Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function

The aim of this particular article is at studying a holomorphic function f de ﬁ ned on the open-unit disc D = f z ∈ ℂ : j z j < 1 g for which the below subordination relation holds z f ′ ð z Þ / f ð z Þ ≺ q 0 ð z Þ = 1 + tan h ð z Þ : The class of such functions is denoted by S ∗ tan h : The radius constants of such functions are estimated to conform to the classes of starlike and convex functions of order β and Janowski starlike functions, as well as the classes of starlike functions associated with some familiar functions.


Introduction
To completely comprehend the mathematical concepts used throughout our key observations, some of the essential literature of the geometric function theory must be described and analyzed here. Let us begin with the symbol A n which describes the family of holomorphic (or analytic) functions in a subset D of the complex plan ℂ having the following series expansion f z ð Þ = z + a n+1 z n+1 + a n+2 z n+2 +⋯: Also, let the family of all univalent functions be denoted by S and is a subset of the class A 1 = A: Next, we define that the subordination between the function belongs to the class A. Let g 1 , g 2 ∈ A: Then, g 1 ≺ g 2 or g 1 ðzÞ ≺ g 2 ðzÞ, the mathematical form of the subordination between g 1 and g 2 , if a holomorphic function w occurs in D with the restriction wð0Þ = 0 and jwðzÞj < 1 in such a way that f ðzÞ = gðwðzÞÞ hold. Further, if g 2 ∈ S in D, then, the following relation holds: Three significant subfamilies of S, which are well studied and have nice geometric interpretations, are the families of starlike S * ðξÞ, convex KðξÞ, and strongly starlike S S * ðζÞ functions of order ξð0 ≤ ξ < 1Þ and ζð0 < ζ ≤ 1Þ, respectively. These families are defined as follows: Particularly, the notations SS * ð1Þ = S * ð0Þ = S * and Kð0Þ = K represent familiar families of starlike and convex functions, respectively. These subfamilies of S satisfy the following relationship The reverse of the above relation hold only under certain restriction of the domain. That is; if f ∈ S in D, then, it was given in [1], Corollary, p. 98, that f maps the disc jzj < r onto a region which is star shaped about the origin for every r ≤ r 0 = tan hðπ/4Þ. The constant r 0 is known as the radius of starlikness for the family S. Also, given in [1], Corollary, p. 44, the radius of convexity for the families S * and S is 2 − ffiffi ffi 3 p . To make a radius statement for other things than starlikeness and convexity, we choose two subfamilies G and H of the set A. The G radius for the family H , represented by R G ðH Þ, is the largest number R such that r −1 f ðrzÞ ∈ G for every 0 < r ≤ R and f ∈ H . Consequently, an alternative formulation of the radius of starlikeness for S is that the S * radius for the family S is R S * ðSÞ = tan hðπ/4Þ: In 1992, Ma and Minda [2] considered the general form of the families as where φ is a holomorphic function with φ ′ ð0Þ > 0 and has positive real part. Also, the function φ maps D onto a starshaped region with respect to φð0Þ = 1 and is symmetric about the real axis. They addressed some specific results such as distortion, growth, and covering theorems. In recent years, several subfamilies of the set A were studied as a special case of the class S * ðφÞ. For example, (i) If we take