On certain classes of harmonic P-valent functions by applying the Ruscheweyh derivatives

Abstract

In this paper we have introduced two new classes HR p (β, λ, k, v), HR p (β, λ, k, v) of complex valued harmonic multivalent functions of the form f = h+g, where h and g are analytic in the unit disk ∆ = {z : |z| < 1} and f (z) satisfying the condition Re (1 − λ) D v f z p + λ(1 − k) (D v f) (z p) + λk (D v f) (z p) > β p. A sufficient coefficient condition for this function in the class HRp(β, λ, k, v) and a necessary and sufficient coefficient condition for the function f in the class HR p (β, λ, k, v) are determined. We investigate inclusion relations, distortion theorem, extreme points, convex combination and other interesting properties for these families.

Ruscheweyh-type harmonic functions with correlated coefficients

“…The operator D λ,τ H for τ = (−1) n was investigated in [21] (see also [7,10,12,14,28]). We say that a function f ∈ H is subordinate to a function F ∈ H, and write f (z) ≺ F(z) (or simply f ≺ F) if there exists a complex-valued function ω which maps U into oneself with ω(0) = 0, such that f (z) = F(ω(z)) (z ∈ U) .…”

Multivalent harmonic functions Involving multiplier transformation

“…For the purpose of this paper, on applying the linear operator If (z), motivated with the class defined in [10] we define a class…”