Some geometric properties of certain meromorphically multivalent functions associated with the first-order differential subordination

* Correspondence: Email: jlliu@yzu.edu.cn. Abstract: A new subclass Gn(A, B, λ) of meromorphically multivalent functions defined by the firstorder differential subordination is introduced. Some geometric properties of this new subclass are investigated. The sharp upper bound on |z| = r < 1 for the functional Re{(1 − λ)zp f (z) − pz f ′(z)} over the class Gn(A, B, 0) is obtained.

For functions f (z) and g(z) analytic in U = {z : |z| < 1}, we say that f (z) is subordinate to g(z) and write f (z) ≺ g(z) (z ∈ U), if there exists an analytic function w(z) in U such that |w(z)| ≤ |z| and f (z) = g(w(z)) (z ∈ U).
In this paper we introduce and investigate the following subclass of Σ n (p).
Definition. A function f (z) ∈ Σ n (p) is said to be in the class G n (A, B, λ) if it satisfies the first-order differential subordination: Recently, several authors (see, e.g., [1][2][3][4][5][6][7][8][10][11][12][13][14][15][16] and the references cited therein) introduced and studied various subclasses of meromorphically multivalent functions. Certain properties such as distortion bounds, inclusion relations and coefficient estimates are given. In this note we obtain inclusion relation, coefficient estimate and sharp bounds on Re (z p f (z)) for functions f (z) belonging to the class G n (A, B, λ). Furthermore, we investigate a new problem, that is, to find where f (z) varies in the class We need the following lemma in order to derive the main results for the class G n (A, B, λ).
2. Geometric properties of functions in class G n (A, B, λ) for f (z) ∈ Q n (A, B, α 2 ). Then the function g(z) is analytic in U with g(0) = p. By using (1.3) and (2.1), we have An application of the above Lemma yields By noting that 0 < α 1 α 2 < 1 and that the function 1+Az 1+Bz is convex univalent in U, it follows from This shows that f (z) ∈ Q n (A, B, α 1 ). The proof of Theorem 1 is completed.
Furthermore, for the function f n (z) given by (2.8), we find that f n (z) ∈ A n (p), and Hence f n (z) ∈ Q n (A, B, α) and, from (2.14), we conclude that the inequalities (2.4) to (2.7) are sharp. The proof of Theorem 2 is completed.
Proof. Let f (z) ∈ Q n (A, B, α) and (2.15) be satisfied. Then, by using (2.5) in Theorem 2, we see that This shows that f (z) is p-valent close-to-convex in U. The proof of the corollary is completed. and Re .
All of the above bounds are sharp.
Proof. It is obvious that Making use of (2.4) in Theorem 2, it follows from (2.19) that From this and (2.19), we obtain (2.18). Furthermore, it is easy to see that the inequalities (2.16)-(2.18) are sharp for the function f n (z) given by (2.8). Now the proof of Theorem 3 is completed.
Proof. Since AB ≤ 1, it follows from (2.9) that By virtue of (2.12) and (2.22), we have, for |z| = r < 1, Theorem 5. Let f (z) ∈ Q 1 (A, B, α) and where the symbol * denotes the familiar Hadamard product of two analytic functions in U.
Proof. Since g(z) ∈ Q 1 (A 0 , B 0 , α 0 ), we find from the inequality (2.18) in Theorem 3 and (2.25) that Re Thus the function g(z) z p has the following Herglotz representation: where µ(x) is a probability measure on the unit circle |x| = 1 and |x|=1 dµ(x) = 1. For f (z) ∈ Q 1 (A, B, α), we have and In view of the fact that the function 1+Az 1+Bz is convex univalent in U, it follows from (2.26) to (2.28) that This shows that ( f * g)(z) ∈ Q 1 (A, B, α). The proof of Theorem 5 is completed.
The result is sharp for each k ≥ n.
Proof. It is known that, if where ϕ(z) is analytic in U and ψ(z) = z + · · · is analytic and convex univalent in U, then |b j | ≤ 1 ( j ∈ N). By using (2.29), we have In view of the fact that the function z 1+Bz is analytic and convex univalent in U, it follows from (2.31) that which gives (2.30).
Next we consider the function f k (z) given by and for each k ≥ n, the proof of Theorem 6 is completed.
The proof of Theorem 7 is completed.

Conclusions
In this paper, we have introduced and investigated some geometric properties of the class G n (A, B, λ) which is defined by using the principle of first-order differential subordination. For this function class, we have derived the sharp upper bound on |z| = r < 1 for the following functional: over the class G n (A, B, 0). We have also obtained other properties of the function class G n (A, B, λ). Motivated by a recently-published survey-cum-expository review article by Srivastava [15], the interested reader's attention is drawn toward the possibility of investigating the basic (or q-) extensions of the results which are presented in this paper. However, as already pointed out by Srivastava, their further extensions using the so-called (p, q)-calculus will be rather trivial and inconsequential variations of the suggested extensions which are based upon the classical q-calculus, the additional paremeter p being redundant or superfluous (see, for details, [15, p. 340]).