Strong Differential Subordinations Obtained with New Integral Operator Defined by Polylogarithm Function

Abstract

By using the polylogarithm function, a new integral operator is introduced. Strong differential subordination and superordination properties are determined for some families of univalent functions in the open unit disk which are associated with new integral operator by investigating appropriate classes of admissible functions. New strong differential sandwich-type results are also obtained.

On a Certain Subclass of Analytic Functions Involving Integral Operator Defined by Polylogarithm Function

“…. This implies that the desired in equality (6) holds. If f (z) = z (|z| = r < 1), then there exist a coefficient 1+c k+c δ a k = 0 for some k ≥ 2.…”

“…Proof. In view of Theorem 1 we need only to prove that f ∈ TΦ δ c (λ, β) satisfies the coefficient inequality (6)…”

“…In [6], Al-Shaqsi noted that the operator defined by (3) can be expressed by series expansion as below:…”

“…Authors like Murugusundaramoorthy and Janani [7], Dziok and Murugusundaramoorthy [8], Obradovic and Joshi [9], Padmanabhan [10] and Ramesha et al [11] have investigated various subclasses of analytic functions. Motivated by aforementioned work, we introduce a new subclass Φ δ c (λ, β) of A involving Al-Shaqsi operator [6] as below:…”

Strong Differential Subordination and Superordination of New Generalized Derivative Operator

“…New strong differential sandwich-type results associated with the generalized derivative operator are also obtained. Using various linear operators, strong differential subordinations were investigated by Jeyaraman et al [8], Cho [5], and AL-Shaqsi [3] and of course many others.…”