Radius and Differential Subordination Results for Starlikeness Associated with Limaçon Class

In this study we investigate the sharp radius of starlikeness of subclasses of Ma and Minda class for the ratio of analytic functions which are related to limaçon functions. This survey is connected also to the ﬁ rst-order di ﬀ erential subordinations. In this context, we get the condition on β for which certain di ﬀ erential subordinations associated with limaçon functions imply Ma and Minda starlike functions. Simple corollaries are provided for certain examples of our results. Finally, we present several geometries related to our study.


Introduction and Preliminaries
Let A be the class of normalized analytic functions f ðzÞ having the seres form f z ð Þ = z + 〠 ∞ n=2 a n z n z ∈ U ≔ z ∈ ℂ : z j j < 1 f g : Let P ðαÞ denotes the class of all functions pðzÞ such that Re pðzÞ > α, z ∈ U, 0 ≤ α < 1. The case P ð0Þ = P is the usual class of Carathéodory functions. Let S denotes the subclasses of A consisting of functions which are univalent in U. We say f ðzÞ ∈ A is subordinate to gðzÞ ∈ A (written as f ≺ g or f ðzÞ ≺ gðzÞ) if there exists a Schwarz function wðzÞ such that f ðzÞ = gðwðzÞÞ for all z ∈ U. The class S is one of the most vital categories of geometric function theory due to its wide applications in sciences and engineering. For example, univalent functions are extensively used in ODEs and PDEs and operators' theory. Also, they are important in image processing techniques. Among the earlier subclasses of S that had recieved tremendous attention are the classes C and S * of convex and starlike functions, respectively.
In 2020, Masih and Kanas [11] explored another novel subclass of S * ðV Þ with V ðzÞ = ð1 + szÞ 2 , 0 < s ≤ 1/ ffiffi ffi 2 p . This class was denoted by S * L and functions in it map U onto a region bounded by limaçon. Saliu et al. [12] furthered the investigation of this class and obtained the bounds of the Hankel determinants, sharp radius, and differential implications associated with it. Also, Kanaga and Ravichandran [13] examined L s ðzÞ = ð1 + szÞ 2 , 0 < s ≤ 1/ ffiffi ffi 2 p and found the smallest disc D a ðR a Þ and the largest disc D a ðr a Þ centered at ða, 0Þ such that This concept was then applied to find the radius of limaçon starlikeness for S * ðαÞ. For more findings associated with L s ðzÞ, we refer to [14][15][16].
Let P and Q be two subclasses of A defined on U. The Q radius of P denoted by R Q ðP Þ ≤ 1 is the largest radius such that f ∈ P implies the function f r , defined by f r ðzÞ = r −1 f ðrzÞ ∈ Q, for all 0 < r ≤ R Q ðP Þ. Radius result with ratio of analytic functions f , g ∈ A satisfying was studied by MacGregor [17], and the ones satisfying was examined by Ratti [18]. Presently, the radius of S * ðV Þ of various choices of V for ratio of analytic functions has attracted the interest of researchers. To this end, Ali et al. [19] considered the functions f ∈ A whose ratio ðf ðzÞÞ/ ðgðzÞÞ, ðgðzÞÞ/ðzpðzÞÞ, and pðzÞ are each subordinate to ffiffiffiffiffiffiffiffiffi ffi 1 + z p or e z , for some analytic functions gðzÞ and pðzÞ. They obtained various radii of starlikeness for these classes. In clases where these are subordinate to z + ffiffiffiffiffiffiffiffiffiffiffi ffi 1 + z 2 p or e z , Yadav et al. [20] also obtained various radii of starlikeness. Zhang et al. [21] found the radius of starlikeness connected with subclasses of S * ðV Þ for the ratio of analytic functions f ðzÞ and gðzÞ satisfying The theory of first-order differential subordinations arose from the work of Goluzin and Robertson in 1935 and 1947, respectively. Later, Miller and Mocanu [22,23] developed and generalized this idea. Using this theory, many results related with the Ma and Minda class have emerged in different directions and perspectives in the literature. For more information in this direction, we refer to the recent work of Cho et al. [24] and Kumar and Gangania [25] with the references therein.
Let PðL s Þ be the class of analytic functions pðzÞ satisfying the subordination: Motivated with these aforecited works, we initiated the following classes of analytic functions: , , Journal of Function Spaces Then, we investigate the sharp radius of starlikeness for various subclasses of Ma and Minda class. Moreover, the first-order differential subordination implications are also studied. Some special cases of our findings are given as a simple corollaries. Finally, we illustrate the geometries of some of our findings.
The following lemmas are required for our investigations.
Proof. From the property of subordination, we have pðzÞ = ð1 + swðzÞÞ 2 , where wðzÞ is a Schwarz function. Therefore, a simple computation and Schwarz lemma ( [26], p. 166) give where ϕðxÞ = ð1 − x 2 Þ/ð1 − sxÞ. It is easy to see that ϕðxÞ is continous on ½0, r. Then, by the elementary theorem of Real analysis, we have that where RðϕÞ denotes the range of ϕðxÞ. Therefore, Lemma 2 (see [23], Theorem 3.4h, p. 132). Let tðzÞ be univalent in U and let ψ and φ be analytic in a domain D containing tðUÞ with ψðwÞ ≠ 0, where w ∈ tðUÞ. Set QðzÞ = zt ′ ðzÞ · ψðtðzÞÞ, hðzÞ = φðtðzÞÞ + QðzÞ, and suppose that either If pðzÞ is analytic in U with pð0Þ = tð0Þ, pðUÞ ⊂ D and then pðzÞ ≺ tðzÞ, and tðzÞ is the best dominant in the sense that p ≺ q ⇒ t ≺ q for all q.

