Majorization and Coefﬁcient Problems for a General Class of Starlike Functions

: In the current paper, we study a majorization issue for a general category S ∗ ( ϑ ) of starlike functions, the region of which is often symmetric with respect to the real axis. For various special symmetric functions ϑ , corresponding consequences of the main result are also presented with some relevant connections of the outcomes rendered here with those obtained in recent research. Moreover, coefﬁcient bounds for some majorized functions are estimated.


Introduction and Preliminaries
Let U denote the unit disk {z ∈ C : |z| < 1} and H represent the class of analytic functions in U. We denote by A the subclass of H consisting of functions f (z) = z + ∞ ∑ n=2 a n z n . (1) Let Φ represent the category of all analytic functions in U that satisfy the requirements of (0) = 0 and | (z)| < 1 for z ∈ U, i.e., we consider Φ the set of Schwarz functions. Definition 1. [1,2] For two analytic functions θ and Θ in the unit disk, we state θ(z) is quasi-subordinate to Θ(z) if there is a function ν(z), analytic in U, so that θ(z)/ν(z) is analytic in U and |ν(z)| ≤ 1 (z ∈ U), where ≺ stands for the usual subordination for analytic functions in U. We denote the above quasi-subordination by θ(z) ≺ q Θ(z) (z ∈ U). (2) It is remarkable that the relation (2) can be rewritten as follows θ(z) = ν(z)Θ( (z)) (z ∈ U), For example, for the function ϑ(z) = (1 + Cz)/(1 + Dz) (−1 ≤ D < C ≤ 1), the class S * (ϑ) becomes the subclass S * [C, D] of the well-known Janowski starlike functions. By replacing C = 1 − 2γ and D = −1 where 0 ≤ γ < 1, we obtain the category S * (γ) of the starlike functions of order γ. Specifically, S * := S * (0) is the well-known category of starlike functions in U. Some special subclasses of the class S * (ϑ) play a significant act in geometric function theory because of their geometric properties. It is fairly common that a function in one of these subclasses is lying in a given region in the right half-plan and the region is often symmetric with respect to the real axis.
Taking ϑ(z) = √ 1 + z we get a category of S * L , which was reviewed by Sokół and Stankiewicz [6] and implies that f ∈ S * L if and only if Moreover, the features of the category S * e := S * (e z ) comprising functions f ∈ A, with the requirement of | log(z f (z)/ f (z))| < 1 was considered by Mendiratta et al. in [7]. In [8] researchers investigated the category S * (h), where and proved that f ∈ S * (h) if and only if z f (z)/ f (z) ∈ R, where R = {w ∈ C : |w 2 − 1| < 2|w|}. Lately, Kanas et al. [9] defined the class ST hpl (b) := S * (q b (z)) and obtained some geometric properties in this class where the function where the branch of the logarithm is considered by q b (0) = 1, maps U onto a region, which is bounded by a right branch of a hyperbola Moreover, q b (U) is symmetric about the real axis, starlike with respect to q b (0) = 1 and convex. Further q b (z) has positive real part in U and q b (0) > 0. Therefore, q b (z) satisfies the classification of Ma-Minda functions.
Recently, Goel and Kumar [10] introduced the class S * SJ and obtained some different problems in this class as follows: The modified sigmoid function maps U onto a domain ∆ SJ := {ξ ∈ C : | log(ξ/(2 − ξ))| < 1}, which is symmetric about the real axis. Also, J(z) is a convex function and so starlike function with respect to J(0) = 1. Moreover, J(z) has positive real part in U and J (0) > 0. Therefore, J(z) satisfies the classification of Ma-Minda functions.
MacGregor [4] and Altintas et al. [11] (see also [12]) studied the majorization issues for the category S * and for specific analytic functions by convex and starlike functions of complex order.
By setting Θ(z) = z, in above outcome we conclude the next well-known result: Recently, several authors have investigated majorization issues for the families of meromorphic and multivalent meromorphic or univalent and multivalent functions including various linear and nonlinear operators, which all are subordinated by the similar function ϑ(z) = (1 + Cz)/(1 + Dz) (for example, see [14][15][16][17][18][19][20]). Lately, Tang et al. [21] studied majorization problem for the subclasses of S * (ϑ), which are relevant to S * (1 + sin z) and S * (cos z), regardless of any linear or nonlinear operators. Hence, in this work, we study a majorization issue for the general category S * (ϑ) with various special consequences of the main result. Also, some suitable relations of the outcomes are presented with those reported in the earlier results. Moreover, coefficient estimates for majorized functions related to the class S * (ϑ) are obtained.

Main Results
We first state and establish a majorization feature for the general category S * (ϑ) and then some consequences of the main result are stated.
Θ(z), then |θ (z)| ≤ |Θ (z)| for all z in the disk |z| ≤ r 1 , where r 1 is the smallest positive root of the equation Differentiating the last equality with respect to z, it follows that Now, let Θ ∈ S * (ϑ), then from the subordination concept, there exists a ∈ Φ with | (z)| ≤ |z| = r so that Since Re(ϑ(z)) > 0 in U, so ϑ(z) = 0 for all z ∈ U. Now, by the minimum modulus principle we conclude min We know that ϑ is a continuous function with Re(ϑ(z)) > 0 in U and so min |z|=r |ϑ(z)| = 0. Therefore, from this point, (4) and the above relation we obtain On the other hand, applying the popular inequality for Schwarz functions, which states that Utilizing (5) and (6) in (3), we obtain Setting |ν(z)| = γ (0 ≤ γ ≤ 1), it follows that In order to determine r 1 , we must choose We know l(r, γ) ≤ 1 if and only if Clearly, the function p(r, γ) chooses its minimum value for γ = 1, that is, where Further, since p(0) = 1 > 0 and p(1) = −2 < 0, there exists r 1 , so that for all r ∈ [0, r 1 ], we have p(r) ≥ 0 where r 1 is the smallest positive root of the above equality and this completes the proof.
The following corollary concludes a majorization property for the subclass ST hpl (b) considering Lemma 2.1 in [9].
Since cos r ≤ | cos z| (|z| = r < 1), the following corollary concludes a majorization property for a subclass S * (cos z) and also we have a correction of the result which was given by Tang et al. in ([21], Theorem 2.2).
Since this inequality holds for all r in the interval 0 < r < 1, it follows that |a n | ≤ 1 + |b 2 | + · · · + |b n |. Now using Lemma 2 we have which completes the proof.

Conclusions
In the current paper, we obtain a majorization result for a general category S * (ϑ) of starlike functions. Also, we investigate coefficient bounds for majorized functions associated with the class S * (ϑ). Furthermore, we can consider some particular functions ϑ in Theorems 2 and 3 to get the corresponding majorization results.

Conflicts of Interest:
The authors declare no conflict of interest.