We introduce new subclasses and of analytic functions with respect to k-symmetric points defined by differential operator. Some interesting properties for these classes are obtained.

1. Introduction

Let A denote the class of functions of the form

(1.1)

which are analytic in the unit disk U = {z ∈ ℂ : |z| < 1}.

Also let ℘ be the class of analytic functions p with p(0) = 1, which are convex and univalent in U and satisfy the following inequality:

(1.2)

A function f ∈ A is said to be starlike with respect to symmetrical points in U if it satisfies

(1.3)

This class was introduced and studied by Sakaguchi in 1959 [1]. Some related classes are studied by Shanmugam et al. [2].

In 1979, Chand and Singh [3] defined the class of starlike functions with respect to k-symmetric points of order α (0 ≤ α < 1). Related classes are also studied by Das and Singh [4].

Recall that the function F is subordinate to G if there exists a function ω, analytic in U, with ω(0) = 0 and |ω(z)| < 1, such that F(z) = G(w(z)), z ∈ U. We denote this subordination by F(z)≺G(z). If G(z) is univalent in U, then the subordination is equivalent to F(0) = G(0) and F(U) ⊂ G(U).

A function f ∈ A is in the class S_k(ϕ) satisfying

(1.4)

where ϕ ∈ ℘, k is a fixed positive integer, and f_k(z) is given by the following:

(1.5)

The classes S_k(ϕ) of starlike functions with respect to k-symmetric points and K_k(ϕ) of convex functions with respect to k-symmetric points were considered recently by Wang et al. [5]. Moreover, the special case

(1.6)

imposes the class S_k(α, β), which was studied by Gao and Zhou [6], and the class S₁(ϕ) = S^*(ϕ) was studied by Ma and Minda [7].

Let two functions given by

and

be analytic in U. Then the Hadamard product (or convolution) f*g of the two functions f, g is defined by

(1.7)

and for several function f₁(z), …, f_m(z) ∈ A,

(1.8)

The theory of differential operators plays important roles in geometric function theory. Perhaps, the earliest study appeared in the year 1900, and since then, many mathematicians have worked extensively in this direction. For recent work see, for example, [8–12].

We now define differential operator as follows:

(1.9)

where λ ≥ 0, C(δ, n) = (δ + 1) _n−1/(n − 1)!, for δ, σ, s ∈ N₀ = {0,1, 2, …}, and (x) _n is the Pochhammer symbol defined by

(1.10)

Here

can also be written in terms of convolution as

(1.11)

where

and

Note that and . When σ = 0, we get the Sǔlǔgean differential operator [9], when λ = s = 0, σ = 1 we obtain the Ruscheweyh operator [8], when s = 0, σ = 1, we obtain the Al-Shaqsi and Darus [11], and when δ = s = 0, we obtain the Al-Oboudi differential operator [10].

In this paper, we introduce new subclasses of analytic functions with respect to k-symmetric points defined by differential operator. Some interesting properties of and are obtained.

Applying the operator

(1.12)

where k is a fixed positive integer, we now define classes of analytic functions containing the differential operator.

Definition 1.1. Let denote the class of functions in A satisfying the condition

(1.13)

where ϕ ∈ ℘.

Definition 1.2. Let denote the class of functions in A satisfying the condition

(1.14)

where ϕ ∈ ℘.

In order to prove our results, we need the following lemmas.

Lemma 1.3 (see [13].)Let c > −1, and let I_c : A → A be the integral operator defined by F = I_c(f), where

(1.15)

Let ϕ be a convex function, with ϕ(0) = 1 and ℜ{ϕ(z) + c} > 0 in U. If f ∈ A and zf^′(z)/f(z)≺ϕ(z), then zF^′(z)/F(z)≺q(z)≺ϕ(z), where q is univalent and satisfies the differential equation

(1.16)

Lemma 1.4 (see [14].)Let κ, υ be complex numbers. Let ϕ be convex univalent in U with ϕ(0) = 1 and ℜ[κϕ + υ] > 0, z ∈ U, and let q(z) ∈ A with q(0) = 1 and q(z)≺ϕ(z). If p(z) = 1 + p₁z + p₂z² + ⋯∈℘ with p(0) = 1, then

(1.17)

Lemma 1.5 (see [15].)Let f and g, respectively, be in the classes convex function and starlike function. Then, for every function H ∈ A, one has

(1.18)

where

denotes the closed convex hull of H(U).

