Some Properties for Certain Subclasses of Analytic Functions Associated with k−Integral Operators

In this paper, the k-integral operators for analytic functions de ned in the open unit disc U = {z ∈ C : |z| < 1} are introduced. Several new subclasses of analytic functions satisfying certain relations involving these operators are also introduced. Further, we establish the inclusion relation for these subclasses. Next, the integral preserving properties of a k-integral operator satis ed by these newly introduced subclasses are obtained. Some applications of the results are discussed. Concluding remarks are also given.


Introduction
Geometric function theory is one of the most important branches of complex analysis which focus on the geometric properties of analytic functions. Geometric function theory was evolved around the turn of the 20 th century and developed deep connections with other elds of mathematics and physics like hyperbolic geometry, theory of partial dierential equations, uid dynamics etc. In this paper, we study certain classes based on some important geometric properties like starlikeness, convexity, close-to-convexity and quasi-convexity of univalent analytic functions and associated with certain k-integral operators.
Let A be the class of analytic functions of the form where f is analytic in the open unit disc U = {z ∈ C : |z| < 1}.
Let P be the class of function h(z) of the form which are analytic and convex in U and satisfy the following condition: Re(h(z)) > 0 (z ∈ U ).
For the analytic functions f and g in U , we say that the function g is subordinate to f in U [10], and write if there exists a Schwarz function w, which is analytic in U with w(0) = 0 and |w(z)| < 1, such that g(z) = f (w(z)) (z ∈ U ).
Using equations (11) and (12), we obtain that the function M α β,k f , dened in the open unit disc U = {z ∈ C : |z| < 1} has the following series representation: which on using equations (9) and (10), gives Using equations (9) and (14), we can verify that the function M α β,k f satises the following recurrence relation: Remark 2.1. Substituting k = 1, β = λ and α = µ in equation (16) Also, we dene another k-integral operator ð σ k by putting α = k and β = σ in equation (2.1), as: which has the following series representation: In view of equations (7), (17) and (18), it is clear that for k = 1 and σ = µ, the k-integral operator ð σ k reduces to the Bernardi integral operator J µ .
First, we need to mention the following lemma [6] to establish the inclusion property of the class S α β,k : Lemma 2.8. Let u, v ∈ C and p(z) be convex and univalent in U such that p(0) = 1 and Re(up(z) + v) > 0.
Next, we establish the following inclusion relation for the class S α β,k (µ; φ): where Proof. Let f ∈ S α β,k (µ; φ), then from Denition 2.2, we have We assume that where u is analytic in U and u(0) = 1. Making use of equation (16) in the above equation, we get Taking Logarithm of equation (43), then dierentiating with respect to z and multiplying the resultant equation with z, yields Using equation (42) in equation (44), we get From subordination (41), we have Since u(0) = 1, therefore in view of Lemma 2.1, the subordination (46) implies with the condition Since α, β, µ ∈ R, therefore the condition (48) is equivalent to the condition (30).
Using equation (42) in subordination (47), we get which in view of the Denition 2.2, gives Hence, we establish the inclusion relation (39) subject to the condition (40).
In view of the Theorem 2.9, we get the following corollary by mathematical induction: Corollary 2.10. Let 0 ≤ µ < 1 and φ ∈ P such that condition (40) holds, then in Corollary 2.10, we obtain the following corollary: Again, taking α = k and β = σ in Theorem 2.9 and Corollary 2.10, we obtain the following result for the k-integral operator ð σ k : Also, taking φ(z) = 1 + Az in Corollary 2.12, we obtain the following corollary: Next, we establish the following inclusion relation for the subclass K α β,k : Theorem 2.14. Let 0 ≤ µ < 1 and φ ∈ P . If the condition (40) holds, then Proof. Making use of relation (35) and Theorem 2.9, we have . Hence, we get the inclusion property (53) .
In view of Theorem 2.14, we get the following corollary by mathematical induction: Corollary 2.15. Let 0 ≤ µ < 1, φ ∈ P with condition (2.28) holds, then Again, taking α = k and β = σ in Corollary 2.15, we obtain the following corollary for the subclass K k σ (µ; φ) associated with the k-integral operator ð σ k : Corollary 2.17. Let 0 ≤ µ < 1, φ ∈ P and the condition (51) holds, then Further, we need to mention the following lemma [11] to establish the inclusion property for the subclass C α β,k (µ, η; φ, ψ) : Lemma 2.19. Let h(z) be convex and univalent function in U and q(z) be analytic in U with Re(q(z)) ≥ 0.
Hence, we establish the inclusion relation (54) subject to the condition (40).
If we take then V is analytic in U with V (0) = 1.
Next, we take with the condition (74), which is equivalent to the condition Since, if we take q(z) = 1 , then from subordination (3.22) and the above inequality Re(q(z)) > 0.