Coefficient Estimate Of p − valent Bazilevič Functions with a Bounded Positive Real Part

O. S. Babu, C. Selvaraj, G. Murugusundaramoorthy, and S.Logu abstract: By considering a p−valent Bazilevič function in the open unit disk △ which maps △ onto the strip domain w with pα < Rw < pβ, we estimate bounds of coefficients and solve Fekete-Szegö problem for functions in this class.

Let P (z) and Q(z) be analytic in △.Then the function P (z) is said to subordinate to Q(z) in △ written by if there exists a function w(z) which is analytic in △ with w(0) = 0 and |w(z)| < 1 (z ∈ △), and such that From the definition of the subordinations, it is easy to show that the subordination (1.1) implies that In particular, if Q(z) is univalent in △, then the subordination (1.1) is equivalent to the condition (1.2).

Bazilevič Functions with bounded positive real part
Motivated by the classes S * (α) and M(β), we define a new class for certain p−valent functions.Definition 2.1.Let α and β be real numbers such that 0 ≤ α < 1 < β.The function f ∈ A p belongs to the class S p λ (α, β) if f satisfies the following inequality By taking λ = 0 we further define a new class S p λ (α, β) ≡ S p (α, β).Definition 2.2.The function f ∈ A p belongs to the class S p (α, β) if f satisfies the following inequality for some real number α(α < 1) and some real number β(β > 1).
Example 2.6.Let us consider the function f (z) given by with α < 1 and β > 1.Then we have It is clear that the function f (z) given by (2.5) satisfies the inequality (2.2), which implies that f (z) ∈ S p (α, β).
Applying the function S α,β (z) defined by (2.3), we give a necessary and sufficient condition for f (z) ∈ A p to belong to the class S p λ (α, β).
By taking λ = 0 we state the following Lemma.
Coefficient Estimate Of P −Valent Bazilevic Functions 67

Some coefficient problems
Using the subordination (2.6), we find sharp bounds on the second and third coefficients for f (z) ∈ S p λ (α, β), by applying the following lemma due to Rogosinki [5].
Applying Lemma 3.1 with where B n is as in (2.8) and using Remark 1.1, we obtain the following theorem.
Moreover, the equality holds in either inequality if and only if for some real number θ (0 ≤ θ < 2π), where S α,β (z) is defined by (2.3).
When λ = 0 we state the following corollary: a p+n z p+n ∈ S p (α, β), then Moreover, the equality holds in either inequality if and only if for some real number θ (0 ≤ θ < 2π), where S α,β (z) is defined by (2.3).
When p = 1, from Theorem 3.2, we state the following corollary: Moreover, the equality holds in either inequality if and only if for some real number θ (0 ≤ θ < 2π), where S α,β (z) is defined by (2.3).
Making use of the following lemma we shall solve the Fekete-Szegö problem for f (z) ∈ S p λ (α, β).
Lemma 3.5.(Keogh and Merkers [2]) Let h(z) = 1+h 1 z +h 2 z 2 +• • • be a function with positive real part in △.Then for any complex number ν, Theorem 3.6.Let 0 ≤ α < 1 < β and let the function f given by f (z) = a p+n z p+n be in the class S p λ (α, β).Then for any complex number µ, Then, since f ∈ S p λ (α, β), we have P (z) ≺ Q(z), where Q(z) is given by (3.1). Let Then h is analytic and has positive real part in the open disk △.We also have We find from the equation (3.2) that which imply that , Applying Lemma 3.5, we obtain 3), we can obtain the results as asserted. ✷ By taking λ = 0 we state the following: Corollary 3.7.Let 0 ≤ α < 1 < β and let the function f given by f (z) = a p+n z p+n be in the class S p (α, β).Then for any complex number µ, Putting p = 1 in Theorem 3.6, we get the following corollary.
Corollary 3.8.Let 0 ≤ α < 1 < β and let the function f given by f a n z n be in the class S λ (α, β).Then for any complex number µ, In this section by making use of the following lemma we deduced some coefficient estimates for f (z) ∈ S p (α, β).A n z n is analytic in △ and satisfies the following subordination Proof.According to the assertion of Lemma 4.1, the function f (z) satisfies the subordination (2.7).Let us define P (z) and Q(z) by Coefficient Estimate Of P −Valent Bazilevic Functions 71 and Then, the subordination (2.7) satisfies (1.1).Note that the function Q(z) defined by (4.2) is convex in △ and has the form where If we let then by Lemma 4.1 we see that the subordination (2.7) implies that where that is, Then, the coefficient of z n−1 in both sides lead to A simple calculation combined with the inequality (4.4) yields that where B 1 is given in (4.5).To prove the assertion of the theorem, we need to show that where B 1 is given in (4.5).This completes the proof of the theorem.✷ Remark 4.3.By specializing the parameters λ = 0, p = 1, the results proved in this paper , leads the results obtained in [3].Further by taking λ = 0 one can deduce the results for functions f ∈ S p (α, λ), hence we omit the details

Let
A p denote the class of all functions f (z) of the form f (z) = z p + ∞ n=1 a p+n z p+n (p ∈ N = {1, 2, 3, . ..}) which are analytic and p−valent in the open unit disk △ = {z : z ∈ C : |z| < 1}.Note that A 1 := Athe class of analytic functions further S the subclass of A consisting of all univalent functions f in △.A function f ∈ A is said to be starlike of order α(0 ≤ α < 1) in △ if it satisfies