A p times continuously differentiable complex-valued function F = u + iv in a domain D⊆ℂ is p-harmonic if F satisfies the p-harmonic equation Δ ⋯ ΔF = 0, where p is a positive integer. By using the generalized Salagean differential operator, we introduce a class of p-harmonic functions and investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points, and convex combination of the class.

1. Introduction

A continuous complex-valued function f = u + iv in a domain D⊆ℂ is harmonic if both u and v are real harmonic in D; that is, Δu = 0 and Δv = 0. Here Δ represents the complex Laplacian operator

()

In any simply connected domain D we can write , where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that in D. See [1, 2].

Denote by SH the class of functions

that are harmonic, univalent, and sense preserving in the unit disk U = {z : |z| < 1} for which f(0) = f_z(0) − 1 = 0. Then for

we may express the analytic functions h and g as

()

The properties of the class SH and its geometric subclasses have been investigated by many authors; see ([1–6]). Note that SH reduces to the class S of normalized analytic univalent functions in U if the coanalytic part of f is identically zero.

A p times continuously differentiable complex-valued function F = u + iv in a domain D⊆ℂ is p-harmonic if F satisfies the p-harmonic equation Δ ⋯ ΔF = 0, where p is a positive integer.

A function F is p-harmonic in a simply connected domain D if and only if F has the following representation:

()

where Δf_p−k+1(z) = 0 in D for each k ∈ {1, …, p}. f_p−k+1 has the form

()

where

()

Denote by SH_p the class of functions F of the form (3) that are harmonic, univalent, and sense-preserving in the unit disk. Apparently, if p = 1 and p = 2, F is harmonic and biharmonic, respectively.

Biharmonic functions have been studied by several authors, such as, [7–9]. Also, biharmonic functions arise in many physical situations, particularly, in fluid dynamics and elasticity problems. They have many important applications in engineering, biology, and medicine, such as in [10, 11].

For a function f in S, differential operator Dⁿ (n ∈ ℕ₀) was introduced by Sălăgean [12]. Al-Oboudi [13] generalized Dⁿ as follows:

()

When λ = 1, we get the Salagean differential operator.

For

given by (2), Li and Liu [14] defined the following generalized Salagean operator

in SH:

()

where

()

For a p-harmonic function F given by (3), we define the following operator:

()

If F is given by (3), then from (10) we see that

()

When p = 1, we get the generalized Salagean operator for harmonic univalent functions defined by Li and Liu [14].

Denote by SH_p(n, λ, α) the class of functions F of the form (3) which satisfy the condition

()

where

is defined by (11).

We let the subclass

of SH_p consist of functions F of the form (3) which include

, where

()

Define

The main object of the paper is to introduce a class of p-harmonic functions by using the generalized Salagean operator which was defined by Li and Liu [14]. We investigate necessary and sufficient coefficient conditions, extreme points, distortion bounds, and convex combination of the class.

2. Main Results

Theorem 1. Let F be a p-harmonic function given by (3). Furthermore, let

()

where λ ≥ 1, n ∈ ℕ, 0 ≤ α < 1, and a_1,p = 1. Then F is sense preserving, p-harmonic, and univalent in U and F ∈ SH_p(n, λ, α).

Proof. Suppose z₁, z₂ ∈ U and z₁ ≠ z₂, so that |z₁| ≤ |z₂| < 1:

()

which proves univalence.

In order to prove that F is sense preserving, we need to show that :

()

for all z ∈ U.

Using the fact that Rew ≥ α if and only if |1 − α + w| ≥ |1 + α − w|, it suffices to show that

()

Substituting for

in (18), we obtain

()

This last expression is nonnegative by (15), and so the proof is complete.

Theorem 2. Let F be given by (13) and (14). Then if and only if

()

where λ ≥ 1, n ∈ ℕ, 0 ≤ α < 1, and a_1,p = 1.

Proof. The “if" part follows from Theorem 1 upon noting that . For the “only if" part, we show that if the condition (20) does not hold.

Note that a necessary and sufficient condition for F given by (13) and (14) to be in is that the condition (12) should be satisfied.

This is equivalent to Re{A(z)/B(z)} ≥ 0, where

()

The above condition must hold for all values of z, |z| = r < 1. Upon choosing the values of z on the positive real axis, where 0 ≤ z = r < 1 we must have

()

If the condition (20) does not hold, then the numerator in (22) is negative for r is sufficiently close to 1. Hence there exist z₀ = r₀ in (0,1) for which the quotient in (22) is negative. This contradicts the required condition for F∈ and so the proof is complete.

Theorem 3. Let F be given by (13) and (14). Then if and only if

()

where

()

and

, X_j,p−k+1 ≥ 0, Y_j,p−k+1 ≥ 0.

In particular, the extreme points of are {h_j,p−k+1(z)} and {g_j,p−k+1(z)}, where j ≥ 1 and 1 ≤ k ≤ p.

Proof. For functions F of the form (13) and (14) we have

()

Then

()

and so

. Conversely, if

, then

()

Set

()

where X_1,p ≥ 0. Then, as required, we obtain

()

Theorem 4. Let . Then for |z| = r < 1 we have

()

Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Let . Taking the absolute value of F we have

()

The following covering result follows from the left-hand inequality in Theorem 4.

Corollary 5. Let F of the form (13) and (14) be so that . Then

()

Theorem 6. The class is closed under convex combinations.

Proof. Let F_i∈ for i = 1,2, …, where F_i is given by

()

Then by (20),

()

For

, 0 ≤ t_i ≤ 1, the convex combination of F_i may be written as

()

Then by (34),

()

This is the condition required by (20) and so

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Properties of a Class of p-Harmonic Functions

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1. Introduction

2. Main Results

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Properties of a Class of p-Harmonic Functions

Abstract

1. Introduction

2. Main Results

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