Harmonic mappings have found several applications in many diverse fields such as operations research, engineering, and other allied branches of applied mathematics. A continous function f = u + iv is a complex-valued harmonic function in a complex domain ℂ if both u and v are real harmonic in D. In any simply connected domain D ⊂ ℂ, we can write

where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h′(z)| > |g′(z)| in D (see [5]). Recently, Jahangiri and Ahuja [9] defined the class ℋ_p (p ∈ ℕ = {1, 2, 3, …}), consisting of all p–valent harmonic functions f = h + ḡ that are sense preserving in the open unit disk U = {z: |z| < 1}, and h, g are of the form:

If g ≡ 0, the harmonic function f = h + ḡ reduces to an analytic function f = h. Let Hp- denote the subclass of ℋ_p consisting of functions fn=h+gn¯ such that h and g_n given by:

The class ℋ₁ = ℋ of harmonic univalent functions studied by Jahangiri et al. [10] (see also [6], [12]). For complex parameters α₁, …, α_q and β1,…,βs(βj∉ℤ0-={0,-1,-2,…},   j=1,2,…,s), n ∈ ℕ₀ = ℕ ∪ {0}, ℕ = {1, 2, …}, ℓ, λ ≥ 0, the operator Ip,q,s,λn,ℓ(α1)f(z) is defined as follows (see El-Ashwah and Aouf [8]):

where

and (θ)_ν is the Pochhammer symbol defined, in terms of the Gamma function Γ, by

For 1 < γ < 2, and for all z ∈ U, let ℋ_p,q,s(n, ℓ, λ, α₁, γ) denote the family of harmonic p–valent functions f = h + ḡ where h and g of the form (1.2) such that

Let Hp,q,s-(n,ℓ,λ,α1,γ) be the subclass of ℋ_p,q,s(n, ℓ, λ, α₁, γ) consisting of harmonic functions fn=h+gn¯ so that h and g_n given by (1.3).

We note that by the special choices of α_i (i = 1, 2, …, q) and β_j (j = 1, 2, …, s), n, ℓ and λ our class Hp,q,s-(n,ℓ,λ,α1,γ) gives rise the following new subclasses of the class ℋ_p:

(i) For p = 1, q = s + 1, α_i = 1(i = 1, …, s + 1), β_j = 1 (j = 1, 2, …, s), we get H1,s+1,s-(n,ℓ,λ,α1,γ)=H1-(n,ℓ,λ,γ)

where Iⁿ(λ, ℓ) is the modified Cata’s operator (see [14]).

(ii) For p = 1, λ = 1, ℓ = 0, q = s + 1, α_i = 1(i = 1, …, s + 1), β_j = 1 (j = 1, 2, …, s), we get H1,s+1,s-(n,0,1,α1,γ)=H1-(n,γ)

where Dⁿ is the modified Salagean operator (see [11]), the differential opertor Dⁿ was introduced by Salagean (see [15]);

(iii) For p = 1, λ = 1, ℓ = 1, q = s + 1, α_i = 1(i = 1, …, s + 1), β_j = 1 (j = 1, 2, …, s), we get H1,s+1,s-(n,1,1,α1,γ)=H1-(n,γ)

where Iⁿ is the modified Uralegaddi-Somanatha operator (see [16]), defined as follows:

(iv) For p = 1, λ = 1, q = s + 1, α_i = 1(i = 1, …, s + 1), β_j = 1 (j = 1, 2, …, s), we get H1,s+1,s-(n,ℓ,1,α1,γ)=H1-(n,ℓ,γ)

where Iℓn is the modified Cho-Kim operator [3] (also see [4]), defined as follows:Iℓnf(z)=Iℓnh(z)+(-1)nIℓngn(z)¯.

(v) For p = 1, ℓ = 0, q = s + 1, α_i = 1(i = 1, …, s + 1), β_j = 1 (j = 1, 2, …, s), we get H1,s+1,s-(n,0,λ,α1,γ)=H1-(n,λ,γ)

where Dλn is the modified Al-Oboudi operator (see [1]), defined as follows:

Unless otherwise mentioned, we assume throughout this article that 1 < γ ≤ 2, ℓ > −p, p ∈ ℕ, λ ≥ 0, n ∈ ℕ₀ and Γ_k(α₁) is given by (1.7). In our first theorem, we introduce a sufficient condition for the coefficient bounds of harmonic functions in ℋ_p,q,s(n, ℓ, λ, α₁, γ).

Theorem 2.1

Let f = h + ḡ where h and g are of the form (1.2). Then f ∈ ℋ_p,q,s(n, ℓ, λ, α₁, γ) if

where a_p = 1.

Theorem 2.2

Letfn=h+gn¯where h and g_n are of the form (1.3). Thenfn∈Hp,q,s-(n,ℓ,λ,α1,γ)if and only if

where a_p = 1.

Remark 2.1

(i) If p = 1, q = s + 1, α_i = 1(i = 1, …, s + 1) and β_j = 1 (j = 1, 2, …, s) in Theorem 2.2, then we get the result obtained by Mostafa et al. [14, Theorem 2].

(ii) If λ = 1, ℓ = 0, p = 1, q = s + 1, α_i = 1(i = 1, …, s + 1), β_j = 1 (j = 1, 2, …, s) and n = 1, in Theorem 2.2, then we get the result obtained by Dixit and Porwal [7, Theorem 2.1].

In the following theorem we give the extreme points of the closed convex hulls of the class Hp,q,s-(n,ℓ,λ,α1,γ) denoted by Hp,q,s-(n,ℓ,λ,α1,γ).

Theorem 3.1

Letfn=h+gn¯where h and g_n are of the form (1.3). Thenfn∈clcoHp,q,s-(n,ℓ,λ,α1,γ)if and only if

where

and

In particular, the extreme points of the classHp,q,s-(n,ℓ,λ,α1,γ)are {h_k(z)} and {g_kn(z)}.

Theorem 3.2

Letfn∈Hp,q,s-(n,ℓ,λ,α1,γ)with |b_p| < γ − 1. Then for |z| = r < 1, we have

Remark 2.2

The bounds given in Theorem 3.2 for functions fn=h+gn¯, where h and g_n are given by (1.3), also hold for functions of the form f = h + g, where h and g are given by (1.2) if the coefficient condition (2.1) is satisfied. The upper bound given for f(z)∈Hp,q,s-(n,ℓ,λ,α1,γ) is sharp and the equality occurs for the functions

and

showing that the bounds given in Theorem 3.2 are sharp.

In the next two theorems, we prove that the class Hp,q,s-(n,ℓ,λ,α1,γ) is invariant under convolution and convex combinations of its members. The convolution of two harmonic functions,

and

is defined as

Using this definition, the next theorem shows that the class Hp,q,s-(n,ℓ,λ,α1,γ) is closed under convolution.

Theorem 4.1

For 1 < β ≤ γ ≤ 2, Letfn∈Hp,q,s-(n,ℓ,λ,α1,γ)andFn∈Hp,q,s-(n,ℓ,λ,α1,β). Then

Theorem 4.2

The familyHp,q,s-(n,ℓ,λ,α1,γ)is closed under convex combination.