Starlikeness of Certain Non-Univalent Functions

We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions $f$ with $f/g\in\mathcal{P}$ and $g/(zp)\in\mathcal{P}$ for some normalized analytic function $g$ and $p\in \mathcal{P}$. The second class is defined by replacing the condition $f/g\in\mathcal{P}$ by $|(f/g)-1|<1$ while the other class consists of normalized analytic functions $f$ with $f/(zp)\in\mathcal{P}$ for some $p\in \mathcal{P}$. We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order $\alpha$, parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.


Introduction
Let D denote the open unit disc in C. Let A := {f : f is analytic on D, f (0) = 0, and f ′ (0) = 1}. Let S := {f ∈ A : f is univalent on D}. An analytic function f is subordinate to another analytic function g, written f ≺ g, if f (z) = g(w(z)) for some analytic function w : D → D that fixes the origin. If g ∈ S, then f ≺ g if and only if the functions f and g takes the origin to the same point as well as the range of f is a subset of the range of g: f (D) ⊆ g(D). Several well known subclasses of starlike and convex functions were characterized by subordination of zf ′ (z)/f (z) or 1 + (zf ′′ (z)/f ′ (z) to some function in P. For a univalent ϕ in unit disc D with Re ϕ(z) > 0, starlike with respect to ϕ(0) = 1, symmetric about the real axis and ϕ ′ (0) > 0, Ma and Minda [22] gave a unified treatment for functions in the class S * (ϕ) = {f ∈ A : zf ′ (z)/f (z) ≺ ϕ(z)} and K(ϕ) = {f ∈ A : 1 + (zf ′′ (z)/f ′ (z)) ≺ ϕ(z)}. Convolution theorems for these two classes in a more general setting was previously studied by Shanmugham [35] with the stronger assumption that ϕ is convex. Several authors considered these classes for various choices of the function ϕ. For −1 ≤ B < A ≤ 1, let ϕ be the bilinear transform that maps the unit disc D onto the disc whose diametric end points are (1 + A)/(1 + B) and (1 − A)/(1 − B); if we impose ϕ(0) = 1, then this mapping is given by ϕ(z) = (1 + Az)/(1 + Bz). For this function ϕ, the classes S * (ϕ) and K(ϕ) reduce respectively to the classes S * [A, B] and K[A, B] of Janowski starlike and convex functions. Other well-known choices for ϕ(z) include √ 1 + z, e z , 1 + sin z, z + √ 1 + z 2 . Readers may refer to [11,39] for brief survey of these classes.
Motivated by the aforestated works, we define three classes of functions by making use of the class P as follows: and We determine radii for functions in the classes G 1 , G 2 , G 3 to belong to several subclasses of A like starlike functions of order α, starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, exponential function, cardioid, lune, nephroid, a particular rational function, modified sigmoid function and parabolic starlike functions. The disc that contains the image of unit disc D under the mapping zf ′ (z)/f (z) aids in determining the radius of various classes and we discuss this mapping in the following section.

Mapping of zf
In this section, we show that the classes G 1 , G 2 , G 3 are non-empty and contains nonunivalent functions. We also determine the disk containing the image of the disc D under the function zf ′ /f when f belong to each of the classes. We shall use the function p 0 : D → C defined by This function p 0 belongs to P and maps the unit disc D onto the right half-plane. For functions p ∈ P(α), we shall use the following inequality [34,Lemma 2]: , |z| ≤ r. 3) The function f 1 satisfies f 1 /g 1 ∈ P and g 1 /zp 0 ∈ P where p 0 ∈ P is given in (2.1). Thus the function f 1 ∈ G 1 and the class G 1 is non-empty. It is an extremal function for the class G 1 . From the coefficients of Taylor series expansion of functions f 1 given by it is evident that the functions f 1 is not univalent. Hence the class G 1 , contain non-univalent functions. As . From Theorem 3.1, it is apparent that the radii of univalence of the class G 1 is √ 10 − 3 and it coincides with its radius of starlikeness.
If the function f ∈ G 1 , then there exists an element g ∈ A and p ∈ P such that f g ∈ P and g zp ∈ P. (2.4) Let p 1 , p 2 be two functions defined on unit disc D by and p 2 (z) = g(z) zp(z) .
Thus the function f 2 ∈ G 2 and the class G 2 is non empty. It is an extremal function for the class G 2 . Note that the Taylor series expansion of f 2 (z) = z + 5z 2 + 13z 3 + 25z 4 + . . . and it is clear that the function f 2 is not univalent function, as it does not satisfy de Brange's theorem. Thus the class G 2 contain non univalent functions. Since the function f ′ 2 vanishes at z = −ζ = −1/5 and it follows that the radius of univalence of the class of functions G 2 is 1/5 by the Theorem 3.1. It also coincides with radius of starlikeness.
