Third Order Differential Subordination of Analytic Function Defined by Fractional Derivative Operator

Abstract

Third order differential subordination results are obtained for analytic functions in the open disc which are associated with the fractional derivative operator. These results are obtained by investigating suitable classes of admissible functions. Certain examples are also discussed.

New Results on Higher-Order Differential Subordination and Superordination for Univalent Analytic Functions Using a New Operator

“…Suppose that 𝛺 is a set in 𝐶 and 𝑞 ∈ 𝜗. The admissible functions class 𝛷 𝑛 [𝛺, 𝑞], (𝑛 ∈ 𝑁 ∖ {2}) consists of those functions 𝜓: 𝐶 5 × 𝑈 → 𝐶 that fulfill the following admissibility condition: 𝜓(𝑟, 𝑠, 𝑡, 𝑢, 𝑏; 𝜈) ∉ 𝛺, wherever 𝑟 = 𝑞(𝜏), 𝑠 = 𝑚𝜏𝑞 ′ (𝜏), 𝑅𝑒 ( 𝑡 𝑠 + 1) ≥ 𝑚𝑅𝑒 (1 + 𝜏𝑞 ′′ (𝜏) 𝑞 ′ (𝜏) ) , 𝑅𝑒 ( 𝑢 𝑠 ) ≥ 𝑚 2 𝑅𝑒 ( 𝜏 2 𝑞 ′′′ (𝜏) 𝑞 ′ (𝜏) ) , 𝑅𝑒 ( 𝑏 𝑠 )…”

“…Third-order differential inequalities in the complex plane were considered in 1992 [3], and the concept of third-order differential subordination was introduced in 2011 by Antonino and Miller [4]. Further investigations were conducted on third-order differential subordination results for univalent analytic functions involving an operator [5][6][7][8] and, continuing the idea, the concept of fourth-order differential subordination was introduced and studied in 2020 [9,10]. Further results were published in 2021 and 2022 [11][12][13][14] regarding the new concepts of higher-order differential subordinations.…”

Third-Order Differential Subordination and Differential Superordination Results for Analytic Functions Involving the Srivastava-Attiya Operator

“…The recent work due to Tang et al [18,20] on third order differential subordination attracted to many researchers in this field. For example see [4,8,9,14,18,19,20,13,11]. In the present paper we considered suitable classes of admissible functions associated with Srivastava-Attiya operator and obtained sufficient conditions on the normalized analytic function f such that Sandwich-type subordination of the following form holds:…”

Applications of third order differential subordination and superordination involving generalized struve function

“…In the recent years, few works have been carried out on results related to the third order differential subordination and superordination in the different context. For example see ( [18,23,24,34,[38][39][40]). In this present investigation our aim is to determine third order differential subordination and superordination of generalized Struve function by using the technique developed in [5] and [38].…”