Third-Order Differential Superordination Involving the Generalized Bessel Functions

Abstract

There are many articles in the literature dealing with the first-order and the second-order differential subordination and differential superordination problems for analytic functions in the unit disk, but there are only a few articles dealing with the third-order differential subordination problems. The concept of third-order differential subordination in the unit disk was introduced by Antonino and Miller, and studied recently by Tang and Deniz. Let be a set in the complex plane C, let p(z) be analytic in the unit disk U = {z : z ∈ C and |z| < 1}, and let ψ : C 4 ×U → C. In this paper, we investigate the problem of determining properties of functions p(z) that satisfy the following third-order differential superordination:As applications, we derive some third-order differential superordination results for analytic functions in U, which are associated with a family of generalized Bessel functions. The results are obtained by considering suitable classes of admissible functions. New third-order differential sandwich-type results are also obtained.

Third-order differential sandwich-type results involving the Liu-Owa integral operator

“…On the other word, Tang et al [33] (see also [34]) obtained the following result for the class of admissible functions Ψ n [Ω, q].…”

“…More recently, Antonino and Miller [6] introduced the notion of third-order differential subordination, and Tang and Deniz [32] studied the third-order differential subordination results for analytic functions involving the generalized Bessel functions. Later on, Tang et al [33] introduced the notion of third-order differential superordination and also studied the corresponding third-order differential superordination involving the generalized Bessel functions. Based on [6] and [33], Tang et al [34] considered third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava operator.…”

“…Later on, Tang et al [33] introduced the notion of third-order differential superordination and also studied the corresponding third-order differential superordination involving the generalized Bessel functions. Based on [6] and [33], Tang et al [34] considered third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava operator. Here, in the present paper, we aim to study third-order differential sandwichtype results of multivalent analytic functions involving the Liu-Owa integral operator Q α β,p defined by (1.2).…”

Third-order differential subordination and superordination involving a fractional operator

“…In 2014, Tang et al [35] investigated some thirdorder differential subordination results for analytic functions involving the generalized Bessel functions. In 2014, Tang et al [36] studied the differential superordination based on analytic functions involving the generalized Bessel functions. In 2014, Farzana et al [37] discussed some third-order differential subordination results for analytic functions which are associated with the fractional derivative operator.…”

“…Proof. Let the function p.z/ be defined by (36) and by (42). Since 2ˆ0 T;2 OE ; q; (43) and (53) From equations (40) and (41), we see that the admissible condition for 2ˆ0 T;2 OE ; q in Definition 4.10 is equivalent to the admissible condition for as given in Definition 2.6 with n D 2: Hence 2 ‰ 0 2 OE ; q; and by using (52) and Theorem 2.7, we have q.z/ p.z/ D T˛f .z/:…”