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


Introduction
Let p, n denote the class of functions f z of the form which are analytic and p-valent in the punctured unit disk U {z : 0 < |z| < 1}.Also let * λ p, n, α and λ c p, n, α denote the subclasses of p, n consisting of all functions f z which are defined, respectively, by Re −e iλ zf z f z > α cos λ z ∈ U , Re −e iλ zf z f z > α cos λ z ∈ U .

1.2
We note that for λ 0 and n 1, the above classes reduce to the well-known subclasses of p consisting of meromorphic multivalent functions which are, respectively, starlike 2 Abstract and Applied Analysis and convex of order α 0 ≤ α < p .For the detail on the subject of meromorphic spiral-like functions and related topics, we refer the work of In this paper, first, we find sufficient conditions for the classes * λ p, n, α and λ c p, n, α and then study some mapping properties of the integral operator given by 1.4 .
To obtain our main results, we need the following Lemmas.
Lemma 1.1 see 20 .If q z ∈ 1, n with n ≥ 1 and satisfies the condition Lemma 1.2 see 21 .Let Ω be a set in the complex plane C and suppose that Ψ is a mapping from C 2 × U to C which satisfies Ψ ix, y, z / ∈ Ω for z ∈ U, and for all real x, y such that y ≤

logarithmically, we have
which gives
Then clearly h z and g z ∈ 1, n .Now

2.13
Differentiating logarithmically and then simple computation gives us

2.14
Therefore, by using Lemma 1.1, we have Re e iλ 1 zf z f z α cos λ .

2.17
Since h z ∈ 0 c 1, n, 0 , so Abstract and Applied Analysis

2.21
Proof.Let us set

2.22
Then h z is analytic in U with p 0 1.Taking logarithmic differentiation of 2.22 and then by simple computation, we obtain

2.24
Now for all real x and y satisfying y ≤ − n/2 1 x 2 , we have

2.25
Reputing the values of A, B, C, and D and then taking real part, we obtain

2.26
where L, M, and N are given in 2.21 .

2.31
Now taking real part on both sides, we obtain

2.32
This further implies that

2.34
Clearly Liu and Srivastava 1 , Goyal and Prajapat 2 , Raina and Srivastava 3 , Xu and Yang 4 , and Spacek 5 and Robertson 6 .For p 1, 1.4 reduces to the integral operator introduced and studied by Mohammed and Darus 12, 13 .Similar integral operators for different classes of analytic, univalent, and multivalent functons in the open unit disk are studied by various authors, see 14-19 .
Making use of 2.27 and Corollary 2.5, one can prove the following result.For j ∈ {1, . . .m}, let f j z ∈ p, n and satisfy 2.27 .If Therefore H m,p z ∈ N p, n, ζ with ζ > p. N p, n, ζ , where ζ > p.