Harmonic Bloch Function Spaces and their Composition Operators
Download PDF
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 δ,pm(α,β,a1,b1)f(z) is newly introduced in this study. The main objective of this paper is to investigate various subordination and superordination related to the aforementioned generalised linear derivative operator. Additionally, the resultant sandwich-type of this operator is also considered.
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.