Scholar

[8] Yang X.-J., Advanced local fractional calculus and its applications, World Science, New York, NY, USA, 2012. Search in Google Scholar

[9] Wu G.-C., Baleanu D., Discrete fractional logistic map and its chaos, Nonlinear Dynamics, 2014, 75, 283-287. 10.1007/s11071-013-1065-7Search in Google Scholar

[10] Chen F.L., A review of existence and stability results for discrete fractional equations, Journal of Computational Complexity and Applications, 2015, 1, 22-53. Search in Google Scholar

[11] Li M., Fractal time series- a tutorial review, Mathematical Problems in Engineering, 2010, 2010 1-26. 10.1155/2010/157264Search in Google Scholar

[12] Yang X. J.,et al., Fractal boundary value problems for integral and differential equations with local fractional operators, Thermal Science, 2015, DOI: 0354-98361300103Y. Search in Google Scholar

[13] Atici F., Eloe P., Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, 2009, 137, 981-989 10.1090/S0002-9939-08-09626-3Search in Google Scholar

[14] Yang X.-J., Local Fractional Functional Analysis & Its Applications, Asian Academic Publisher Limited Hong Kong, 2011 Search in Google Scholar

[15] Ibrahim R.W., On generalized Hyers-Ulam stability of admissible functions, Abstract and Applied Analysis, 2012, 2012, 1-10. 10.1155/2012/749084Search in Google Scholar

[16] Ibrahim R.W., Modified fractional Cauchy problem in a complex domain, Advances in Difference Equations, 2013, 2013, 1-10. 10.1186/1687-1847-2013-149Search in Google Scholar

[17] Ibrahim R.W., Jahangiri J., Boundary fractional differential equation in a complex domain, Boundary Value Problems 2014, 2014, 1-11. 10.1186/1687-2770-2014-66Search in Google Scholar

[18] Ibrahim R.W., Sokol J., On a new class of analytic function derived by fractional differential operator, Acta Mathematica Scientia, 2014, 34B(4), 1-10. 10.1016/S0252-9602(14)60093-XSearch in Google Scholar

[19] Ibrahim R.W., Fractional Cauchy problem in sense of the complex Hadamard operators, Mathematics Without Boundaries, Springer, 2014, 259-272. 10.1007/978-1-4939-1106-6_10Search in Google Scholar

[20] Ibrahim R.W., Studies on generalized fractional operators in complex domain, Mathematics Without Boundaries, Springer, 2014, 273-284. 10.1007/978-1-4939-1106-6_11Search in Google Scholar

[21] Branges L., A proof of the Bieberbach conjecture, Acta Math., 1985, 154, 137–152. 10.1007/BF02392821Search in Google Scholar

[22] Srivastava H. M., et al., Generating functions for the generalized Gauss hypergeometric functions, Appl. Math. Comput., 2014, 247, 348–352. 10.1016/j.amc.2014.08.105Search in Google Scholar

[23] Luo M. J., et al., Some results on the extended beta and extended hypergeometric functions, Appl. Math. Comput., 2014, 248, 631–651. 10.1016/j.amc.2014.09.110Search in Google Scholar

[24] Srivastava H. M., Choi J., Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York. 2012. 10.1016/B978-0-12-385218-2.00002-5Search in Google Scholar

[25] Agarwal P., et al., Extended Riemann-Liouville fractional derivative operator and its applications, J. Nonlinear Sci. Appl., 2015, 8, 451–466. 10.22436/jnsa.008.05.01Search in Google Scholar

[26] S. Ruscheweyh, New criteria for univalent functions, Proceedings of the American Mathematical Society, 1975, 49, 109–115. 10.1090/S0002-9939-1975-0367176-1Search in Google Scholar

[27] Noor K. L., On new classes of integral operators, Journal of Natural Geometry, 1999, 16, 71–80. Search in Google Scholar

[28] Noor K. L., Integral operators defined by convolution with hypergeometric functions, Appl. Math. Comput., 2006, 182, 1872–1881. 10.1016/j.amc.2006.06.023Search in Google Scholar

[29] Ibrahim R. W., Darus M., New classes of analytic functions involving generalized Noor integral operator, Journal of Inequalities and Applications, 2008, 390435, 1–14. 10.1155/2008/390435Search in Google Scholar

[30] Miller S. S., Mocanu P. T., Differential Subordinations, Theory and applications, Monographs and Textbooks in Pure and Applied Mathematics, 225, Dekker, New York, 2000. 10.1201/9781482289817Search in Google Scholar

[31] Miller S. S., Mocanu P. T., Subordinants of differetial superordinations, Complex Var. Theory Appl., 2003, 48 (10), 815–826. 10.1080/02781070310001599322Search in Google Scholar

[32] Brickman L., like analytic functions, Bulletin of the American Mathematical Society, 1973, 79 (3), 555–558. 10.1090/S0002-9904-1973-13200-8Search in Google Scholar

[33] Ruscheweyh St., A subordination theorem for like functions, J. London Math. Soc., 1976, 2(13), 275–280.10.1112/jlms/s2-13.2.275Search in Google Scholar

[34] Bulboaca T., Classes of first-order differential superordinations, Demonstratio Mathematica, 2002, 35(2), 287–292. 10.1515/dema-2002-0209Search in Google Scholar

[35] Ravichandran V., Jayamala M., On sufficient conditions for Caratheodory functions, Far East Journal of Mathematical Sciences, 2004, 12, 191–201. Search in Google Scholar

[36] Ali R., et al., Differential sandwich theorems for certain analytic functions, Far East Journal of Mathematical Sciences, 2005, 15, 87–94. Search in Google Scholar

[37] Ibrahim R. W., et al., Third-order differential subordination and superordination involving a fractional operator, Open Mathematics, 2015, To appear. 10.1515/math-2015-0068Search in Google Scholar

[38] Yang X. -J., et al., Local Fractional Integral Transforms and Their Applications, Elsevier, 2015. 10.1016/B978-0-12-804002-7.00004-8Search in Google Scholar

[39] Yang X. J., Srivastava H. M., Cattani C., Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Romanian Reports in Physics, 2015, 67(3), 752-761. Search in Google Scholar

[40] Yang X. J., Machado J. T., Hristov J., Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow. Nonlinear Dynamics, 2015, 1-5. 10.1007/s11071-015-2085-2Search in Google Scholar

[41] Yang X. J., Srivastava H. M., An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives. Communications in Nonlinear Science and Numerical Simulation, (2015), 29(1), 499-504. 10.1016/j.cnsns.2015.06.006Search in Google Scholar

Received: 2015-7-29
Accepted: 2015-9-10
Published Online: 2015-11-4

©2015 Ibrahim et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 27.2.2024 from https://www.degruyter.com/document/doi/10.1515/math-2015-0071/html
Scroll to top button