ON A SUBCLASS OF α-CONVEX λ-SPIRAL FUNCTIONS

Let H denote the class of functions f(z) = z+∑∞k=2akzk which are analytic in the unit disc ∆ = {z : |z| < 1}. In this paper, we introduce the class Mλ α[A,B] of functions f ∈ H with f(z)f ′(z)/z ≠ 0, satisfying for z ∈ ∆ : {(eiλ −αcosλ)(zf ′(z)/f(z))+αcosλ(1+ zf ′′(z)/f ′(z))} ≺ cosλ((1+Az)/(1+Bz))+isinλ, where ≺ denotes subordination, α and λ are real numbers, |λ|<π/2 and −1≤ B <A≤ 1. Functions in Mλ α[A,B] are shown to be λ-spiral-like and hence univalent. Integral representation, coefficients bounds, and other results are given.

In this paper, we introduce and study a subclass of SC(α, λ) defined by using subordination to convex functions.Definition 1.1.Let f ∈ H with f (z)f (z)/z ≠ 0 in ∆.Then f is said to belong to the class MS λ α [A, B] if and only if for z ∈ ∆, where ≺ denotes subordination, α and λ are real numbers, |λ| < π/2 and A and B are arbitrary fixed numbers such that It is clear from Definition 1.1 that a function f ∈ MS λ α [A, B] if and only if there exists a function w(z) analytic in ∆ and satisfying w(0) = 0 and |w(z , the subclass of αconvex functions introduced by Kim and Jung [5] and the subclass of spiral-like functions introduced by Dashrath and Shukla [2].
In this paper, we show that functions in M λ α [A, B] are spiral-like and hence univalent in ∆.Integral representation, coefficient bounds, and other results are given.

Spiral-likeness.
To derive our main result, we prove the following lemma. where It is clear that the function q satisfies |q(z)| < 1. Hence (2.1) follows from (2.4).Conversely, suppose that (2.1) holds.Then We note that condition (2.1) can be written as As B → −1 and A = 1, the above condition reduces to the necessary and sufficient condition for f to belong to MS λ α [1, −1] (see [8]).
The following lemma is due to Jack [3].
Lemma 2.2.Let w be a nonconstant and analytic function in ∆, w(0) = 0. Then if |w(z)| attains its maximum value on the circle |z| = r < 1 at z 0 we can write where φ is a real number such that φ ≥ 1.

Integral representation Theorem 3.1. A necessary and sufficient condition for the function f to be in MS
for some g ∈ Sp λ [A, B], where the powers are assumed to be principal values.
Proof.Let f ,g ∈ H and f be given as in (3.1).Differentiating both sides and simplifying we get Differentiating (3.2) logarithmically and multiplying both sides by ze iλ we obtain Remark 3.2.Using the integral representation and the external function of the class Sp λ [A, B] (see [2]) we get the external function of the class MS λ α [A, B] as Coefficients bounds.To derive our next result, we need the following lemma (see [4]).
The equality may be attained with the functions w(z) = z 2 and w(z) = z.
The sharpness of (4.9) follows from the sharpness of inequality (4.1).Proof.Let w = f (z) ∈ MS λ α [A, B] given by f (z) = z + ∞ k=2 a k z k , and let w 0 be any complex number such that f (z) ≠ w 0 for z ∈ ∆.Then

.Theorem 5 . 1 .
In this section, we discuss the covering theorem of the class MS λ α [A, B], that is, we find the radius of the largest disk covered by the image of the unit disk ∆ under the mapping f ∈ MS λ α [A, B].We also find the α-convex β-spiral radius of functions in MS λ α [A, B].Let f ∈ MS λ α [A, B].Then the disk ∆ is mapped onto a domain that contains the disk |w| < e iλ + α cos λ 2 e iλ + α cos λ + (A − B) cos λ .(5.1)