If the function f ∈ G 2 , then there exists a normalized analytic function g and p ∈ P such that f g ∈ P and g zp ∈ P (1/2) . (2.9) Let h 1 and h 2 be two functions, h 1 , h 2 : D −→ C defined as and h 2 (z) = g(z) zp(z) . (2.10) Using (2.9) and (2.10), we have f (z) = zh 1 (z)h 2 (z)p(z), it can be shown that As h 1 , p ∈ P and h 2 ∈ P(1/2), it follows from (2.2) and (2.11) that It would be interesting to find the boundary of the set ∪ f ∈G 2 {zf ′ (z)/f (z) : |z| ≤ r}. This would help in finding sharp radii to some of the problem where we have got only a lower bound.
2.3. The class G 3 . The function g 1 , defined by (2.3), belongs to the class G 3 and therefore the class G 3 is non-empty. It is an extremal function for the class G 3 . From the coefficients of Taylor series expansion of functions g 1 given by it is evident that the functions g 1 is not univalent. Hence the class G 3 , contain non-univalent functions. As . From Theorem 3.1, it is apparent that the radii of univalence of the class G 3 coincide with its radius of starlikeness. We now discuss the mapping of zf ′ (z)/f (z) when the function f ∈ G 3 . Let p ∈ P and p 1 be a function defined on unit disc D such that It is clear from (2.13) that f (z) = zp 1 (z)p(z). Then it follows that (2.14) Applying (2.2) in (2.14), we obtain (2.15) Using the equations (2.7), (2.12), (2.15), we investigate several radius problems associated with functions in the classes G 1 , G 2 and G 3 in the next section.
Theorem 3.1. The following sharp results hold for the class S * (α): Thus the radius is sharp.
is the root of the equation n(r) = α. Let f ∈ G 2 and let h 1 , p ∈ P and h 2 ∈ P(1/2) be the functions defined in Section 2.2. For h ∈ P, by [4, Using (3.1) and (3.2) in (2.11), it follows that Thus the radius is sharp.
Thus the radius is sharp.
The class S * L = S * ( √ 1 + z) and it represents the collection of functions in the class A whose zf ′ (z)/f (z) lies in the region bounded by the lemniscate of Bernoulli |w 2 − 1| = 1. Various studies on S * L can be seen in [38,3,28].
Using this lemma, we obtain radii results for the classes G 1 , G 2 , G 3 to be in the class S * L in the following theorem.
Theorem 3.2. The following results for the class S * L are sharp.
For the class G 1 , the centre of the disc is 1, therefore the disc obtained in (2.7) is contained in the region bounded by lemniscate, by Lemma 3.3. For the function For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by lemniscate, by Lemma 3.3. For the function For the class G 3 , the centre of the disc is 1, therefore the disc obtained in (2.15) is contained in the region bounded by lemniscate, by Lemma 3.3. For the function |w − 1|} is a parabolic region, the functions in the class S p := S * (ϕ P AR ) are known as parabolic starlike functions. These functions are studied by authors in [9,21,32]. Shanmugam and Ravichandran [36, pp.321] had proved that for 1/2 < a < 3/2, then The following theorem gives the radius of parabolic starlikeness of the three classes G 1 , G 2 and G 3 .
The following results hold for the class S p : For the class G 1 , the centre of the disc is 1, therefore the disc obtained in (2.7) is contained in the region bounded by parabola, by Lemma 3.4. For the function is the root of the equation n(r) = 1/2. For 0 < r ≤ R Sp (G 2 ), we have n(r) ≥ 1/2. That is, For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by parabola, by Lemma 3.4. This shows that R Sp (G 2 ) is at least ρ.
For the class G 3 , the centre of the disc is 1, therefore the disc obtained in (2.15) is contained in the region bounded by parabola, by Lemma 3.4. For the function In 2015, Mendiratta et al. [27] introduced the class of starlike functions associated with the exponential function as S * e = S * (e z ) and it satisfies the condition |log zf ′ (z)/f (z)| < 1. They had also proved that, for e −1 ≤ a ≤ (e + e −1 )/2, {w ∈ C : |w − a| < a − e −1 } ⊆ {w ∈ C : | log w| < 1}. (3.5) Theorem 3.4. The following results hold for the class S * e : For the class G 1 , the centre of the disc is 1, therefore the disc obtained in (2.7) is contained in the region bounded by exponential function, by Lemma 3.5. For the function f 1 defined in (2.3), at z = R S * e (G 1 ) = ρ, (ii) The function n(r) : [0, 1) −→ R defined by n(r) = (1 − 5r − 2r 2 )(1 − r 2 ) −1 + 1 is a decreasing function. Let ρ = R S * e (G 2 ) is the root of the equation n(r) = 1/e. For 0 < r ≤ R S * e (G 2 ), we have n(r) ≥ 1/e. That is, For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by the exponential function, by Lemma 3.5. This shows that R S * e (G 2 ) is at least ρ. (iii) The function defined by s(r) = 1 − (4r(1 − r 2 ) −1 ), 0 ≤ r < 1 is a decreasing function.

