On Starlike and Convex Functions with Respect to K-symmetric Points

We introduce new subclasses S σ,s k λ, δ, φ and K σ,s k λ, δ, φ of analytic functions with respect to k-symmetric points defined by differential operator. Some interesting properties for these classes are obtained.


Introduction
Let A denote the class of functions of the form a n z n , 1.1 which are analytic in the unit disk U {z ∈ : |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: 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 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 imposes the class S k α, β , which was studied by Gao and Zhou 6 , and the class S 1 φ S * φ was studied by Ma and Minda 7 .
Let two functions given by f z z ∞ n 2 a n z n and g z z ∞ n 2 b n z n be analytic in U. Then the Hadamard product or convolution f * g of the two functions f, g is defined by and for several function 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 .

1.10
Here D σ,s λ,δ f z can also be written in terms of convolution as When σ 0, we get the S ǔl ǔgean 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 ksymmetric points defined by differential operator.Some interesting properties of S σ,s k λ, δ, φ and K σ,s k λ, δ, φ are obtained.
Applying the operator D σ,s λ,δ f z where k is a fixed positive integer, we now define classes of analytic functions containing the differential operator.
Definition 1.1.Let S σ,s k λ, δ, φ denote the class of functions in A satisfying the condition where φ ∈ ℘.

International Journal of Mathematics and Mathematical Sciences
Definition 1.2.Let K σ,s k λ, δ, φ denote the class of functions in A satisfying the condition where φ ∈ ℘.
In order to prove our results, we need the following lemmas.
Let φ be a convex function, with φ 0 , where q is univalent and satisfies the differential equation q z zq z q z c φ z .

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 Ê κφ υ > 0, z ∈ U, and let q z ∈ A with q 0 1 and q z ≺ φ z .If 1.17 where co H U denotes the closed convex hull of H U .
Proof.Let f ∈ S σ,s k λ, δ, φ , then by Definition 1.1 we have Substituting z by ε ν z, where ε k 1 ν 0, 1, . . ., k − 1 in 2.1 , respectively, we have According to the definition of f k and ε k 1, we know f k ε ν z ε ν f k z for any ν 0, 1, . . ., k − 1, and summing up, we can get Hence there exist ζ ν in U such that and the operator D σ,s λ,δ f can be written as D σ,s λ,δ f g * f.Then from the definition of the differential operator D σ,s λ,δ , we can verify
By using Theorems 2.2 and 2.1, we get the following.

International Journal of Mathematics and Mathematical Sciences
Proof.Let f ∈ K σ,s k λ, δ, φ .Then Theorem 2.2 shows that zf ∈ S σ,s k λ, δ, φ .We deduce from Theorem 2.1 that zf k ∈ S σ,s λ, δ, φ .From zf k zf k , Theorem 2.2 now shows that where and q is the univalent solution of the differential equation From the definition of D σ,s λ,δ , we see that which implies that 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 Ê{φ} > 0 with Ê 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 ∈ N 0 .Then where p is analytic function with p 0 1.By using the equation we get and then differentiating, we get

2.19
Applying 2.16 for the function f k we obtain
zp z q z p z ≺ φ z .

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.
Some other works related to other differential operators with respect to symmetric points for different types of problems can be seen in 16-21 .

1 . 2 A 3 International
function f ∈ A is said to be starlike with respect to symmetrical points in U if it satisfiesÊ zf z f z − f −z > 0, z ∈ U.1.Journal of Mathematics and Mathematical Sciences This class was introduced and studied by Sakaguchi in 1959 1 .Some related classes are studied by Shanmugam et al. 2 .

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