First-Order Differential Subordinations and Their Applications

: In this paper, we consider some relations related to the representations of starlike and convex functions, and obtain some sufﬁcient conditions for starlike and convex functions by using the theory of differential subordination. Actually, we generalize a result by Suffridge for analytic functions with missing coefﬁcients and then we apply that generalization for obtaining the different methods to the implications of starlike or convex functions. Our results generalize and improve the previous results in the literature.


Introduction
We let H denote the class of analytic functions in the open unit disk U = {z ∈ C : |z| < 1} and define H[a, n] = { f ∈ H : f (z) = a + a n z n + a n+1 z n+1 + . . .}, where a ∈ C and n is a positive integer number.Furthermore, we introduce the subclass A n of H as follows: A n = { f ∈ H : f (z) = z + a n+1 z n+1 + a n+2 z n+2 + . . .}.
In particular, we set A 1 ≡ A. As usual, let the subclass S of A be the class of all univalent functions in the open unit disk U. A function f ∈ A is said to be starlike of order γ(0 ≤ γ < 1), written f ∈ S * (γ), if it satisfies Specifically, we put S * (0) ≡ S * .Every element in S * is called a starlike function.Furthermore, a function f ∈ A is said to be convex of order γ(0 ≤ γ < 1), written f ∈ K(γ), if it satisfies In particular, we put K(0) ≡ K. Every element in K is called a convex function.Now for analytic functions in U with the fixed initial coefficient, we define the class H β [a, n] as follows: H β [a, n] = { f ∈ H : f (z) = a + βz n + a n+1 z n+1 + . . .}, where n is a positive integer number, a ∈ C and β ∈ C are fixed numbers.Moreover we assume A n,b = { f ∈ H : f (z) = z + bz n+1 + a n+2 z n+2 + . . .}, where n is a positive integer number and b ∈ C is a fixed number.In addition, we set A 1,b ≡ A b .Assume f and g be in H.We say that the function f is subordinate to g, denoted by f ≺ g, if there exists an analytic function ω in U, with ω(0) = 0 and |ω(z)| ≤ |z| < 1, such that f (z) = g(ω(z)).Moreover, if g is an univalent function in U, then f ≺ g if and only if f (0) = 0 and f (U) ⊂ g(U).
By considering the function 1+Az 1+Bz , we can generalize the class of starlike functions as follows: Let Through ( 1), it can be easily observed that 1+Az 1+Bz with −1 The theory of differential subordination has a key role in the study of geometric function theory.In 1935 Goluzin [1] considered the subordination zp (z) ≺ h(z) and proved that if h is convex then p(z) ≺ q(z) = z 0 h(t)t −1 dt.Furthermore, in 1970 Suffridge [2] showed that Goluzin's result is true if h is starlike.Moreover, Miller and Mocanu by writing many research papers in this direction extended the concept of differential subordination (see for example [3] and references therein).Further, many authors have recently using different combinations of the representations of starlike and convex functions have obtained the simple conditions for starlikeness and convexity of analytic functions.In [4], Silverman gained the results for analytic functions including the terms as the quotient of the analytic representations of convex and starlike functions.For instance, Silverman proved that , where 0 < b ≤ 1 and Next, Obradovic and Tuneski [5], in view of 1 1+b ≥ , improved the work of Silverman.Indeed, they established for 0 < b ≤ 1. Nunokawa et al. [6], by applying the Silverman's quotient function [4], proved that f could be strongly starlike, strongly convex or starlike in U.In [7][8][9][10], the authors have studied some conditions for the analytic functions to belong to the class S * [A, B].In [11], some results related to the above discussion, with respect to n-symmetric points of functions, have been given.Inspired by [9,10], in this paper, we will extend and improve some results obtained in [9,10] and then we will determine some conditions which by means of them, a function belongs to the class S * [A, B].
The contents of this article are regulated as follows: In Section 3, initially, we will prove a theorem that is the extension of a little change to the Suffridge theorem [3].Next, we bring some applications of this theorem as the main results, making the functions be in the class S * [A, B].These results extend and improve some results in [10].In Section 4, we intend to bring some sufficient conditions for starlikeness of analytic functions.We also produce the functions belonging to the class S * [A, B] by considering other conditions, and so we include some corollaries from the result acquired.Furthermore, these results extend and improve some results in [9].Moreover, Suffridge's result is used in recent investigations like [12][13][14][15].Note that some results related to this article for analytic functions with fixed initial coefficients are also mentioned.
In the continuation of the argument, in order to prove the main results, we require to remind a definition and two basic lemmas: Definition 1. (see [3]) Let Q denote the set of functions q that are analytic and injective on U\E(q), where and are such that q (ζ) = 0 for ζ ∈ ∂U\E(q).Lemma 1. (see [3] ) Let q ∈ Q with q(0) = a, and let p(z) = a + a n z n + . . .be analytic in U with p(z) ≡ a and n ≥ 1.If p is not subordinate to q, then there exist points z 0 = r 0 e iθ 0 ∈ U and ζ 0 ∈ ∂U\E(q), and an m ≥ n ≥ 1 for which p(U r 0 ) ⊂ q(U), . Lemma 2. (see [16]) Let q ∈ Q with q(0) = a and p ∈ H c [a, n] with p(z) ≡ a.If there exist a point z 0 ∈ U such that p(z 0 ) ∈ q(∂U) and p({z

Main Results
First, we mention a lemma which is slightly different from the original one, ([3], Th 3.4 h).
Lemma 3. Let q be univalent in U and functions θ and φ be analytic in a domain D containing q(U) and φ(z) = 0 for z ∈ q(U).Moreover, let (i) Q(z) = zq (z)φ[q(z)] be starlike and then p ≺ q and p(z) = q(z n ) is the best dominant.

Proof. Let us define
It easy to verify that the conditions (i) and (ii) imply that h is close-to-convex and hence univalent in U. Now using the same argument as the proof of ([3], Th 3.4h), we get our result and we omit the details of the proof.
By considering θ(z) ≡ 0 in Lemma 3, we extend a little change of the Suffridge theorem [3] as follows: Corollary 1.Let q be univalent in U and q(0) = a.Moreover, let φ be analytic in a domain D containing q(U) and let p ∈ H and q is the best (a, n)-dominant.
Using the same argument of Lemma 3 and applying Lemma 1, we obtain the following theorem, and we omit its proof: Theorem 1.Let q be univalent in U and q(0) = a.Furthermore, let φ be analytic in a domain D containing q(U) and By putting φ ≡ 1 in Lemma 3 and Theorem 1, we reach to the following corollaries: then p ≺ q.
Corollary 3. Let q be convex univalent in U with q(0) Corollary 4. Let p ∈ H [1, n].Suppose that A and B are real numbers with −1 then p ≺ 1+Az 1+Bz .
By applying the same argument as Corollary 4, and using Corollary 3 we have: and A and B be real numbers with −1 ≤ then p ≺ 1+Az 1+Bz .
Setting p(z) = z f (z) f (z) in Corollary 4, we obtain: Corollary 6.Let f ∈ A n .Moreover, let A and B be real numbers with −1 ≤ B < A ≤ 1.If Setting p(z) = z f (z) f (z) in Corollary 5, we obtain: , and A and B be real numbers with Putting A = 1 in Corollary 6, we get to the following corollary: Corollary 8. Let f ∈ A n .Moreover, let B be a real number with −1 ≤ B < 1.If Putting A = 1 in Corollary 7, we come to the following corollary:

Further Results about Analytic Functions to Settle in the Class S * [A, B]
It is well known that for f ∈ A, the condition |z f (z)/ f (z)| < 2 is sufficient for starlikeness of f .In this section, we will extend this result and will also try to bring other sufficient conditions for starlikeness.

Theorem 2. Let A and B be real numbers with −1 ≤
Proof.Let us define θ(z) = z − 1, φ(z) = 1 z and q(z) = 1+Az 1+Bz .For proving this theorem, it is sufficient to show that the conditions of Lemma 3 hold.However, we note that the condition (i) is equivalent to where we attain the assertion of condition (i).On the other hand, from we observe (ii).Moreover, the condition (2) is correct and consequently the proof is completed.
By putting A = 1, B = −1 and p(z) = z f (z) f (z) in Theorem 2 we obtain: . By some calculations, one can observe that h(e it ) = −1 + (n+1)+cos t sin t i.However, the function g(t) := n+1+cos t sin t takes the the minimum value at the point t 0 = cos −1 ( −1 n+1 ) and so |g(t)| ≥ |g(t 0 )| = n(n + 2).Hence, h maps unit disk onto the complement of Ω and the proof is completed.
By using Corollary 10, we have: , then f is starlike and the result is sharp for the function f (z) = 1+z n 1−z n .
By putting A = 0, B = −1 and p(z) = z f (z) f (z) in Theorem 2, we obtain: then f is starlike of order 1/2 and the result is sharp for the function f (z) = 1 1−z n .
We remark that Corollary 12 is the generalization of Marx-Strohh äcker Theorem (see [17]).By putting in Theorem 2 we gain: and the result is sharp for the function f (z) = 1  1−bz n . where In particular, if then f is convex in U.
With some calculations and so utilizing (5), one can obtain By letting ζ 0 = e it , where 0 ≤ t ≤ 2π and using (8), we have Letting x = cos t and defining we have In view of (9) we deduce that g is an increasing function and takes its minimum at the point −1.Hence which contradicts (8), and so this give the result.Since z f (z) f (z) ≺ 1+Az 1+Bz , we have Now combining (5), ( 6) and (10), we have and so f is convex.
With the same approach as the previous theorem and by applying Lemma 1, we attain the following theorem which we omit the proof of.
In particular, if b = A−B n and then f is convex in U.
Corollary 16.Let f ∈ A be an odd function with f Proof.Since f is odd function, we have f (z) = z + ∑ ∞ k=1 a 2k+1 z 2k+1 .Putting n = 2 in Corollary 15, the desired result is obtained.
We know that f is an odd analytic function.On the other hand, one can see that , then f is convex.
Finally we prove the following result: then f is a starlike function.

Conclusions
In this paper, we have proved a result similar to Suffridge's theorem and given some applications related to this result.Moreover, we have investigated some sufficient conditions for starlikeness of analytic functions and functions that are in the class S * [A, B].
Moreover for the different values of A and B, other types of the class A such as the class S * [1, −1], which is equivalent to the class S * , also for 0 Therefore making use of Corollary 16, if 1 2 ≤ α < 1 and |λ| ≤ arctan In addition, if 4 5 ≤ α < 1 and |λ| ≤ arctan * (α).