On Certain Sufficiency Criteria for p-Valent Meromorphic Spiralike Functions

and Applied Analysis 3 2. Some Properties of the Classes ∑∗ λ p, n, α and ∑λ c p, n, α Theorem 2.1. If f z ∈ ∑ p, n satisfies ∣ ∣ ∣ ∣ ( zf z )eiλ/ p−α cosλ { e zf ′ z f z α cosλ ip sinλ } ( p − α cosλ ∣ ∣ ∣ ∣ < n √ n2 1 ( p − α cosλ z ∈ U , 2.1 then f z ∈ ∑∗λ p, n, α . Proof. Let us set a function h z by h z 1 z ( zf z )eiλ/ p−α cosλ 1 z ean ( p − α cosλ n · · · 2.2 for f z ∈ ∑ p, n . Then clearly 2.2 shows that h z ∈ ∑ 1, n . Differentiating 2.2 logarithmically, we have h′ z h z e ( p − α cosλ [ f ′ z f z p z ] − 1 z 2.3 which gives ∣ ∣ ∣z2h′ z 1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( zf z )eiλ/ p−α cosλ 1 ( p − α cosλ { e zf ′ z f z α cos λ ip sinλ } 1 ∣ ∣ ∣ ∣ ∣ . 2.4 Thus using 2.1 , we have ∣ ∣ ∣z2h′ z 1 ∣ ∣ ∣ ≤ n √ n2 1 z ∈ U . 2.5 Hence, using Lemma 1.1, we have h z ∑∗ 0 1, n, 0 . From 2.3 , we can write zh′ z h z 1 ( p − α cosλ [ eiλ zf ′ z f z ( α cosλ ip sinλ ) ] . 2.6 Since h z ∈ ∑∗0 1, n, 0 , it implies that Re −zh′ z /h z > 0. Therefore, we get 1 ( p − α cosλ [ Re ( −eiλ zf ′ z f z ) − α cosλ ] Re ( − ′ z h z ) > 0 2.7 4 Abstract and Applied Analysis or Re ( −eiλ zf ′ z f z ) > α cosλ, 2.8 and this implies that f z ∈ ∑∗λ p, n, α . If we take λ 0, we obtain the following result. Corollary 2.2. If f z ∈ ∑ p, n satisfies ∣ ∣ ∣ ∣ ( zf z )1/ p−α { e zf ′ z f z α } ( p − α ∣ ∣ ∣ ∣ < 1 √ 2 ( p − α z ∈ U , 2.9 then f z ∈ ∑∗ p, n, α . Theorem 2.3. If f z ∈ ∑ p, n satisfies ∣ ∣ ∣ ∣ ∣ ∣ ( z 1f ′ z −p )eiλ/ p−α cosλ{ e ( zf ′′ z f ′ z 1 ) α cos λ ip sinλ } ( p − α cosλ ∣ ∣ ∣ ∣ ∣ ∣ < n 1 ( p − α cosλ √ n 1 2 1 z ∈ U , 2.10 then f z ∈ ∑λc p, n, α . Proof. Let us set