Some Properties of Subclasses of Multivalent Functions

and Applied Analysis 3 On the other hand, for a function f defined by 1.6 and in the class C p, α , Lemma 1.2 yields ap 1 ≤ p ( p − α ( p 1 )( p 1 − α . 1.10 In view of the coefficient inequalities 1.9 and 1.10 , it would seem to be natural to introduce and study here two further classes T ∗ ε p, α and Cε p, α of analytic and p-valent functions, where T ∗ ε p, α denotes the subclass of T ∗ p, α consisting of functions of the form f z z − ( p − αε p 1 − α p 1 − ∞ ∑ k 2 ap kz p k ap k ≥ 0, p ∈ N, k ∈ N − {1}; 0 ≤ α < p; 0 ≤ ε < 1 ) 1.11 and Cε p, α denotes the subclass of C p, α consisting of functions of the form f z z − p ( p − αε ( p 1 )( p 1 − α p 1 − ∞ ∑ k 2 ap kz p k ap k ≥ 0, p ∈ N, k ∈ N − {1}; 0 ≤ α < p; 0 ≤ ε < 1 ) . 1.12 The classes T ∗ ε p, α and Cε p, α are studied by Aouf et al. 4 . The classes T ∗ ε α : T ∗ ε 1, α , Cε α : Cε 1, α 1.13 were considered earlier by Silverman and Silvia 5 . Now, we give the following equalities for the functions f z belonging to the class A p, n : D0f z f z , D1f z Df z z ( D0f z )′ z [ pzp−1 ∞ ∑ k n ( p k ) ap kz p k−1 ] pz ∞ ∑ k n ( p k ) ap kz p , D2f z D ( Df z ) z ( D1f z )′ z [ pz ∞ ∑ k n p k ap kz k ]′ p2zp ∞ ∑ k n ( p k )2 ap kz p , .. DΩf z D ( DΩ−1f z ) pΩzp ∞ ∑ k n ( k p )Ω ap kz p . 1.14 4 Abstract and Applied Analysis We define ℘ : A p, n → A p, n such that ℘ ( Ω, λ, p ) ( 1 pΩ − λ ) DΩf z λ p z ( DΩf z )′ ( 0 ≤ λ ≤ 1 pΩ , Ω ∈ N ∪ {0} ) . 1.15 A function f z ∈ A p, n is said to be in the class I Ω, λ, p, α if it satisfies the inequality Re { z ( ℘ ( Ω, λ, p ))′ ℘ ( Ω, λ, p ) } Re { z ( 1/pΩ ( 1/p − 1λDΩf z )′ λ/pzDΩf z )′′ ( 1/pΩ − λDΩf z λ/pzDΩf z )′ } > α, 1.16 for some α 0 ≤ α < p , 0 ≤ λ ≤ 1/pΩ, Ω ∈ N ∪ {0} and for all z ∈ U. IfΩ 0 and λ 0, we obtain the condition 1.2 . Furthermore, we obtain the condition 1.3 for Ω 0 and λ 1. We denote by T p, n the subclass of the class A p, n consisting of functions of the form f z z − ∞ ∑ k n ak pz k p ak p ≥ 0; n, p ∈ N ) , 1.17 and define the class I∗ Ω, λ, p, α by I∗ ( Ω, λ, p, α ) I ( Ω, λ, p, α ) ∩ Tp, n. 1.18 Furthermore, we denote by Iε Ω, λ, p, α the subclass of I ∗ Ω, λ, p, α consisting of functions of the form f z z − ( p − αε ( p n /p Ω1 λnpΩ−1 )( n p − α z n − ∞ ∑ k n 1 ap kz p k 0 ≤ ε < 1 . 1.19 Themain object of the present paper is to investigate interesting properties and characteristics of the classes I∗ Ω, λ, p, α and Iε Ω, λ, p, α . Also, the partial sums is defined for f function defined by 1.19 . 2. A Coefficient Inequality for the Class I∗ Ω, λ, p, α and Some Theorems for the Class Iε Ω, λ, p, α First, we give a coefficient inequality for the class I∗ Ω, λ, p, α . Theorem 2.1. Let the function f be defined by 1.17 . Then, f is in the class I∗ Ω, λ, p, α if and only if ∞ ∑ k n ( k p p )Ω( 1 λkpΩ−1 )( k p − αap k ≤ p − α. 2.1 Abstract and Applied Analysis 5 Proof. Suppose that f z ∈ I∗ Ω, λ, p, α . Then, we find from 1.16 thatand Applied Analysis 5 Proof. Suppose that f z ∈ I∗ Ω, λ, p, α . Then, we find from 1.16 that Re { z ( ℘ ( Ω, λ, p ))′ ℘ ( Ω, λ, p ) } Re { z ( 1/pΩ ( 1/p − 1λDΩf z )′ λ/pzDΩf z )′′ ( 1/pΩ − λDΩf z λ/pzDΩf z )′ } Re ⎧ ⎨ ⎩ pz −∞k n ( k p )Ω 11/pΩ λk/p ) ap kz p k zp −∞k n ( k p Ω1/pΩ λk/p ) ap kz k ⎫ ⎬ ⎭ > α. 2.2 If we choose z to be real and let z → 1−, we get ⎧ ⎨ ⎩ p −∞k n ( k p )Ω 11/pΩ λk/p ) ap k 1 −∞k n ( k p Ω1/pΩ λk/p ) ap k ⎫ ⎬ ⎭ ≥ α 2.3