Sufficiency Criteria for a Class of p-Valent Analytic Functions of Complex Order

and Applied Analysis 3 Proof. Let p(z) be given by (11), which clearly belongs to the class A(n). Now differentiating (11), we have p 󸀠 (z) = ( f (z) ∗ g (z) z ) 1/(p+b−1) × 1 p + b − 1 { z(f (z) ∗ g (z)) 󸀠 f (z) ∗ g (z) + b − 1} (20) which gives 󵄨󵄨󵄨󵄨 arg p (z)󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 arg( f (z) ∗ g (z) z ) 1/(p+b−1) + arg{ 1 p+b−1 ( z(f (z) ∗ g (z)) 󸀠 f (z) ∗ g (z) +b−1)} 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 . (21) Thus using (19), we have 󵄨󵄨󵄨󵄨 argp (z)󵄨󵄨󵄨󵄨 ≤ π 2 δn (z ∈ U) , (22) where δn is the root of (8). Hence, using Lemma 2, we have p(z) ∈ S(n, 1). From (20), we can write zp 󸀠 (z) p (z) = 1 p + b − 1 [ z(f (z) ∗ g (z)) 󸀠 f (z) ∗ g (z) − p] + 1. (23) Since p(z) ∈ S(n, 1), it implies that Re(zp(z)/p(z)) > 0. Therefore, we get (16), and hence f(z) ∈ Mp(n, b; g(z)). Making n = 1, b = 1 − α with 0 ≤ α < p and g(z) = z p /(1 − z), we have the following. Corollary 9. If f(z) ∈ Ap satisfies 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 arg( f (z) z ) + (p − α) arg{ zf 󸀠 (z) f (z) − α} 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < π 2 δ1 (p − α) (z ∈ U) , (24) where δ1 is the unique root of (8) with n = 1, then f(z) ∈ S p (α), the class of p-valent starlike functions of order α. Also if we take n = 1, b = 1 − α with 0 ≤ α < p and g(z) = z p /(1 − z) 2p in Theorem 8, we obtain the following result. Corollary 10. If f(z) ∈ Ap satisfies 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 arg( f 󸀠 (z) pz ) + (p − α) arg { zf 󸀠󸀠 (z) f (z) + 1 − α} 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < π 2 δ1 (p − α) (z ∈ U) , (25) where δ1 is the unique root of (8) with n = 1, then f(z) ∈ Cp(α), the class of p-valent convex functions of order α. Remark 11. For putting p = 1, α = 0 in Corollary 10 and p = 1 in Corollary 9, we obtain the results proved byMocanu [10] and Uyanık et al. [1], respectively. Theorem 12. If f(z) ∈ Ap(n) satisfies Re[ 1 p+b−1 { z(f (z)∗g (z)) 󸀠 f (z)∗g (z) (ρ z(f (z)∗g (z)) 󸀠󸀠 (f (z)∗g (z)) 󸀠 + 1)}