Abstract
In this paper we introduce and study new classes VM_{p,η}(λ,α,β) and VN_{p,η}(λ,α,β) of multivalent functions with varying arguments of coefficients. We obtain coefficients inequalities, distortion theorems and extreme points for functions in these classes. Also, we investigate several distortion inequalities involving fractional calculus. Finally, results on partial sums are considerd.
Highlights
0), where f (z) is an analytic function in a -connected region of the complex z−plane containing the origin and the multiplicity of (z − ζ)λ−1 is removed by requiring log(z − ζ) to be real when z − ζ > 0
1), where f (z) is an analytic function in a -connected region of the complex z−plane containing the origin and the multiplicity of (z − t)−λ is removed by requiring log(z − ζ) to be real when z − ζ > 0
Silverman [12] used the concept of varying arguments of the coefficients to introduce and study the class V∗(α), which is a subclass of V consisting of starlike functions of order α
Summary
We assume throughout this paper that 0 ≤ α < p, β ≥ 0, 0 ≤ λ < 1, p ∈ N, z ∈ U and φλp,n. Letting r → 1−, we obtain the required result and the proof of the inequality (2.1) is completed. F (z) ∈ VMp,η(λ, α, β) satisfies the equality in (2.1). The equality in (2.2) holds true for p(p − α)eiθp+n zp+n. This completes the proof of Theorem 2.3. Let f (z) defined by (1.1) be in the class VMp,η(λ, α, β), for z ∈ U, we have. Since the equality in (3.1) is satisfied by f (z) given by (3.2), the proof is completed. Using similar arguments to those in the proof of the Theorem 3.1, we obtain the following theorem. Let f (z) defined by (1.1) be in the class VNp,η(λ, α, β), for z ∈ U, we have. |z|p+1 φλp, (1 + β)φλp,1 − (pβ + α) φλp, (1 + β)φλp,1 − (pβ + α) p(p − α)eiθp+1 zp+1
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