2. Main Results

Theorem 2.1. Let . Then f_k defined by (1.5) is in .

Proof. Let , then by Definition 1.1 we have

(2.1)

Substituting z by ε^νz, where ε^k = 1 (ν = 0,1, …, k − 1) in (2.1), respectively, we have

(2.2)

According to the definition of f_k and ε^k = 1, we know f_k(ε^νz) = ε^νf_k(z) for any ν = 0,1, …, k − 1, and summing up, we can get

(2.3)

Hence there exist ζ_ν in U such that

(2.4)

for ζ₀ ∈ U since ϕ(U) is convex. Thus f_k ∈ S^σ,s(λ, δ, ϕ).

Theorem 2.2. Let f ∈ A and ϕ ∈ ℘. Then

(2.5)

Proof. Let

(2.6)

and the operator

can be written as

Then from the definition of the differential operator , we can verify

(2.7)

Thus

if and only if

By using Theorems 2.2 and 2.1, we get the following.

Corollary 2.3. Let . Then f_k defined by (1.5) is in .

Proof. Let . Then Theorem 2.2 shows that . We deduce from Theorem 2.1 that (zf^′) _k ∈ S^σ,s(λ, δ, ϕ). From , Theorem 2.2 now shows that .

Theorem 2.4. Let ϕ ∈ ℘, λ > 0 with ℜ[ϕ(z) + (1/λ) − 1] > 0. If , then

(2.8)

where

and q is the univalent solution of the differential equation

(2.9)

Proof. Let . Then in view of Theorem 2.1, f_k ∈ S^σ,s(λ, δ, ϕ), that is,

(2.10)

From the definition of

, we see that

(2.11)

which implies that

(2.12)

Using (2.10) and (2.12), we see that Lemma 1.3 can be applied to get (2.8), where c = (1/λ) − 1 > −1 and ℜ{ϕ} > 0 with ℜ[ϕ(z) + (1/λ) − 1] > 0 and q satisfies (2.9). We thus complete the proof of Theorem 2.4.

Theorem 2.5. Let ϕ ∈ ℘ and s ∈ N₀. Then

(2.13)

Proof. Let . Then

(2.14)

Set

(2.15)

where p is analytic function with p(0) = 1. By using the equation

(2.16)

we get

(2.17)

and then differentiating, we get

(2.18)

Hence

(2.19)

Applying (2.16) for the function f_k we obtain

(2.20)

Using (2.20) with

, we obtain

(2.21)

Since

, then by using (2.14) in (2.21) we get the following.

(2.22)

We can see that q(z)≺ϕ(z), hence applying Lemma 1.4 we obtain the required result.

By using Theorems 2.2 and 2.5, we get the following.

Corollary 2.6. Let ϕ ∈ ℘ and s ∈ N₀. Then

(2.23)

Now we prove that the class , ϕ ∈ ℘, is closed under convolution with convex functions.

Theorem 2.7. Let , ϕ ∈ ℘, and φ is a convex function with real coefficients in U. Then .

Proof. Let , then Theorem 2.1 asserts that , where ℜ{ϕ} > 0. Applying Lemma 1.5 and the convolution properties we get

(2.24)

Corollary 2.8. Let , ϕ ∈ ℘, and φ is a convex function with real coefficients in U. Then .

Proof. Let , ϕ ∈ ℘. Then Theorem 2.2 shows that . The result of Theorem 2.7 yields , and thus .

Some other works related to other differential operators with respect to symmetric points for different types of problems can be seen in ([16–21]).

Acknowledgments

The work presented here was partially supported by UKM-ST-06-FRGS0244-2010, and the authors would like to thank the anonymous referees for their informative and critical comments on the paper.

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On Starlike and Convex Functions with Respect to k-Symmetric Points

Abstract

1. Introduction

2. Main Results

Acknowledgments

References

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