Starlikeness of Analytic Functions with Subordinate Ratios

Let $h$ be a non-vanishing analytic function in the open unit disc with $h(0)=1$. Consider the class consisting of normalized analytic functions $f$ whose ratios $f(z)/g(z)$, $g(z)/z p(z)$, and $p(z)$ are each subordinate to $h$ for some analytic functions $g$ and $p$. The radius of starlikeness is obtained for this class when $h$ is chosen to be either $h(z)=\sqrt{1+z}$ or $h(z)=e^z$. Further $\mathcal{G}$-radius is also obtained for each of these two classes when $\mathcal{G}$ is a particular widely studied subclass of starlike functions. These include $\mathcal{G}$ consisting of the Janowski starlike functions, and functions which are parabolic starlike.


Classes of Analytic Functions
Let A denote the class of normalized analytic functions f (z) = z + ∞ k=2 a k z k in the unit disc D = {z ∈ C : |z| < 1}. A prominent subclass of A is the class S * consisting of functions f ∈ A such that f (D) is a starlike domain with respect to the origin. Geometrically, this means the linear segment joining the origin to every other point w ∈ f (D) lies entirely in f (D). Every starlike function in A is necessarily univalent.
Since f ′ (0) does not vanish, every function f ∈ A is locally univalent at z = 0. Further, each function f ∈ A mirrors the identity mapping near the origin, and thus in particular, maps small circles |z| = r onto curves which bound starlike domains. If f ∈ A is also required to be univalent in D, then it is known that f maps the disc |z| < r onto a domain starlike with respect to the origin for every r ≤ r 0 := tanh(π/4) (see [4,Corollary,p. 98]). The constant r 0 cannot be improved. Denoting by S the class of univalent functions f ∈ A, the number r 0 = tanh(π/4) is commonly referred to as the radius of starlikeness for the class S.
Another informative description of the class S is its radius of convexity. Here it is known that every f ∈ S maps the disc |z| < r onto a convex domain for every r ≤ r 0 := 2 − √ 3 [4,Corollary,p. 44]. Thus the radius of convexity for S is r 0 = 2 − √ 3. To formulate a radius description for other entities besides starlikeness and convexity, consider in general two families G and M of A. The G-radius for the class M, denoted by R G (M), is the largest number R such that r −1 f (rz) ∈ G for every 0 < r ≤ R and f ∈ M. Thus, for example, an equivalent description of the radius of starlikeness for S is that the S * -radius for the class S is R S * (S) = tanh(π/4).
In this paper, we seek to determine the radius of starlikeness, and certain other Gradius, for particular subclasses G of A. Several widely-studied subclasses of A have simple geometric descriptions; these functions are often expressed as a ratio between two functions. Among the very early studies in this direction is the class of close-to-convex functions introduced by Kaplan [9], and Reade's class [22] of close-to-starlike functions. Close-to-convex functions are necessarily univalent, but not so for close-to-starlike functions. Several works, for example those in [2, 11-14, 25, 27], have advanced studies in classes of functions characterized by the ratio between functions f and g belonging to given subclasses of A.
In this paper, we examine two different subclasses of functions in A satisfying a certain subordination link of ratios. Interestingly, these classes contain non-univalent functions.
An analytic function f is subordinate to an analytic function g, written f ≺ g, if for some analytic self-map w in D with |w(z)| ≤ |z|. The function w is often referred to as a Schwarz function. Now let h be a non-vanishing analytic function in D with h(0) = 1. The classes treated in this paper consist of functions f ∈ A whose ratios f (z)/g(z), g(z)/zp(z), and p(z), are each subordinate to h for some analytic functions g and p: When p is the constant one function, then the class contains functions f ∈ A satisfying the subordination of ratios For h(z) = (1 + z)/(1 − z), and other appropriate choices of h, these functions have earlier been studied, notably by MacGregor in [11][12][13][14], and Ratti in [19,20]. Recent investigations include those in [2,25,27].
