Properties of Differential Subordination and Superordination for Multivalent Functions Associated with the Convolution Operators

Abstract

Using convolution (or Hadamard product), we define the El-Ashwah and Drbuk linear operator, which is a multivalent function in the unit disk U=w:w<1 and w∈₵, and satisfy its specific relationship to derive the subordination, superordination, and sandwich results for this operator by using properties of subordination and superordination concepts.

$K^{th}$-order Differential Subordination Results of Analytic Functions in the Complex Plane

“…The theory of differential subordination in C is the generalization of differential inequality in R. Many of the significant works on differential subordination have been pioneered by Miller and Mocanu, and their monograph [10] compiled their great efforts in introducing and developing the same. In recent years, various authors have successfully applied the theory of first and second order differential subordination to address many important problems in this field for example (see [5,6,9,12,14,[18][19][20][21]).…”

Third-Order Differential Subordination for Generalized Struve Function Associated with Meromorphic Functions

“…(𝑧 ∈ 𝑈𝐷) ⇔ 𝑓 ̌(0) = 𝑔 ̌(0) and 𝑓 ̌(𝑈𝐷) ⊆ 𝑔 ̌(𝑈𝐷). [1][2][3] Let ϓ(𝑡, 𝑢, 𝑣, 𝑤; 𝑧): ℂ 4 x 𝑈𝐷 → ℂ and ħ(𝑧) be univalent in unit disk 𝑈𝐷. If ℳ(𝑧) is analytic in 𝑈𝐷 satisfies:…”