φðzÞ = ð1 + LzÞ/ð1 + MzÞ with −1 ≤ M < L ≤ 1, then, we achieved the class S * ½L, M ≡ S * ðð1 + LzÞ/ð1 + MzÞÞ which is described by the functions of the Janowski starlike class investigated in [3]. Furthermore, S * ðξÞ ≔ S * ½1 − 2ξ,−1 is the familiar starlike function family of order ξ with 0 ≤ ξ < 1 (ii) The family S * L ≔ S * ðφðzÞÞ with φðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p was developed in [4] by Sokol and Stankiewicz. The function φðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p maps the region D onto the the image domain which is bounded by jw 2 − 1j < 1 (iii) The class S * car ≔ S * ðφðzÞÞ with φðzÞ = 1 + ð4/3Þz + ð2/3Þz 2 was examined by Sharma and his coauthors [5] which consists of function f ∈ A in such a manner that ðzf ′ ðzÞÞ/ð f ðzÞÞ is located in the region bounded by the cardioid given by (iv) The family S * R ≔ S * ðφðzÞÞ with φðzÞ = 1 + ðz/JÞ ðJ + z/J − zÞ, J = ffiffi ffi 2 p + 1 is studied in [6] while S * cos ≔ S * ðcos ðzÞÞ and S * cos h ≔ S * ðcosh ðzÞÞ were contributed by Raza and Bano [7] and Alotaibi et.al [8], respectively (v) By choosing φðzÞ = 1 + sin z, we obtain the class S * sin ≔ S * ðφðzÞÞ which was established in [9]. The authors determined radius problems in this article for the defined class S * sin (vi) The class S * e ≔ S * ðe z Þ was explored recently in [10]. For such a class S * e , the authors calculated Hankel determinant bounds of order three in [11]. Also, the class S * RL ≔ S * ðh RL ðzÞÞ with was contributed by Mendiratta et al. [12] in which they investigated the radius problems (vii) The family S * C ≔ S * ðφðzÞÞ with φðzÞ = z + ffiffiffiffiffiffiffiffiffiffiffi ffi 1 + z 2 p was introduced and studied by Raina and Sokół [13] (viii) By considering the function φðzÞ = 1 + sin h −1 z, we get the recently examined family S * ρ ≔ S * ð1 + sin h −1 zÞ introduced by Kumar and Arora [14]. They discussed relationships of this class with the already known classes. For more particular classes, see the articles [15][16][17][18][19][20] In the present paper, we consider a trigonometric function φ 1 ðzÞ = 1 + tan hz with φ 1 ð0Þ = 1: Also, one can easily obtain that Reφ 1 ðzÞ > 0: By using this function, we define the below family of functions as In other words, a function f ∈ S * tan h if and only if there exists a holomorphic function q, fulfilling qðzÞ ≺ q 0 ðzÞ ≕ 1 + tan hz, such that Now, we construct some examples of our newly described family S * tan h . For this, consider the following functions Since q 0 ðzÞ = 1 + tanh z is univalent in D, q i ð0Þ = 1 = q 0 ð0Þ,ði = 1, 2, 3, 4Þ, and q i ðDÞ ⊆ q 0 ðDÞ, this implies that for each i = 1, 2, 3, 4, the relation q i ≺ q 0 holds. Thus, from (8), the functions corresponding to the functions q 1 ðzÞ,q 2 ðzÞ,q 3 ðzÞ, and q 4 ðzÞ, respectively, belong to the family S * tanh : By taking qðzÞ = q 0 ðzÞ = 1 + tan hz in (8), we get the below function that plays a role of the extremal in many problems of the class S * In this article, we work on determining the radius of starlikeness and convexity and S * tanh radius for some subfamilies of starlike functions, mentioned above in which mostly have simple geometric interpretation. Besides these subfamilies, we also discuss the S * tanh radius for some families of A, whose functions have been expressed as a ratio between two functions.

