New Criteria for Meromorphic Starlikeness and Close-to-Convexity

: The main purpose of current paper is to obtain some new criteria for meromorphic strongly starlike functions of order α and strongly close-to-convexity of order α . Furthermore, the main results presented here are compared with the previous outcomes obtained in this area


Introduction and Preliminaries
For two analytic functions f and F in U := {z : z ∈ C and |z| < 1}, it is stated that the function f is subordinate to the function F in U, written as f (z) ≺ F(z), if there exists a Schwarz function , which is analytic in U with (0) = 0 and | (z)| < 1 (z ∈ U), such that f (z) = F (z) for all z ∈ U.In particular, if F be a univalent function in U, then we have below equivalence: f (z) ≺ F(z) ⇐⇒ f (0) = F(0) and f (U) ⊂ F(U).
Let Σ n denote the category of all functions analytic in the punctured open unit disk U * given by U * := {z : z ∈ C and 0 < |z| < 1} = U \ {0}, which have the form A function f ∈ Σ, where Σ is the union of Σ n for all positive integers n, is said to be in the class MS * (α) of meromorphic strongly starlike functions of order α if we have the condition arg − z f (z) f (z) < απ 2 (z ∈ U * ; 0 < α 1).
In particular, MS * := MS * (1) is the class of meromorphic starlike functions in the open unit disk U.
Let A n be the category of all functions analytic in U which have the following form The class A 1 is denoted by A.
Let S * (α) be the subcategory of A defined as follows The classes S * (α) will be called the class of strongly starlike functions of order α.In particular, S * := S * (1) is the class of starlike functions in U.
By means of the principle of subordination between analytic functions, the above definition is equivalent to Furthermore, let CC(α) denote the category of all functions in A which are strongly close-to-convex of order α in U if there exists a function g ∈ S * such that In particular, CC := CC(1) is the class of close-to-convex functions in U.
In the year 1978, Miller and Mocanu [1] introduced the method of differential subordinations.Because of the interesting properties and applications possessed by the Briot-Bouquet differential subordination, there have been many attempts to extend these results.Then, in recent years, several authors obtained several applications of the method of differential subordinations in geometric function theory by using differential subordination associated with starlikeness, convexity, close-to-convexity and so on (see, for example, [2][3][4][5][6][7][8][9][10][11][12][13]).Furthermore, based on the generalized Jack lemma, the well-known lemma of Nunokawa and so on, certain sufficient conditions were derived in [14][15][16] considering concept of arg, real part and imaginary part for function to be p-valently starlike and convex one in the unit disk.
The aim of the current paper is to obtain some new criteria for univalence, strongly starlikeness and strongly close-to-convexity of functions in the normalized analytic function class A n in the open unit disk U and meromorphic strongly starlikeness in the punctured open unit disk U * by using a lemma given by Nunokawa (see [17,18]).Further, the current results are compared with the previous outcomes obtained in this area.
In order to prove our main results, we require the following lemma.Lemma 1. (see [17,18]) Let the function p(z) given by be analytic in U with p(0) = 1 and p(z) = 0 (z ∈ U).

Main Results
Theorem 1.Let p be an analytic function in U, given by and p(z) = 0 for z ∈ U. Let α 0 is the only root of the equation Proof.To prove our result we suppose that there exists a point z 0 ∈ U so that Then, Lemma 1, gives us that zp (z 0 ) where [p(z 0 )] and k is given by ( 3) or (4).For the case arg p(z 0 ) = απ 2 when [p(z 0 )] which contradicts with condition (5).
Next, for the case arg p(z 0 ) = − απ 2 when [p(z 0 )] with k −m, applying the similar method as the above, we can get which is a contradiction to (5).Therefore, from the two mentioned contradictions, we obtain This completes our proof.
Let ψ(r, s, t; z) : C 3 × U → C and let h be univalent in U.If p is analytic in U and satisfies the (second order) differential subordination then p is called a solution of the differential subordination.The univalent function q is called a dominant of the solution of the differential subordination or more simply a dominant, if p ≺ q for all p satisfying (6).A dominant q satisfying q ≺ q for all dominants q of ( 6) is said to be the best dominant of (6).The best dominant is unique up to a rotation of U.
The following result, which is one of the types of differential subordinations was expressed in [1].
Remark 1.The form (5) cannot be used to obtain in inequality (7).Therefore, Theorem 1 is a small extension of Theorem 2.
Next, for the case arg p(z 0 ) = − απ 2 when [p(z 0 )] applying the similar method as the above, we can get which is a contradiction to condition (10).Therefore, from the two mentioned contradictions, we obtain This completes the proof of Theorem 3.
The condition (10) can be written as a generalized Briot-Bouquet differential subordination.However, It is remarkable that the condition (12) among the outcomes on the generalized Briot-Bouquet differential subordination collected in ( [19], Ch. 3) is not taken into account the case γ = 0, β = i which we have in (10).
Theorem 5. Let p be an analytic function in U, given by and p(z) = 0 for z ∈ U.Then, from Lemma 1, it follows that where [p(z 0 )] and k is given by ( 3) or (4).For the case arg p(z 0 ) = απ 2 when [p(z 0 )] which contradicts our hypothesis in (13).
Next, for the case arg p(z 0 ) = − απ 2 when [p(z 0 )] with k −m, applying the similar method as the above, we can get which is a contradiction to (13).Therefore, from the two mentioned contradictions, we obtain This completes the proof of Theorem 5.By choosing in Theorem 6, we obtain a sufficient condition for strongly meromorphic starlikeness as follows.
Let α > 0 and β > 0 satisfy the inequality Then f is meromorphic strongly starlike function of order α.Theorem 6.Let p be an analytic function in U with p(0) = 1, p (0) = 0 and p(z) = 0 for z ∈ U that satisfies the following inequality where Proof.To prove the result asserted by Theorem 6, we suppose that there exists a point z Then, from Lemma 1, it follows that where [p(z 0 )] which is a contradiction to the assumption of Theorem 6.Therefore, from the two mentioned contradictions, we obtain arg p(z) < απ 2 (z ∈ U).
This completes the proof of Theorem 6.
By setting p(z) := z f (z) g(z) = 0, in Theorem 6, we obtain a sufficient condition for strongly close-to-convexity as follows.

Corollary 1 .Theorem 3 .
Let p be an analytic function in U, given by p(z) = 1 + z) = 0 for z ∈ U. Let arg p 2 (z) − 2p(z)zp (z) < Let p be an analytic function in U, given by p(z) = 1 + z) = 0 for z ∈ U. Let α 0 be the smallest positive root of the equation 2 π arctan