COEFFICIENT ESTIMATE OF CERTAIN SUBCLASSES OF CONVEX p−VALENT FUNCTIONS WITH A BOUNDED POSITIVE REAL PART

We estimate the bounds of coefficients and solve Fekete-Szegö problem for p−valent Mocanu-convex and Pascu-type functions in the open unit disk △ which maps △ onto the strip domain w with pα < Rw < pβ. AMS Subject Classification: 30C45, 30C50

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.This class is denoted by S * (α) and S * (0) = S * .The class S * (α) was introduced by Robertson [4].It is well-known that S * (α) ⊂ S * ⊂ S. Furthermore, let M(β) be the class of functions f ∈ A which satisfy ℜ zf ′ (z) f (z)
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 P (z) = Q(w(z)) (z ∈ △).From the definition of the subordinations, it is easy to show that the subordination (1) implies that In particular, if Q(z) is univalent in △, then the subordination (1) is equivalent to the condition (2).Motivated by the classes S * (α) and M(β), we define a new class for certain p−valent functions.Definition 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 (α, β) if f satisfies the following inequality Definition 2. Let α and β be real numbers such that 0 ≤ α < 1 < β.The function f ∈ A p belongs to the class C p λ (α, β) if f satisfies the following inequality Definition 3. Let α and β be real numbers such that 0 ≤ α < 1 < β.The function f ∈ A p belongs to the class M p λ (α, β) if f satisfies the following inequality Remark 4. When p = 1, M p λ (α, β) reduces to M λ (α, β), the class of Mocanu-convex functions with bounded positive real part.Further we note that ), the class of p−valent convex functions with bounded positive real part and M 1 1 (α, β) = C(α, β), the class of convex functions with bounded positive real part.
Definition 5. Let α and β be real numbers such that 0 ≤ α < 1 < β.The function f ∈ A p belongs to the class N p λ (α, β) if f satisfies the following inequality Remark 6.When p = 1, N p λ (α, β) reduces to N λ (α, β), the class of Pascu-type functions with positive real part.Further we note that

Coefficient Estimates for
Applying the function S α,β (z) defined by (7), we give a necessary and sufficient condition for f (z) ∈ A p to belong to the class M p λ (α, β).
We note that where Using the subordination (8), we find sharp bounds on the second and third coefficients for f (z) ∈ M p λ (α, β), by applying the following lemma due to Rogosinki [5]. Proof.Let where B n is as in (11).Applying Lemma 10 we can get the results as asserted.
When λ = 0 and λ = 1 we state the following corollaries respectively: When p = 1, from Theorem 11, we state the following corollary: Making use of the following lemma we shall solve the Fekete-Szegö problem for f a p+n z p+n be in the class M p λ (α, β).Then for any complex number µ, , where Q(z) is given by (12).Let Then h is analytic and has positive real part in the open disk △.We also have We find from the equation ( 13) that which imply that Applying Lemma 15, we obtain β−α in (14), we can obtain the results as asserted.
By taking λ = 0 and λ = 1 we state the following corollaries respectively: Corollary 17. [1] Let 0 ≤ α < 1 < β and let the function f given by a p+n z p+n be in the class S p (α, β).Then for any complex number µ, Corollary 18.Let 0 ≤ α < 1 < β and let the function f given by a p+n z p+n be in the class C p (α, β).Then for any complex number µ, Putting p = 1 in Theorem 16, we get the following corollary.
Corollary 19.Let 0 ≤ α < 1 < β and let the function f given by a n z n be in the class M λ (α, β).Then for any complex number µ, First, by applying the function S α,β (z) defined by (7), we give a necessary and sufficient condition for f (z) ∈ A p to belong to the class N p λ (α, β).Lemma 20.Let f (z) ∈ A p and 0 ≤ α < 1 < β.Then f (z) ∈ N p λ (α, β) if and only if Remark 21.For λ = 0 and λ = 1, we get Lemma 8 and Lemma 9 respectively.
Using the subordination (15), we find sharp bounds on the second and third coefficients for f (z) ∈ N p λ (α, β), by applying Lemma 10. and where B n is as in (11).Applying Lemma 10 we can get the results as asserted.
When p = 1, from Theorem 22, we state the following corollary: Again using Lemma 15 we shall solve the Fekete-Szegö problem for f (z) ∈ N p λ (α, β).Theorem 25.Let 0 ≤ α < 1 < β and let the function f given by a p+n z p+n be in the class N p λ (α, β).Then for any complex number µ, Putting p = 1 in Theorem 25, we get the following corollary.