Differential Subordinations for Certain Meromorphically Multivalent Functions Defined by Dziok-Srivastava Operator

and Applied Analysis 3 Corresponding to a function hp α1, . . . , αq; β1, . . . , βs; z defined by hp ( α1, . . . , αq; β1, . . . , βs; z ) z−pqFs ( α1, . . . , αq; β1, . . . , βs; z ) , 1.11 we now consider a linear operator Hp ( α1, . . . , αq; β1, . . . , βs ) : Σ ( p ) −→ Σp, 1.12 defined by means of the Hadamard product or convolution as follows: Hp ( α1, . . . , αq; β1, . . . , βs ) f z : hp ( α1, . . . , αq; β1, . . . , βs; z ) ∗ f z . 1.13 For convenience, we write Hp,q,s α1 : Hp ( α1, . . . , αq; β1, . . . , βs ) . 1.14 Thus, after some calculations, we have z ( Hp,q,s α1 f z )′ α1Hp,q,s α1 1 f z − ( α1 p ) Hp,q,s α1 f z . 1.15 The operator Hp,q,s α1 is popularly known as the generalized Dziok-Srivastava operator. Many interesting subclasses of multivalent functions, associated with the operator Hp,q,s α1 and its various special cases, were investigated recently by e.g. Dziok and Srivastava 2–4 , Liu 5 , Liu and Srivastava 6, 7 , Patel et al. 8 , Wang et al. 9 , and others. Let P be the class of functions h z with h 0 1, which are analytic and convex univalent in U. Definition 1.1. A function f z ∈ Σ p is said to be in the class Tp,q,s α1, λ;h if it satisfies the subordination condition λ − 1 p z 1 ( Hp,q,s α1 f z )′ λ p ( p 1 )z 2 ( Hp,q,s α1 f z )′′ ≺ h z , 1.16 where λ is a complex number and h z ∈ P . The main object of this paper is to present a systematic investigation of the class Tp,q,s α1, λ;h defined above by means of the generalized Dziok-Srivastava operator Hp,q,s α1 . For our purpose, we shall need the following lemmas to derive our main results for the class Tp,q,s α1, λ;h . Lemma 1.2 see 10 . Let g z be analytic in U and h z be analytic and convex univalent in U with h 0 g 0 . If g z 1 μ zg ′ z ≺ h z , 1.17 4 Abstract and Applied Analysis where Re μ > 0, then g z ≺ h̃ z μz−μ ∫z v tμ−1h t dt ≺ h z 1.18 and h̃ z is the best dominant of 1.17 . Lemma 1.3 see 1 . Let α < 1, f z ∈ S∗ α and g z ∈ R α . Then, for any analytic function F z in U, g ∗ fF g ∗ f U ⊂ co F U , 1.19 where co F U denotes the closed convex hull of F U . 2. Properties of the Class Tp,q,s α1, λ;h Theorem 2.1. Let λ1 < λ2 ≤ 0. Then Tp,q,s α1, λ1;h ⊂ Tp,q,s α1, λ2;h . Proof. Let λ1 < λ2 ≤ 0 and suppose that g z − p 1Hp,q,s α1 f z )′ p 2.1 for f z ∈ Tp,q,s α1, λ1;h . Then the function g z is analytic inUwith g 0 1. Differentiating both sides of 2.1 with respect to z and using 1.16 , we have λ1 − 1 p z 1 ( Hp,q,s α1 f z )′ λ1 p ( p 1 )z 2 ( Hp,q,s α1 f z )′′ g z − λ1 p 1 zg ′ z ≺ h z . 2.2 Hence an application of Lemma 1.2 yields g z ≺ h z . 2.3 Noting that 0 < λ2/λ1 < 1 and that h z is convex univalent in U, it follows from 2.1 to 2.3 that λ2 − 1 p z 1 ( Hp,q,s α1 f z )′ λ2 p ( p 1 )z 2 ( Hp,q,s α1 f z )′′ λ2 λ1 ( λ1 − 1 p z 1 ( Hp,q,s α1 f z )′ λ1 p ( p 1 )z 2 ( Hp,q,s α1 f z )′′ ) ( 1 − λ2 λ1 ) g z ≺ h z . 2.4 Thus f z ∈ Tp, q, s α1, λ2;h and the proof of Theorem 2.1 is completed. Abstract and Applied Analysis 5 Theorem 2.2. Let 0 < b1 < b2. Then Tp,q,s b2, λ;h ⊂ Tp,q,s b1, λ;h . Proof. Define a function g z byand Applied Analysis 5 Theorem 2.2. Let 0 < b1 < b2. Then Tp,q,s b2, λ;h ⊂ Tp,q,s b1, λ;h . Proof. Define a function g z by g z z ∞ ∑ n 1 b1 n b2 n z 1 z ∈ U; 0 < b1 < b2 . 2.5 Then z hp b1, α2, . . . , αs, 1; b2, α2, . . . , αs; z g z ∈ A, 2.6 where hp b1, α2, . . . , αs, 1; b2, α2, . . . , αs; z 2.7 is defined as in 1.11 , and z 1 − z b2 ∗ g z z 1 − z b1 . 2.8 By 2.8 , we see that z 1 − z b2 ∗ g z ∈ S∗ ( 1 − b1 2 ) ⊂ S∗ ( 1 − b2 2 ) 0 < b1 < b2 , 2.9