Radii of Starlikeness and Convexity
In this portion, we examined the radius of starlikeness and convexity for the family S * tanh : Proof. If f ∈ S * tan h , then, by virtue of (7), a Schwarz function w exists with jwðzÞj ≤ jzj such as Now, let wðzÞ = Re iv with R ≤ jzj = r,−π ≤ v ≤ π: After easy computation, we get The equation Ψ ′ ðvÞ = 0 has five roots in ½−π, π, namely, 0,±π and ±ðπ/2Þ. Since ΨðvÞ = Ψð−vÞ, it is sufficient to show that v ∈ ½0, π. Furthermore, we can see that Ψð0Þ = tan h 2 R = ΨðπÞ, Ψðπ/2Þ = tan 2 R, and Thus, we have The radius of starlikeness of order ξ, for the family S * tan h , is the smallest positive root r 0 ∈ ð0, 1Þ of 1 − tan r − ξ = 0: Taking ξ = 0 in the above Theorem 1, we obtain the following corollary. Corollary 1. The S * radius, for the family S * tan h , is r 0 = tan −1 ð1Þ ≈ 0:78: The KðξÞ radius r 0 , for the family f ∈ S * tan h , is r 0 = min fr 1 , r 2 g, where r 1 is the smallest root of the equation and r 2 is such that 1 − tan r 2 > 0: Proof. If f ∈ S * tanh , then, a holomorphic function w occurs with wð0Þ = 0 and jwðzÞj ≤ jzj, such that By simple computation, it gives From (18), we get

Journal of Function Spaces
Assume that wðzÞ = Re iv , with R ≤ jzj = r,−π ≤ v ≤ π for calculating the minumum value of the right side of the last inequality. A simple calculation reveals that where y = sin v,x = cos v, and x, y ∈ ½−1, 1: It is easy to observe that Consequently, we have Now, consider that The equation ϕ ′ ðvÞ = 0 attained has five roots in ½−π, π, namely, 0, ±π and ±ðπ/2Þ. Also, ϕðvÞ = ϕð−vÞ; it is enough to consider only those roots which lie in ½0, π. Furthermore, we seen that ϕð0Þ = sec h 4 R = ϕðπÞ, and ϕðπ/2Þ = sec 4 R; therefore Hence, Also, Using the above facts along with the well-known inequality of Schwarz functions wðseeradii19Þ, we have Using (19), we obtain The last inequality is true if ð1 − ξ − tan hr sec 2 rÞð1 − r 2 Þ ð1 − tan rÞ − r sec 2 r ≥ 0 with tan r < 1 holds.
Hence, KðξÞ radius r 0 for the family S * tan h is the minumum of r 1 and r 2 , where r 1 is the smallest positive root of the equation and r 2 is such that tan r 2 < 1: The K radius, for the family S * tan h , is r 0 ≈ 0:33286: Remark 1. The result in the last Theorem is not the best one.
Considering the function f 0 described by (11) provides a sharp result. For the function f 0 , we have and ϕðrÞ = 0:

Radius Problems
To address our main results in this portion, first, we consider a few well-known families as follows.
Also, for n ∈ ℕ, Journal of Function Spaces If we put pðzÞ ≕ ðzf ′ ðzÞÞ/ð f ðzÞÞ, for f ∈ A n , then, the family P n ½L, M is reduced to S * n ½L, M and S * n ðξÞ ≕ S * n ½1 − 2ξ,−1: Let the family MðβÞ contains the functions f ∈ A n satisfying that Re ððzf ′ ðzÞÞ/ð f ðzÞÞÞ < β, for β > 1: Furthermore, let Ali et al. [21] recently studied the below families and calculated S * L,n radii for certain families. Further, they achieved the conditions on L and M such that S * n ½L, M ⊂ S * L,n : In this portion, S * tan h,n radii for the family of Janowski starlike function and some other geometrically described families are explored. To get our results, we employ the following lemmas.
Lemma 1 [22]. If p ∈ P n ðξÞ, then, for jzj = r, Lemma 2 [23]. If p ∈ P n ½L, M, then, for jzj = r, In particular, if p ∈ P n ðξÞ, then, for jzj = r, The aim of the following lemma is at finding the largest and the smallest disks centered at ða, 0Þ and (1,0), respectively, such that the domain Ω tan h ≔ q 0 ðDÞ, where q 0 ðzÞ ≔ 1 + tan hz, is contained in the smallest disk and contains the largest disk.

Lemma 3. Let
And r a = tan h1 − ja − 1j: Then, the following inclusions holds Proof. Since wðzÞ = Re iv with R ≤ jzj = r, we have with First, we consider the square of the distance from ða, 0Þ to a point on the boundary of Ω tan h , which is given by To show that jw − aj < r a is the largest disk contained in Ω tan h , it is sufficient to show that min −π≤v≤π dðvÞ = r a . But since hðvÞ = hð−vÞ, therefore, we consider the range 0 ≤ v ≤ π. Now, it can easily be obtained that h′ðvÞ = 0 has three roots 0, π, and v 0 ∈ ð0, πÞ: The root v 0 is dependent on a. The graph of hðvÞ shows that it is decreasing in ½v 0 , π and increasing in the interval ½0, v 0 . Hence, the minimum of hðvÞ is calculated on either π or 0. A computation provides Thus, we get Therefore, we can write that For the circle of the minimum radius centered at ð1, 0Þ, which contains f ðDÞ = 1 + tanh z, we find the maximum distance from ð1, 0Þ to a point on the boundary of f ðDÞ = Ω tan h and the square of this distance function is given by It is easy to verify that ϕðvÞ achieves its maximum value at π/2, which is ϕðπ/2Þ = tan 2 1: Hence, the radius of the smallest disk which contains Ω tan h is tan 1:☐ In the following examples, we apply Lemma 3 , to find the necessary and sufficient conditions for two specific functions that belong to the family S * tan h : (a) The function if and only if if and only if b j j ≤ tan h1 2 + tan h1 Proof.
