New Results for Some Generalizations of Starlike and Convex Functions

The purpose of the current paper is to investigate several various problems for the categories ST L ð s Þ , S ∗ N e , and other related categories such as various new outcomes for the coe ﬃ cients of f , together with majorization issues, the Hankel determinant, and the logarithmic coe ﬃ cients with sharp inequalities and di ﬀ erential subordination implications.


Introduction and Preliminaries
Let U ≔ fz ∈ ℂ : jzj < 1g denote the open unit disk in the complex plane ℂ. Let A be the category of functions f analytic in U that has the following representation and denoted by S the subclass of all functions of A which are univalent in U. Then, the logarithmic coefficients γ n of f ∈ S are defined as the coefficients of the series expansion These coefficients play an important role for various estimates in the theory of univalent functions (see for example [1][2][3] and [4], Chapter 2), and note that we will use the notation γ n instead of γ n ð f Þ.
Utilizing the principle of subordination, Ma and Minda [5] introduced the classes S * ðφÞ and CðφÞ, where we make here the assumptions that the function φ is univalent in the unit disk U and satisfies φð0Þ = 1, with the power series expansion of the form They considered the abovementioned classes as follows: where the symbol "≺" stands for the usual subordination. Some special subclasses of the class S * ðφÞ and CðφÞ play a significant role in the Geometric Function Theory because of their geometric properties. For instance, the categories S * ðφÞ and CðφÞ reduce to the categories S * ½A, B and C½A, B of the popular Janowski starlike and Janowski convex functions for φðzÞ = ð1 + AzÞ/ ð1 + BzÞ with −1 ≤ B < A ≤ 1, respectively. By replacing A = 1 − 2αð0 ≤ α < 1Þ and B = −1 in these function families, we get the classes S * ðαÞ and CðαÞ of the starlike functions of order α and convex functions of order a, respectively. Especially, S * ≔ S * ð0Þ and C ≔ Cð0Þ are the family of starlike functions and of convex functions in the unit disk U, respectively.
Further, for φðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p , we get the family S * L defined by Sok'ol and Stankiewicz [6], including functions f such that w = zf ′ ðzÞ/f ðzÞ stands in the region bounded by the righthalf branch of the lemniscate of Bernoulli given by jw 2 − 1j < 1. Moreover, the properties of the classes S * e ≔ S * ðe z Þ consisting of functions f ∈ A satisfying the condition jlog ðzf ′ ðzÞ/f ðzÞÞj<1, z ∈ U were considered by Mendiratta et al. in [7]. Raina and Sokół [8] studied the family S * c ≔ S * ðhÞ, where hðzÞ = z + ffiffiffiffiffiffiffiffiffiffiffi ffi 1 + z 2 p maps U onto the crescentshaped region fw ∈ ℂ : jw 2 − 1j < 2jwj, Re w > 0g.
Lately, Masih and Kanas [11] introduced and studied the categories ST L ðsÞ and CV L ðsÞ by and investigated some outcomes regarding the behavior of the functions of these classes. The function L s maps the unit disk onto a domain bounded by the limacon ½ðu − 1Þ 2 + v 2 − s 4 2 = 4s 2 ½ðu − 1 + sÞ 2 + v 2 . Further, L s ðUÞ is symmetric respecting the real axis, and L s is starlike with respect to L s ð0Þ = 1. Also, L s ′ ð0Þ > 0 and Re L s ðzÞ > 0 in U (see for more details [11]); hence, L s satisfies the conditions of the category of the Ma-Minda functions (see [5]). In addition, for n = 1, 2, ⋯ the functions play as extremal functions for some problems for the classes ST L ðsÞ and CV L ðsÞ, respectively. For instance, the quantity zf ′ðzÞ/f ðzÞðor1 + zf ″ðzÞ/f ′ ðzÞÞ lies in a domain bounded by the lima¸con ½ðu − 1Þ 2 + υ 2 − s 4 2 = 4s 2 ½ðu − 1 + sÞ 2 + υ 2 in the category ST L ðsÞ ðor CV L ðsÞÞ while zf ′ ðzÞ/f ðzÞðor1 + zf ″ ðzÞ/f ′ ðzÞÞ lies in a domain bounded by a right branch of a hyperbola ρðsÞ= s jφj < ðπs/2Þð2 cos ðϕ/sÞÞ where in the category ST hpl ðsÞðor CV hpl ðsÞÞ. Therefore, it is observed that these categories have different structures and geometric properties (see for more details [9,11]).
Recently, Wani and Swaminathan [12] investigated the new Ma-Minda-type function classes S * Ne and C Ne and obtained some characteristic properties of these classes defined by The function φ Ne maps U onto the interior of the nephroid, a 2-cusped kidney-shaped curve, Further, for n = 2, 3, ⋯, the functions Journal of Function Spaces play the role of extremal functions for several problems for the categories S * Ne and C Ne , respectively. Finding the upper bound for coefficients has been one of the central topics of research in the Geometric Function Theory as it gives several properties of functions. In particular, the bound for the second coefficient gives growth and distortion theorems for the functions of the class S. In [13], Ebadian et al. studied some coefficient problems for the categories ST hpl ðsÞ, CV hpl ðsÞ, S * SG and related categories like sharp bounds for initial coefficients, logarithmic coefficients, Hankel determinants, and Fekete-Szegö problems. Also, they investigated some geometric properties as applications of differential subordinations.
According to the abovementioned issues, motivated essentially by the recent work [13], this paper is aimed at investigating some various problems for the categories ST L ðsÞ, S * Ne , and other related categories like various new outcomes for the coefficients of the power series expansions of the functions that belong to these classes, together with majorization issue, the Hankel determinant, and the logarithmic coefficients with sharp inequalities, and differential subordination implications.

