Research article

Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept

  • Received: 24 August 2020 Accepted: 21 October 2020 Published: 02 November 2020
  • MSC : 30C55, 30C45

  • In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by $ \alpha [\varphi(z) \varphi" (z) +(\varphi' (z))^2]+ a_m \varphi^m(z)+a_{m-1} \varphi^{m-1}(z)+...+ a_1 \varphi(z)+ a_0 = 0. $ The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of $e^z.$ Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.

    Citation: Rabha W. Ibrahim, Dumitru Baleanu. Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept[J]. AIMS Mathematics, 2021, 6(1): 806-820. doi: 10.3934/math.2021049

    Related Papers:

  • In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by $ \alpha [\varphi(z) \varphi" (z) +(\varphi' (z))^2]+ a_m \varphi^m(z)+a_{m-1} \varphi^{m-1}(z)+...+ a_1 \varphi(z)+ a_0 = 0. $ The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of $e^z.$ Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.


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    [1] G. G. Gundersen, Research questions on meromorphic functions and complex differential equations, Comput. Meth. Funct. Th., 17 (2017), 195-209. doi: 10.1007/s40315-016-0178-7
    [2] G. Yongyi, X. Zheng, F. Meng, Painleve analysis and abundant meromorphic solutions of a class of nonlinear algebraic differential equations, Math. Probl. Eng., 2019 (2019), 1-11.
    [3] R. W. Ibrahim, R. M. Elobaid, S. J. Obaiys, On the connection problem for Painleve differential equation in view of geometric function theory, Mathematics, 8 (2020), 1-11.
    [4] R. W. Ibrahim, J. M. Jahangiri, Boundary fractional differential equation in a complex domain, Bound. Value Probl., 2014 (2014), 1-11. doi: 10.1186/1687-2770-2014-1
    [5] R. W. Ibrahim, M. Darus, On a new solution of fractional differential equation using complex transform in the unit disk, Math. Comput. Appl., 19 (2014), 152-160.
    [6] R. W. Ibrahim, Fractional algebraic nonlinear differential equations in a complex domain, Afrika Mat., 26 (2015), 385-397. doi: 10.1007/s13370-013-0212-0
    [7] N. Steinmetz, Nevanlinna theory, normal families, and algebraic differential equations, Cham: Springer, 2017.
    [8] Q. Wang, M. Liu, N. Li, On algebraic differential equations concerning the Riemann-zeta function and the Euler-gamma function, arXiv preprint arXiv: 2005.02707, 2020.
    [9] C. Fernando, P. Chartier, A. Escorihuela-Tomàs, Y. Zhang, Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations, J. Comput. Appl. Math., 381 (2020), 1-19.
    [10] B. Edwin, S. Majid, Quantum complex structures, Quantum Riemannian Geometry, Springer, Cham, (2020), 527-564.
    [11] D. H. Phong, Geometric partial differential equations from unified string theories, arXiv preprint arXiv: 1906.03693, (2019), 1-22.
    [12] Y. I. Brodsky, Elements of geometric theory of complex systems behavior, 2020. DOI: 10.37394/23203.2020.15.3-CorpusID:211851348.
    [13] W. M. Seiler, M. Seiss, Singular initial value problems for scalar quasi-linear ordinary differential equations, arXiv preprint arXiv: 2002.06572, 2020.
    [14] F. Aaron, The complex geometry of the free particle, and its perturbations, arXiv preprint arXiv: 2008.03836, 2020.
    [15] V. V. Kravchenko, S. Morelos, S. M. Torba, Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems, Appl. Math. Comput., 273 (2016), 321-336.
    [16] Y. Gu, C. Wu, X. Yao, W. Yuan, Characterizations of all real solutions for the KdV equation and WR, Appl. Math. Lett., 107 (2020), 106446. doi: 10.1016/j.aml.2020.106446
    [17] Y. Gu, Y. Kong, Two different systematic techniques to seek analytical solutions of the higher-order modified Boussinesq equation, IEEE Access, 7 (2019), 96818-96826. doi: 10.1109/ACCESS.2019.2929682
    [18] Y. Gu, F. Meng, Searching for analytical solutions of the (2+1)-dimensional KP equation by two different systematic methods, Complexity, 2019 (2019), 1-11.
    [19] S. S. Miller, P. T. Mocanu, Differential subordinations: Theory and applications, Marcel Dekker Inc. Press, New York, 2000.
    [20] G. Shweta, S. Kumar, V. Ravichandran, First order differential subordinations for Caratheodory functions, Kyungpook Math. J., 58 (2018), 257-270.
    [21] C. D. Michael, Majorization-subordination theorems for locally univalent functions. III, Trans. Amer. Math. Soc., 198 (1974), 297-306. doi: 10.1090/S0002-9947-1974-0349987-5
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