(a) We know that f ðzÞ = z + d 2 z 2 ∈ S * , if and only if j maps D onto the disk Since then, from Lemma 3 , the above disk will be contained in The above two inequalities give respectively. Thus, we have ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi tan h1 1 + tan h1 (b) Logarithmic differentiation of the function yields that maps D onto the disk Journal of Function Spaces The disk above is contained in Ω tan h , in Lemma 3, whenever The above two inequalities give respectively. Thus, we have b j j ≤ min ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi tan h1 2 + tan h1 r ; tan h1 2 + tan h1 This completes the required proof.
Theorem 3. The sharp S * tan h,n radius for the family S n is given by Proof. Suppose that f ∈ S n : Consider the function h : D ⟶ ℂ described by Using logarithmic differentiation, we get Implementing Lemma 1 , we have According to Lemma 3 , if the following inequality holds, the image of jzj ≤ r under the function ðzf ′ ðzÞÞ/ðf ðzÞÞ lies on disk Ω tan h : or equivalently Thus, S * tan h,n radius of S n is the smallest positive root of in (0,1). Assume the function Then, it is clear to see that Reððf 0 ðzÞÞ/zÞ > 0 in the unit disk D: Hence, f 0 ∈ S n and Further, f 0 assures the sharpness of the results since at z = R S * tan h,n ðS n Þ, we obtain This completes the proof.
Theorem 4. The sharp S * tan h,n radius for the family CS n ðξÞ is given by Proof. Let f ∈ CS n ðξÞ and describe a function where g ∈ S * n ðξÞ: Then, h ∈ P n : According to the definition of h, we get 7 Journal of Function Spaces Utilizing Lemma 1 and Lemma 2, we conclude that Since it follows from Lemma 3 and (78) that the function f ∈ S * tan h,n if the following holds: or equivalently, the inequality holds. Thus, the S * tan h,n radius for the class CS n ðξÞ is the smallest positive root of Now, assume the functions described by Then, we get Furthermore, it is obvious that in the unit disk D. Therefore, f 0 ∈ CS n ðξÞ: The function f 0 described in (83), at z = R S * tan h,n ðCS n ðξÞÞ satisfies that Hence, the verified result is sharp.
Theorem 5. The S * tan h,n radius for the family S * n ½L, M is given by where Proof. Let f ∈ S * n ½L, M: Then, by Lemma 2 , we get where center of the disk is b = ð1 − LMr 2n Þ/ð1 − M 2 r 2n Þ, jzj = r: Applying Lemma 3 , it is easy to see that b ≥ 1 for M < 0 and we achieved After some simple calculation, we have In addition, if M = 0 for b = 1 and from (89), we get Implementing Lemma 3 with a = 1 leads to f ∈ S * tan h,n , if For 0 < M < L ≤ 1, we get b < 1: Thus, from (89) and Lemma 3 , we see that f ∈ S * tan h,n , if the following holds: This completes the proof.☐ Proof. Let pðzÞ = ðzf ′ ðzÞÞ/ð f ðzÞÞ: Since f ∈ S * n ½L, M, using Lemma 2, we get Therefore, either 1 − LM/1 − M 2 ≤ 1 or 1 − LM/1 − M 2 ≥ 1: For ð1 − LMÞ/ð1 − M 2 Þ ≤ 1, using Lemma 3 , we see that f ∈ S * tan h,n , if the following holds: which, upon simplification, reduces to the condition stated in ðaÞ: For ð1 − LMÞ/ð1 − M 2 Þ ≥ 1, again, applying Lemma 3 , we see that f ∈ S * tan h,n , if the following holds: