Harmonic Bloch Function Spaces and their Composition Operators

Authors: S. ESMAEILI, Y. ESTAREMI AND A. EBADIAN
DOI: 10.46793/KgJMat2105.699E
Abstract:
Owning to the importance and great interest of linear operators, a generalisation of linear derivative operator

Keywords:
Analytic functions, starlike functions, linear operator, superordination, subordination, Mittag-Leffler function.
References:
[1] F. M. Al-Oboudi, On univalent functions defined by a generalized Sǎlǎgean operator, Int. J. Math. Math. Sci. 27 (2004), 1429–1436.
[2] O. Al-Refai and M. Darus, Main differential sandwich theorem with some applications, Lobachevskii J. Math. 30 (2009), 1–11.
[3] M. K. Aouf, A. O. Mostafa and R. El-Ashwah, Sandwich theorems for p-valent functions defined by a certain integral operator, Math. Comput. Modelling 53 (2011), 1647–1653.
[4] A. A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat 30 (2016), 2075–2081.
[5] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984), 737–745.
[6] J. Dziok and H. S. Srivastava, Classes of analytic functions associated with the generalised hypergeometric function, Appl. Math. Comput. 103 (1999), 1–13.
[7] S. Elhaddad, H. Aldweby and M. Darus, On certain subclasses of analytic functions involving differential operator, Jnanabha 48(I) (2018), 55–64.
[8] S. Elhaddad, H. Aldweby and M. Darus, Majorization properties for subclass of analytic p-valent functions associated with generalized differential operator involving Mittag-Leffler function, Nonlinear Functional Analysis and Applications 23(4) (2018), 743–753.
[9] S. Elhaddad and M. Darus, On meromorphic functions defined by a new operator containing the Mittag-Leffler function, Symmetry 11(2) (2019), Article ID 210.
[10] S. Elhaddad and M. Darus, Some properties of certain subclasses of analytic function associated with generalized differential operator involving Mittag-Leffler function, Transylvanian Journal of Mathematics and Mechanics 10(1) (2018), 1–7.
[11] J. E. Hohlov, Operators and operations on the class of univalent functions, Izvestiya Vysshikh Uchebnykh Zavedenii Matematika 10 (1978), 83–89.
[12] R.W. Ibrahim and M. Darus, Subordination and superordination for functions based on Dziok-Srivastava linear operator, Bull. Math. Anal. Appl. 2 (2010), 15–26.
[13] S. S. Miller and P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65 (1978), 298–305.
[14] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (1981), 157–171.
[15] S. S. Miller and P. T. Mocanu, Differential Subordinations. Theory and Applications, Marcel Dekker, New York, 2000.
[16] S. S. Miller and P. T. Mocanu, Subordinants of differential superordinations, Complex Variables, Theory and Application 84 (2003), 815–826.
[17] G. M. Mittag-Leffler, Sur la nouvelle fonction Eα(x), C. R. Math. Acad. Sci. Paris 137(2) (1903), 554–558.
[18] G. M. Mittag-Leffler, Sur la representation analytique d’une branche uniforme d’une fonction monogene, Acta Math. 29(1) (1905), 101–181.
[19] N. M. Mustafa and M. Darus, Differential subordination and superordination for a new linear derivative operator, International Journal of Pure and Applied Mathematics 70 (2011), 825–835.
[20] H. Rehman, M. Darus, and J. Salah, Coefficient properties involving the generalized K-Mittag-Leffler functions, Transylvanian Journal of Mathematics and Mechanics 9(2) (2017), 155–164.
[21] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115.
[22] G. S. Sǎlǎgean, Subclasses of Univalent Functions, Lecture Notes in Mathematics 1013, Springer-Verlag, Heidelberg, 1983, 362–372.
[23] J. Salah and M. Darus, A note on generalized Mittag-Leffler function and application, Far East Journal of Mathematical Sciences 48(1) (2011), 33–46.
[24] T. N. Shanmugam, S. Sivasubramanian and H. M. Srivastava, Differential sandwich theorems for certain subclasses of analytic functions involving multiplier transformations, Integral Transforms Spec. Funct. 17(12) (2006), 889–899.
[25] H. M. Srivastava, B. A. Frasin and V. Pescar, Univalence of integral operators involving Mittag- Leffler functions, Appl. Math. Inf. Sci. 11(3) (2017), 635–641.
[26] H. M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), 198–210.
[27] A. Wiman, Über den fundamentalsatz in der teorie der funktionen Eα(x), Acta Math. 29(1) (1905), 191–201.