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.
Let [Formula: see text] be an associative ring equipped with an automorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. In this note, we characterize when a skew inverse Laurent series ring [Formula: see text] and a skew inverse power series ring [Formula: see text] are 2-primal, and we obtain partial characterizations for those to be NI.
For a ring endomorphism [Formula: see text], a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, [Formula: see text]-[Formula: see text]), is introduced and studied. It is shown that the [Formula: see text]-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.
In this paper, we discuss the p-valent functions that satisfy the differential subordinations z(I p (r,λ)f (z)) (j+1) (p−j)(I p (r,λ)f (z)) (j) ≺ a+(aB+(A−B)β)z a(1+Bz). We also obtain coefficient inequalities, extreme points, integral representation and arithmetic mean. Further we investigate some interesting properties of operators defined on Ap(r, j, β, a, A, B).
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