Differential inequalities for spirallike and strongly starlike functions

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{eiβp(z)} > γ or | arg{p(z) – γ }| < δ for all z ∈ D, where β ∈ (–π /2,π /2), γ ∈ [0, cosβ), δ ∈ (0, 1] and D := {z ∈C : |z| < 1}. The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D.

A function p in P γ (0) is said to be a Carathéodory function of order γ in D. In particular, P 0 (0) ≡ P is the well-known class of Carathéodory functions. Also, a function p in P 0 (β) is said to be a tilted Carathéodory function by angle β [27]. We note that holds, by (1.1). Let A denote the class of functions f in H normalized by f (0) = 0 = f (0) -1. And let S be the subclass of A consisting of all univalent functions. Further we denote by S * γ (β) and SS * γ (δ) the subclass of A consisting of β-spirallike functions of order γ [8,II,p. 89] (see also [16,24]) and strongly starlike functions of order δ and type γ [9]. That is, a function f ∈ A belongs to the class S * γ (β) if f satisfies and belongs to the class SS * γ (δ) when f satisfies Thus we have is the class of starlike functions of order γ , and S * 0 (β) ≡ SP(β) is the class of β-spirallike functions. It is well known [24] (or [8, Vol. I, p. 149]) that S * (γ ) and SP(β) are the subclasses of S. See [7,12,28] for sufficient conditions for spirallike functions. We also note that SS * γ (δ) ⊂ S * γ (0) ⊂ S. Especially, SS * 0 (δ) ≡ SS * (δ) which is the class of strongly starlike functions of order δ [4,25]. Refer to [5, 6, 11, 13, 14, 17-20, 23, 26] for various sufficient conditions for strongly starlike functions.
In the present paper we investigate new sufficient conditions for functions in P γ (β) or Q γ (δ). As direct consequences of these results, we will obtain several sufficient conditions for spirallike functions or strongly starlike functions in D.
For analytic functions f and g, we say that f is subordinate to g, denoted by f ≺ g, if there is an analytic function ω : D → D with |ω(z)| ≤ |z| such that f (z) = g(ω(z)). Further, if g is univalent, then the definition of subordination f ≺ g simplifies to the conditions f (0) = g(0) and f (D) ⊆ g(D) (see [21, p. 36]).
Let D = {z ∈ C : |z| ≤ 1} and ∂D = {z ∈ C : |z| = 1} be the closure and boundary of D, respectively. We denote by R the class of functions q that are analytic and injective on D \ E(q), where E(q) = ζ : ζ ∈ ∂D and lim z→ζ q(z) = ∞ , and are such that Furthermore, let the subclass of R for which q(0) = a be denoted by R(a). We recall the following lemma which will be used for our results. Lemma 1.1 ([15, p. 24]) Let q ∈ R(a) and let p(z) = a + a n z n + · · · (n ≥ 1) be an analytic function in D with p(0) = a. If p is not subordinate to q, then there exist points z 0 ∈ D and ζ 0 ∈ ∂D \ E(q) for which

Main results
Throughout this section, let β and γ be real numbers such that -π/2 < β < π/2 and 0 ≤ γ < cos β unless we mention it. We define a function ϕ β,γ : D → C by Then it is easy to check that the bilinear function ϕ β,γ maps the unit disk D onto the halfplane γ (β). By using the function ϕ β,γ we obtain the following results.
Proof Let us define functions q and h : D → C by and where ϕ β,γ is the function defined by (2.1). Then the functions q and h are analytic in D with Suppose now that q is not subordinate to h. Then, by Lemma 1.1, there exist points which implies that Using (2.3) and (2.5), we have (2.7) Let α = α 1 + iα 2 with α 1 ≥ 0 and α 2 ∈ R. Then we have Furthermore it is easy to see that Since m ≥ 1, from (2.8), we have Since σ < 0, α 1 ≥ 0, and cos β > γ , inequality (2.9) implies Finally, from (2.7), (2.10), and (2.11), we obtain This inequality contradicts hypothesis (2.2). Therefore, we obtain q ≺ h in D and the inequality Re{e iβ /p(z)} > γ holds for all z ∈ D.
We remark that the hypothesis in Theorem 2.1 implies also 1/p ∈ P γ (β). And we also remark that Theorem 2.1 reduces the result [13] when α = 1.
By the above remark, taking γ = 1/2 in Theorem 2.1 gives the following corollary. Taking p(z) = zf (z)/f (z), f ∈ A, in Corollary 2.1 gives the following result.
Example 2.1 Let a ∈ C be given, and let f a (z) = z/(1az), z ∈ D. Then a computation shows that Hence if then g a is strongly starlike of order 2(cot β)/π in D.
This contradicts (2.21). Therefore we obtain q ≺ h in D, and it follows that the inequality Re{e iβ p(z)} > γ holds for all z ∈ D.
then f is strongly starlike of order 1 -(2/π)β and type γ in D.

Concluding remarks and observations
In the present investigation, we have found several conditions for Carathéodory functions by using a technique of the first-order differential subordination. In particular, one can obtain conditions for Carathéodory functions of order γ (0 < γ ≤ 1) and for tilted Carathéodory functions by angle β (-π/2 < β < π/2). We have applied these results to obtain new criteria for geometric properties such as spirallikeness and strongly starlikeness, and several examples were given here. We conclude this paper by remarking that the results here reduce the earlier conditions [13] for Carathéodory functions. Also, as the examples in this paper show, the first-order differential subordination with the conformal mapping ϕ β,γ defined by (2.1) gives some nice criteria for spirallike functions and strongly starlike functions. This observation will indeed apply to any attempt to produce the conditions for other geometric properties such as convexity, q-starlikeness, etc. [1-3, 10, 22, 29, 30].