In this paper, two specific choices of the function h are made: h(z) = √ 1 + z, and h(z) = e z .
The Class T 1 . This is the class given by This class is non-empty: let f 1 , g 1 , p 1 : D → C be given by Then The function f 1 will be shown to play the role of an extremal function for the class T 1 . Since f ′ 1 vanishes at z = −2/5, the function f 1 is non-univalent, and thus, the class T 1 contains non-univalent functions. Incidentally, f 1 demonstrates the radius of univalence for T 1 is at most 2/5. In Theorem 2.1, the radius of starlikeness for T 1 is shown to be 2/5, whence T 1 has radius of univalence 2/5.
The following is a useful result in investigating the starlikeness of the class T 1 .
The Class T 2 . This class is defined by Let f 2 , g 2 , p 2 : D → C be given by Evidently, f 2 (z)/g 2 (z) ≺ e z , g 2 (z)/zp 2 (z) ≺ e z so that f 2 ∈ T 2 , and the class T 2 is nonempty. Similar to f 1 ∈ T 1 , the function f 2 plays the role of an extremal function for the class T 2 . The Taylor series expansion for f 2 is Comparing the second coefficient, it is clear that f 2 is non-univalent. Hence the class T 2 contains non-univalent functions. The derivative f ′ 2 vanishes at z = −1/3, which shows the radius of univalence for T 2 is at most 1/3. From Theorem 2.1, the radius of starlikeness is shown to be 1/3, and so the radius of univalence for T 2 is 1/3.
The function w also satisfy the sharp inequality (see [4,Corollary,p. 199 This inequality is sharp for p(z) = e z and r = |z| ≤ √ 2 − 1. It is also sharp in the remaining interval for the function p(z) = e w(z) , where w is the extremal function for which equality holds in (1.8).
In this paper, we shall adopt the commonly used notations for subclasses of A. First, for 0 ≤ α < 1, let S * (α) denote the class of starlike functions of order α consisting of functions f ∈ A satisfying the subordination The case α = 0 corresponds to the classical functions whose image domains are starlike with respect to the origin. Various other starlike subclasses of A occurring in the literature can be expressed in terms of the subordination for suitable choices of the superordinate function ϕ. When ϕ : are known as Janowski starlike. When ϕ(z) : , the subclass is denoted by S * p , and its functions are called parabolic starlike.
In Section 2 of this paper, the radius of starlikeness, Janowski starlikeness, and parabolic starlikeness are found for the classes T i , with i = 1, 2. Section 3 deals with the determination of the G-radius for the class T i with i = 1, 2, for certain other subclasses G occurring in the literature. These classes are associated with particular choices of the superordinate function ϕ in (1.11). As mentioned earlier, the G-radius for a given class M, denoted by R G (M), is the largest number R such that r −1 f (rz) ∈ G for every 0 < r ≤ R and f ∈ M. It will become apparent in the forthcoming proofs that there are common features in the methodology of finding the G-radius for each of these subclasses.
Corollary 2.3. The radius of parabolic starlikeness for T 1 and T 2 is respectively equal to its radius of starlikeness of order 1/2. Thus,

Further radius of starlikeness
In this section, we find the G-radius for the class T i with i = 1, 2, for certain other widely studied subclasses G. These are associated with particular choices of the superordinate function ϕ in (1.11).
Denote by S * exp := S * (e z ) the class associated with ϕ(z) := e z in (1.11). This class was introduced by Mendiratta et al. [16], and it consists of functions f ∈ A satisfying the condition | log(zf ′ (z)/f (z))| < 1. The following result gives the radius of exponential starlikeness for the classes T 1 and T 2 .
This proves that R S * C (T 2 ) ≤ 2/9. In 2019, Cho et al. [3] studied the class S * sin := S * (1 + sin z) consisting of functions f ∈ A satisfying the condition zf ′ (z)/f (z) ≺ 1 + sin z. We find the S * sin -radius for the classes T 1 and T 2 .
Corollary 3.4. The following are the S * -radius for each class T 1 and T 2 : Proof. It was shown by Gandhi and Ravichandran [5,Lemma 2 We complete the proof by .