The Class H 1
Define the function f 1 , g 1 : U ⟶ ℂ by It is easy to see that f 1 ∈ H 1 and, thus, H 1 ≠ ∅. Also, the class contains nonunivalent functions, since varnishes at z = −1/7s. Hence, the radius of univalence for H 1 is 1/7s for 1/7 < s ≤ 1/√2. It is observed that for α = 0 in Corollary 4 (i), this radius coincides with the radius of starlikeness for the class.
for p, p 1 , p 2 ∈ PðL s Þ. Therefore, a computation yields Then, from Lemma 1, it follows that

The Class H 2
Let the function f 2 : U ⟶ ℂ be defined by Then, the functions f 2 ðzÞ and g 1 ðzÞ satisfy for some p = ð1 + szÞ 2 . Thus, f 2 ∈ H 2 and H 2 ≠ ∅. Since 3 Journal of Function Spaces the class H 2 contains some nonunivalent functions. Hence, the radius of univalence for this class is 2/ð5s + 2 + ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25s 2 − 4s + 4 p Þ. Let f ∈ H 2 . Then, there exists an analytic function gðzÞ such that for some p ∈ PðL s Þ. Since the disc jw − 1j < 1 implies Re 1/ ðwÞ > 1/2, then Re gðzÞ/f ðzÞ > 1/2. Let p 1 , Then, and a computation gives It is known from Lemma 2 of [27] that for p ∈ PðαÞ, we have Using this fact for the case α = 1/2 and Lemma 1 in (27), we arrive at It also follows from (29) that provided r ≤ R S * ðH 2 Þ = 2/ð5s + 2 + ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 25s 2 − 4s + 4 p Þ. For f 2 ðzÞ, we have at z = −R S * ðH 2 Þ. This shows that the radius is sharp. Hence, the radius of starlikeness for the class H 2 is the same as the radius of univalence for the class.

The Class H 3
Let f 3 ðzÞ = g 1 ðzÞ. Then, f 3 ∈ H 3 . It is obvious that H 3 contains nonunivalent functions, since varnishes at z = −1/5s for 1/5 < s ≤ 1/ ffiffi ffi 2 p . Hence, the radius of univalence for H 3 is 1/5s for 1/5 < s ≤ 1/ ffiffi ffi 2 p . We notice that this radius coincides with the radius of starlikess of the class for the choice of α = 0 in Corollary 4 (iii). Let Then, an obvious calculation yields In view of Lemma 1, we arrive at

