Difference formula defined by a new differential symmetric operator for a class of meromorphically multivalent functions

Rabha W Ibrahim,Ibtisam Aldawish

doi:10.1186/s13662-021-03442-5

Rabha W Ibrahim, Ibtisam Aldawish

Open Access

https://doi.org/10.1186/s13662-021-03442-5

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Abstract

Symmetric operators have benefited in different fields not only in mathematics but also in other sciences. They appeared in the studies of boundary value problems and spectral theory. In this note, we present a new symmetric differential operator associated with a special class of meromorphically multivalent functions in the punctured unit disk. This study explores some of its geometric properties. We consider a new class of analytic functions employing the suggested symmetric differential operator.

Highlights

The study of the operator is narrowly connected with problems in the theory of functions
The realization in studying multiplication operators is seemed in Toeplitz operators, in the Bergman space of holomorphic functions
The geometric function theory is likewise ironic covering a long list of operators, counting differential, integral, and convolution operators

Summary

Introduction

The study of the operator is narrowly connected with problems in the theory of functions. Ibrahim and Darus (see [1] and for applications see [2,3,4,5]) offered new symmetric differential, integral, and linear symmetric operators for a class of normalized functions in the open unit disk. We proceed to consider a differential symmetric operator (DSO) associated with a class of meromorphically multivalent functions in the punctured unit disk. 2.1 Differential symmetric operator (DSO) In this place, we state a few definitions and a lemma that we shall need . Definition 2.1 For functions φ ∈ k(℘), define the symmetric differential operator as follows: 0φ(z) = φ(z) = z–℘ + φnzn–℘ , n=k αφ(z) = α (1 – α)(–1)℘+1 zφ (z) +. We will need the following subordination definition for our class of meromorphic functions. If c > 0 and h ∈ H[1, n], there are constants λ1 > 0 and λ2 > 0 such that the inequality h(z) + czh (z) ≺ 1 + z λ1 1–z implies h(z) ≺

Main results

Conclusion