Proof. (i) The function defined by m(r)
For the class G 1 , the centre of the disc is 1, therefore the disc obtained in (2.7) is contained in the region bounded by the cardioid, by Lemma 3.6. For the function f 1 defined in (2.3), at z = R S * c (G 1 ) = ρ, (ii) The function n(r) : [0, 1) −→ R defined by n(r) = (1 − 5r − 2r 2 )(1 − r 2 ) −1 + 1 is a decreasing function. Let ρ = R S * c (G 2 ) is the root of the equation n(r) = 1/3. For 0 < r ≤ R S * c (G 2 ), we have n(r) ≥ 1/3. That is, For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by the cardioid, by Lemma 3.6. This shows that For the class G 3 , the centre of the disc is 1, therefore the disc obtained in (2.15) is contained in the region bounded by the cardioid, by Lemma 3.6. For the function In 2019, Cho et al. [5] considered the class of starlike functions associated with sine function where the class S * sin is defined as S * sin = {f ∈ A : zf ′ (z)/f (z) ≺ 1 + sin z := q 0 (z)} for z ∈ D. For |a − 1| ≤ sin 1, they had established the following inclusion: Here Ω s := q 0 (D) is the image of the unit disk D under the mappings q 0 (z) = 1 + sin z.
Theorem 3.6. The following results are sharp for the class S * sin .
For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by the sine function, by Lemma 3.7. For the function For the class G 3 , the centre of the disc is 1, therefore the disc obtained in (2.15) is contained in the region bounded by the sine function, by Lemma 3.7. For the function In 2015, Raina and Sokó l [29] introduced the class S * = S * (z + √ 1 + z 2 ). They showed that a function f ∈ S * if and only if zf ′ (z)/f (z) belongs to a lune shaped region L := {w ∈ C : |w 2 − 1| < 2|w|}. Gandhi Theorem 3.7. The following results hold for the class S * : For the class G 1 , the centre of the disc is 1, therefore the disc obtained in (2.7) is contained in the region bounded by the intersection of disks w : |w − 1| < √ 2 and w : |w + 1| < √ 2 , by Lemma 3.8. For the function f 1 defined in (2.3), at (ii) The function n(r) For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by the lune, by Lemma 3.8. This shows that R S * (G 2 ) is at least ρ.
For the class G 1 , the centre of the disc is 1, therefore the disc obtained in (2.7) is contained in the region bounded by the rational function, by Lemma 3.9. For the function (ii) The function n(r) is the root of the equation n(r) = 2( √ 2 − 1). The function defined by For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by the rational function, by Lemma 3.9. This shows that For the class G 3 , the centre of the disc is 1, therefore the disc obtained in (2.15) is contained in the region bounded by the rational function, by Lemma 3.9. For the function Mendiratta et al. [26] studied the subclass of starlike function associated with left half of shifted lemniscate of Bernoulli, given by (w − √ 2) 2 − 1 < 1. The class S * RL is defined as For √ 2/3 ≤ a < √ 2, they had proved the following inclusion: where . Using this result, we obtain S * RL -radii of the classes G 1 , G 2 , G 3 in the following theorem.
For the class G 1 , the centre of the disc is 1, therefore the disc obtained in (2.7) is contained in the region bounded by the reverse lemniscate, by Lemma 3.10. This shows that R S * RL (G 1 ) is at least ρ. For the function f 1 defined in (2.3), the radius is sharp.
(ii) The function defined by n(r) = (5r For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by reverse lemniscate, by Lemma 3.10. This shows that R S * RL (G 2 ) is at least ρ. The obtained radius is sharp for the function f 2 defined in (2.8).
(iii) The function defined by s(r) = 4r(1 − r 2 ) −1 + 1, 0 ≤ r < 1 is an increasing function. Let For the class G 3 , the centre of the disc is 1, therefore the disc obtained in (2.15) is contained in the region bounded by reverse lemniscate, by Lemma 3.10. This shows that R S * RL (G 3 ) is at least ρ. The obtained radius is sharp for the function g 1 defined in (2.3). The sharpness of the results can be shown using the software Wolfram Mathematica.
For the class G 1 , the centre of the disc is 1, therefore the disc obtained in (2.7) is contained in the region bounded by the nephroid, by Lemma 3.11. For the function where ∂Ω N e denotes the boundary of nephroid domain. (ii) The function n(r) : [0, 1) −→ R defined by n(r) = (1 − 5r − 2r 2 )(1 − r 2 ) −1 + 1 is a decreasing function. Let ρ = R S * Ne (G 2 ) is the root of the equation n(r) = 1/3. For 0 < r ≤ R S * Ne (G 2 ), we have n(r) ≥ 1/3. That is, For the class G 2 , the centre of the disc is 1, therefore the disc obtained in (2.12) is contained in the region bounded by the nephroid, by Lemma 3.11. For the function f 2 defined in (2.8) (iii) The function defined by s(r) = 1 − (4r(1 − r 2 ) −1 ), 0 ≤ r < 1 is a decreasing function.