Radius of Starlikeness
In this section, we obtain the radii of starlikeness of the classes H 1 , H 2 , and H 3 for different Ma and Minda starlike classes of functions.
Silverman ([28], pp. 50-51) showed that holds if and only if ja − cj ≤ jb − dj. Using this fact, we obtain the radii of Janowski starlikeness for H 1 and H 3 . Proof.
Theorem 5. The following sharp results hold for S * p : Proof.
(i) For a = 1, relation (50) implies that disc (20) lies in Δ p provided . The function f 1 ðzÞ shows that the radius of parabolic starlike functions is sharp. For this function, we have At z = −R S * p ðH 1 Þ, a calculation shows that (ii) Since the center of disc (29) is 1, then it stays inside or equivalently r ≤ R S * p ðH 2 Þ = 2/ð3ð1 + 2sÞ + ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 + 10s + 81s 2 p Þ. For f 2 ðzÞ at z = −R S * p ðH 2 Þ, a computation gives (iii) Since the center of disc (35) is a = 1, then inclusion (50) holds provided which implies r ≤ R S * p ðH 3 Þ = 1/7s, 1/7 < s ≤ 1/ ffiffi ffi 2 p . For the function f 3 ðzÞ at z = −R S * p ðH 3 Þ, we have Re The class S * e of the exponential starlike functions f ðzÞ satisfies the inequality This class was initiated by Mendiratta et al. [6] in 2015. They also proved that for e −1 < a ≤ ðe + e −1 Þ/2, Theorem 6. The following sharp results hold for S * e : Proof.
(i) For a = 1, relation (59) implies that disc (20) lies inside Δ e whenever . The function f 1 ðzÞ shows that this radius is best possible since and at z = −R S * e ðH 2 Þ, we have Journal of Function Spaces (ii) Since e −1 < 1 = a < ðe −1 + eÞ/2, inclusion (59) implies that disc (29) is in Δ e provided which at the same time means At z = −R S * e ðH 2 Þ for f 2 ðzÞ, we arrive at or ð5e − 1Þsr + 1 − e ≤ 0. Thus, the S * e of the function f ∈ H 3 is R S * e ðH 3 Þ = ðe − 1Þ/ð5e − 1Þs, ðe − 1Þ/ð5e − 1Þ < s ≤ 1/ ffiffi ffi 2 p . The sharpness is seen from the function f 3 ðzÞ at z = −R S * e ðH 3 Þ, i.e., In 2016, Sharma et al. [9] introduced the class S * c of functions that map the open unit disc onto a Cardiod domain. A function f ∈ S * c if it satisfies the subordination: The following inclusion relation was also established in [9]: Theorem 7. The following sharp results hold for the class S * c : Proof.
To prove that this radius is sharp, we consider 7 Journal of Function Spaces the function f 2 ðzÞ such that at z = −R S * c ðH 2 Þ, we arrive at which gives r ≤ R S * c ðH 3 Þ = 1/7s, 1/7 < s ≤ 1/ ffiffi ffi 2 p . For the function f 3 ðzÞ, we see that which shows that the radius is sharp In 2019, Cho et al. [4] considered and investigated the class S * sin = ff ∈ A : zf ′ðzÞ/f ðzÞ ≺ 1 + sin ðzÞg. They also proved the following: (i) It is easy to see from (77) that disc (20) stays inside the region Δ sin provided which implies that r ≤ R S * sin ðH 1 Þ = sin ð1Þ/ð6 + sin ð1ÞÞs, sin ð1Þ/ð6 + sin ð1ÞÞ < s ≤ 1/ ffiffi ffi 2 p . To show that this radius is best possible, we consider the function f 1 ðzÞ so that At z = −R S * sin ðH 1 Þ, we arrive at (ii) Since 1 − sin ð1Þ < 1 = a < 1 + sin ð1Þ, then disc (29) satisfies relation (77) provided This gives r ≤ R S * sin ðH 2 Þ = 2 sin ð1Þ/ðð4 + sin ð1ÞÞs + 1 + sin ð1Þ + ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4 + sin ð1ÞÞ 2 s 2 + ð8 q − 10 sin ð1Þ − 2 sin 2 ð1ÞÞs + ð1 + sin ð1ÞÞ 2 Þ: For the function f 2 ðzÞ, the result is sharp. Indeed, at z = −R S * sin ðH 2 Þ, we have (iii) Proceeding as in the above cases and using inclusion (77), we find that f ∈ S * sin if which holds for r ≤ R S * sin ðH 3 Þ = sin ð1Þ/ð4 + sin ð1ÞÞ s, sin ð1Þ/ð4 + sin ð1ÞÞ < s ≤ 1/ ffiffi ffi 2 p . The sharpness of the radius is assumed by the function f 3 ðzÞ Kumar and Arora [5] investigated the subclass S * sin h −1 of Ma and Minda class of functions that map U onto a petal domain. They also proved the inclusion relation where Δ sin h −1 is the image of U under the function V 3 ðzÞ. In view of the procedure of Theorem 8, (20), (29), and (35), we have the following theorem. In 2021, Bano and Mohsan [3] introduced the subclass S * cos of analytic functions characterized by the subordination They also proved that where Δ cos is the image domain of U under the mapping V 1 ðzÞ = cos ðzÞ. Proof.