Extension of Nunokawa Lemma for Functions with Fixed Second Coefficient and Its Applications

In this paper, we study some properties of analytic functions with ﬁ xed initial coe ﬃ cients. The methodology of di ﬀ erential subordination is used for modi ﬁ cation and improvements of several well-known results for subclasses of univalent functions by restricting the functions with ﬁ xed initial coe ﬃ cients. Actually, by extending the Nunokawa lemma for ﬁ xed initial coe ﬃ cient functions, we obtain some novel results on subclasses of univalent functions, such as di ﬀ erential inequalities for univalency or starlikeness of analytic functions. Also, we provide some new su ﬃ cient conditions for strongly starlike functions. The results of this paper extend and improve the previously known results by considering functions with ﬁ xed second coe ﬃ cients.


Introduction and Preliminaries
Let H be the class of analytic functions in the unit disc U = fz : jzj < 1g. For a ∈ ℂ and n ∈ ℕ, let us define two wellknown classes of analytic functions as follows: H a, n ½ = f ∈ H : f z ð Þ = a + a n z n +⋯,z ∈ U f g , We denote by A = A 1 and S ⊂ A, as the class of univalent functions. Also, we denote by S * ðαÞ, the set of starlike function of order α (α < 1), as follows: which is introduced by Robertson in [1]. Let S * ð0Þ = S * be the classes of starlike functions. Brannan and Kirwan in [2] introduced SS * ðλÞ the class of strongly starlike function of order λ, 0 < λ < 1, by Also, Takahashi and Nunokawa in [3] defined the following subclasses of A: It is easy to see that SS * ðmin fα, λgÞ ⊂ SS * ðα, λÞ ⊂ S S * ðmax fα, λgÞ. Also, if α = λ, then SS * ðα, λÞ ⊂ S * ⊂ S. Then, they are the subclasses of univalent functions in the unit disc U.
Recently, Ali et al. [4] have extended the theory of second-order differential subordination for functions with a fixed initial coefficient. We denote by H β ½a, n the class of analytic functions with a fixed initial coefficient as follows: H β a, n ½ = p ∈ H : p z ð Þ = a + βz n + a n+1 z n+1 +⋯ È É : Further, let and also, let where b ∈ ℂ is fixed. We denote by A b = A 1,b , and we assume that β and b are positive real numbers. The importance of the second coefficient of analytic functions was shown in the monograph [5], for example, coefficient estimates in two well-known growth and distortion theorems for functions in the class S. On the other hand, one of the significant tools in geometric function theory is the theory of differential subordination due to Miller and Mocanu [6]. Recently, Ali et al. [4] improved the theory of differential subordination by this assumption that the second coefficient of analytic function is fixed. For some applications of this improvement, see [7,8]. In addition, Nunokawa has proved a theorem known as Nunokawa lemmas [9] which have had many applications in the geometric function theory. So, it is natural to ask if it is possible to extend this lemma for analytic functions with a fixed second coefficient.
In this paper, we extend this lemma, and then, it will be applied to obtain several new results by restricting the functions to have a fixed second coefficient. It is a remarkable fact when we restrict ourselves to the functions with a fixed second coefficient; then, it will expand the validity domain of the results. As a byproduct of this idea, one can also get results for other class mapping. Even though, in this paper, we are not going into details of all such applications, we remark that other applications will appear in other papers of the authors.
In [10], Silverman examined the class G b , consisting of normalized functions f ∈ S that satisfy the condition for some positive b. He has proved that G b ⊆ S * ð2/1 + ffiffiffiffiffiffiffiffiffiffiffiffi For f ðzÞ and gðzÞ in H , we say that f ðzÞ is subordinate to gðzÞ if there exists a function wðzÞ ∈ H with wð0Þ = 0, jwðzÞj < 1ðz ∈ UÞ, such that f ðzÞ = gðwðzÞÞðz ∈ UÞ denote by f ðzÞ ≺ gðzÞ. When gðzÞ is univalent in U, then we can see This article is arranged as follows. In Section 2, we extend the lemma due to Nunokawa [9] for fixed second coefficient functions and then will bring its nice applications for finding sufficient conditions for strongly starlike functions. In Section 3, by considering the special cases of this extension, we will provide some sufficient conditions for starlikeness and univalency of analytic functions. Also, we discuss about preserving starlikeness of the general integral operator I δ,γ . Finally, in Theorem 26, we state a new result which is improvement of some well-known results in the literature.
We just need a definition and a fundamental lemma due to Ali et al. [4] to get the results. Definition 1 ([6], Definition 1, p. 158). Let Q be the class of functions q that are analytic and injective in and are such that q′ðζÞ ≠ 0 for ζ ∈ ∂U \ EðqÞ.

Main Results
The Nunokawa lemma is a very useful tool in the theory of differential subordination that was proved for the first time by Nunokawa [9]. At first, we mention extension of this lemma with its proof because we need it in the next theorems.

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where Proof. Let us define Also, q ∈ Q with EðqÞ = 1. According to properties of q and assumption (14), we have pðz 1 Þ ∈ qð∂UÞ, pðz 2 Þ ∈ qð∂UÞ, and pðfz : |z|<|z 0 | gÞ ⊂ qðUÞ. Set Then, it can be verified that q 1 ðUÞ is the right half plane Refwg > 0, p 1 ðz 1 Þ = −ix 1 with x 1 > 0, p 2 ðz 2 Þ = ix 2 with x 2 > 0 and p 1 ðfz : jzj < jz 0 jgÞ ⊂ q 1 ðUÞ. We notice that and so, by considering c = exp pðiπα/ðλ + αÞÞ, we can rewrite the functions p 1 and q 1 as Now, taking the derivative of q 1 and calculating the inverse of q 1 yields Since p 1 ∈ H c 1 ½a, n with and so, from Lemma 2, we deduce that there exist two complex numbers ζ 1 and ζ 2 on ∂U with p 1 ðz 1 Þ = q 1 ðζ 1 Þ and p 1 ðz 2 Þ = q 1 ðζ 2 Þ such that where But hence, But the function takes its minimum at the point x = 1, and we have f ð1Þ = tan πλ/2ðα + λÞ. In view of q 1 ′ ð0Þ = 2 sin απ/ðα + λÞ and (25), we obtain 3 Journal of Function Spaces thus, we showed that In a similar way, by noting that pðz 2 Þ = ix 2 with x 2 > 0 and letting we can see that Now, doing analogous above, we find that Thus, we showed that which is (16); therefore, the proof is complete. ☐ Remark 4. Nunokawa [9] provided a similar result to Theorem 3 with this assumption that p ∈ H ½1, n. But, if we fix the second initial coefficient, then we may have some reform as seen above.
Proof. Suppose that f ∈ G n,β b : Let us define with Then, it is clear that p ∈ H nβ ½1, n, p 1 ∈ H c 1 ½a, n with Also, it is easy to see that If f ∉ SS * n,β ðα, λÞ, then p 1 ðUÞ is not contained in the right half plane Rew > 0; hence, there exists a point z 1 ∈ U such that p 1 ðfz : jzj < jz 1 jgÞ is contained in the right half plane Rew > 0 while, In the first step, let p 1 ðz 1 Þ = −ix 1 , then by applying the same argument as Theorem 3, we obtain where k 1 ≥ m 1 = n + sin πα/2δ − nβ/2δ sin πα/2δ + nβ/2δ tan πλ 4δ : ð42Þ

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Also, from the relation of p and p 1 , we have |pðz 1 Þ | = x δ 1 , and so, by making use of (41), it yields If we define then one can verify that g takes its minimum atã 1 ; hence, applying (42) and (43), we have which contradicts with the assumption f ∈ G n,β b . If p 1 ðz 0 Þ = ix 2 , where x 2 > 0, using a similar argument as above leads the contradiction. Indeed, applying the previous considerations, we obtain where k 2 ≥ m 2 = n + sin πα/2δ − nβ/2δ sin πα/2δ + nβ/2δ tan πα 4δ : ð47Þ Notice that jpðz 1 Þj = x δ 2 , and so, by making use of (46), it yields If we define then by some calculation, we find that h takes its minimum at a 2 ; hence, applying (47) and (48), we have which contradicts with the assumption f ∈ G n,β b . So, in both cases, we come to a contradiction, and the proof is complete. ☐ Letting α = λ in the Theorem 5, we get the following corollary.
In the following theorem, we mention a strong result which provides sufficient conditions to strongly starlike functions.

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Now, for 0 < θ < π, we have with t ≥ 1, and for π < θ < 2π, with t ≥ 1. Hence, gðUÞ is a whole complex plane except the half lines ω 1 ðtÞ and ω 2 ðtÞ. Suppose that with Then, it is clear that p ∈ H nβ ½1, n, p 1 ∈ H c 1 ½a, n with Also, it is easy to see that If f ∉ SS * n,β ðα, λÞ, then p 1 ðUÞ is not contained in the right half plane Rew > 0; hence, there exists a point z 1 ∈ U such that p 1 ðfz : jzj < jz 1 jgÞ is contained in the right half plane Rew > 0 while p 1 ðz 1 Þ = −ix 1 , or p 1 ðz 1 Þ = ix 2 , where x 1 , x 2 > 0. In the first step, let p 1 ðz 1 Þ = −ix 1 ; then, using the same argument as Theorem 3, we obtain where k 1 ≥ m 1 = n + sin πα/2δ − nβ/2δ sin πα/2δ + nβ/2δ Also, from the relation of p and p 1 , we have and so, by making use of (64), it yields which contradicts with the assumption. For the case p 1 ðz 1 Þ = ix 2 , where x 2 > 0, using the same argument as Theorem 3, we obtain where Notice that and so, by making use of (68), we have which contradicts with the assumption. So, in both cases, we come to a contradiction, and the proof is complete. ☐

Starlikeness of Analytic Functions
There are many differential inequalities in geometric function theory which is used for univalency or starlikeness of analytic functions. For example, Ozaki [11] has proved that if f ∈ A satisfies the condition Ref1 + zf ″ ðzÞ/f ′ ðzÞg > −1/2 , then f is univalent. Also, the well-known result known as the Mark-strohhäcker [5] states that f ∈ A, then Hence, it is natural to extend the similar results to analytic functions with a fixed initial coefficient.
In this section, we try to obtain some inequalities in analytic functions with second fixed coefficients which improve 6 Journal of Function Spaces earlier results obtained in the literature. At first, we bring the following corollary.