Some New Inclusion and Neighborhood Properties for Certain Multivalent Function Classes Associated with the Convolution Structure

We use the familiar convolution structure of analytic functions to introduce two new subclasses of multivalently analytic functions of complex order, and prove several inclusion relationships associated with the (𝑛,𝛿)-neighborhoods for these subclasses. Some interesting consequences of these results are also pointed out.


Introduction and preliminaries
Let A p n denote the class of functions of the form We denote by T p n the subclass of A p n consisting of functions of the form which are p-valent in U.
For a fixed function g z ∈ A p n defined by we introduce a new class S λ p g; n, b, m of functions belonging to the subclass of T p n , which consists of functions f z of the form 1.5 , satisfying the following inequality: with the parameters α 1 , . . ., α q , β 1 , . . ., β s 1.12 being so chosen that the coefficients b k in 1.6 satisfy the following condition: then the class S λ p g; n, b, m transforms into a presumably new class S λ p n, b, m defined by The operator H q s α 1 f z : H q s α 1 , . . ., α q ; β 1 , . . ., β s f z , 1.15 involved in 1.14 , is the Dziok-Srivastava linear operator see for details 6 ; see also 7, 8 which contains such well-known operators as the Hohlov linear operator, Saitoh generalized linear operator, Carlson-Shaffer linear operator, Ruscheweyh derivative operator as well as its generalized version, the Bernardi-Libera-Livingston operator, and the Srivastava-Owa fractional derivative operator.One may refer to 7 or 6 for further details and references for these operators.The Dziok-Srivastava linear operator defined in 6 has further been generalized by Dziok and Raina 7 see also 8, 9 .Following a recent investigation by Frasin and Darus 10 , let f z ∈ T p n , δ ≥ 0, then a q, δ -neighborhood of the function f z is defined by It follows from the definition 1.16 that if e z z p p ∈ N , 1.17 International Journal of Mathematics and Mathematical Sciences then We observe that where N δ f and M δ f denote, respectively, the δ-neighborhoods of the function defined by Ruscheweyh 11 and Silverman 12 .
Finally, for a fixed function 21 let P λ p g; n, b, m denote the subclass of T p n consisting of functions f z of the form 1.5 which satisfy the following inequality:

1.22
The object of the present paper is to investigate the various properties and characteristics of functions belonging to the above-defined subclasses S λ p g; n, b, m , P λ p g; n, b, m 1.23 of p-valently analytic functions in U. Apart from deriving coefficient inequalities for each of these function classes, we establish several inclusion relationships involving the n, δneighborhoods of functions belonging to these subclasses.

Coefficient bound inequalities
We begin by proving a necessary and sufficient condition for the function f z ∈ T p n to be in each of the classes Theorem 2.1.Let f z ∈ T p n be given by 1.5 .Then f z is in the class S λ p g; n, b, m if and only if Proof.Assume that f z ∈ S λ p g; n, b, m .Then, in view of 1.5 -1.7 , we get

2.4
Putting z r 0 ≤ r < 1 in 2.4 , the denominator expression on the left-hand side of 2.4 remains positive for r 0, and also for all r ∈ 0, 1 .Hence, by letting r→1 − , through real values, inequality 2.4 leads to the desired assertion 2.2 of Theorem 2.1.
Conversely, by applying the hypothesis 2.2 of Theorem 2.1, and letting |z| 1, we find that

2.5
Hence, by the maximum modulus principle, we infer that f z ∈ S λ p g; n, b, m , which completes the proof of Theorem The following results concerning the class of functions P λ p g; n, b, m can be proved on similar lines as given above for Theorem 2.1.

Inclusion properties
We now obtain some inclusion relationships for the function classes Applying the assertion 2.2 of Theorem 2.1 again in conjunction with 3.5 , we obtain In the analogous manner, by applying the assertion 2.10 of Theorem 2.3 instead of the assertion 2.2 of Theorem 2.1 to the functions in the class P λ p g; n, b, m , we can prove the following inclusion relationship.
Proof.Suppose that f z ∈ N q n,δ h .We then find from 1.16 that Next, since h z ∈ S λ p g; n, b, m , we have in view of 3.5 that The proof of Theorem 4.2 below is similar to that of Theorem 4.1 above, and its proof details are, therefore, omitted here.

International
; n, p ∈ N {1, 2, . . .} , 1.1 which are analytic and p-valent in the open unit disk U z; z ∈ C : |z| < 1 .1.2 If f ∈ A p n is given by 1.1 and g ∈ A p n is given by Journal of Mathematics and Mathematical Sciences then the Hadamard product or convolution f * g of f and g is defined as usual by f * g z : z p ∞ k n a k b k z k : g * f z .1.4

1 . 7
We note that there exist several interesting new or known subclasses of our function class S λ p g; n, b, m .For example, if λ 0 in 1.7 , we obtain the class S p g; n, b, m studied very recently by Prajapat et al. 1 .On the other hand, if the coefficients b k in 1.6 are chosen as follows: k ≥ n; r, p, n ∈ N , 1.8 and n is replaced by n p in 1.4 and 1.5 , then we obtain the class S p n,m μ, r, λ, b of p-valently analytic functions involving the multiplier transformation operator I p r, μ defined in 2 which was studied recently by Srivastava et al. 3 .Also, if we set λ 0 in 1.7 and if the arbitrary sequence b k in 1.6 is selected as follows:

Theorem 3 . 1 .Proof.
n, δ -neighborhood defined by 1.18 .If b k ≥ b n k ≥ n and δ : n λ p − m − 1 1 |b| p m n − p |b| λ n − m − 1 Let f z ∈ S λ p g; n, b, m .Then, in view of assertion 2.2 of Theorem 2.1, and the given condition b k ≥ b n k ≥ n , we get n λ p − m − 1 1 |b| p m n − p |b| λ n − m − 1 1 n m b n : δ p > |b| , 3.7 which by virtue of 1.18 establishes the inclusion relation 3.3 of Theorem 3.1.

Theorem 4 . 2 .Remark 4 . 3 .
If h z ∈ P λ p g; n, b, m andα p − δ n q 1 λ n − p 1 n − m n m b n λ n − p 1 n − m n m b n − p − m |b| − 1 /mApplying the parametric substitutions listed in 2.6 , Theorems 4.1 and 4.2 would yield the corresponding results of Srivastava et al. 3, Theorems 5 and 6, page 6 .Also using substitutions as mentioned above in 2.8 , we get the results due to Orhan and Kamali 13, Theorem 3, page 60; Theorem 4, page 61 . 2.1.

10
Remark 2.4.Making use of the same substitutions as mentioned above in 2.6 , Theorem 2.3 yields another known result due to Srivastava et al. 3, Theorem 2, page 4 .Also, using the same substitutions as mentioned above in 2.8 , we get the result of Orhan and Kamali 13, Lemma 2, page 58 .