Logarithmic Coefficients, Coefficient Estimates, and Majorization Issue
We obtain the estimates for the logarithmic coefficients, the first three coefficients, and majorization issue (see [14]) for the functions belonging to ST L ðsÞ, S * Ne and similar classes. For this purpose, suppose Ω represents the set of all analytic functions ω in U, with ωð0Þ = 0 and jωðzÞj < 1 for z ∈ U, i.e., Ω is the set of Schwarz functions. Also, we recall the following lemmas: Lemma 1 (see [15], Theorem 1). Let the function f ∈ S * ðφÞ. If φ is starlike with respect to 1, then the logarithmic coefficients of f satisfy the inequality: The above inequality is sharp for any n ∈ ℕ, for the function f n given by zf n ′ ðnÞ/f n ðzÞ = φðz n Þ.

Journal of Function Spaces
Cho et al. [19] studied the majorization issue for the category S * ðφÞ of starlike functions as follows: Lemma 4 (see [19], Theorem 2). If g ∈ A, f ∈ S * ðφÞ with g ðzÞ ≪ f ðzÞ, then jg′ðzÞj ≤ j f ′ðzÞj for all z in the disk | z | ≤r1, where r1 is the smallest positive root of the equation Setting φ ≕ L s and φ ≕ φ Ne in Lemma 1, since we obtain the following two results: This inequality is sharp for n = 1 for the function Φ s,1 .

Theorem 6.
If the function f ∈ S * Ne , then This inequality is sharp for n = 1 for the function Ω 2 .

Theorem 7.
If the function f ∈ CV L ðsÞ, then These bounds are sharp for f = K s,n and n = 1, 2, 3, respectively.

Theorem 8.
If the function f ∈ C Ne , then
Proof. Applying Lemma 3 for φ = φ Ne , we obtain the first two bounds, and these results are sharp. To find the upper bound for jγ 3 j, by Lemma 3, we have where Therefore, ðq q , q 2 Þ ∈ D 1 and from Lemma 3, we get Since we conclude that the bounds are sharp for f = Λ n+1 and n = 1, 2, 3, respectively.

Theorem 9.
If the function f ∈ ST L ðsÞ has the power expansion series given by (1), then and where q 1 ≕ 4s and q 2 ≕ 7s 2 /2. The first two bounds and the first and last bound for ja 4 j are sharp.
Finally, according to the definition of D 10 , from the above explanations, it follows that ðq 1 , q 2 Þ ∈ D 10 for s ∈ ½0:55177 ⋯ , 2/ ffiffiffiffiffi 13 p = 0:55470. Hence, applying the mentioned outcomes from (37), we conclude that Journal of Function Spaces The extremal function for ja 2 j is f = Φ s,1 , and for ja 3 j in the first and second inequalities is given by f = Φ s,2 and f = Φ s,1 , respectively. Also, the extremal function for ja 4 j in the first and fourth above inequalities is given by f = Φ s, 3 and f = Φ s,1 , respectively.
For a function f ∈ A, we have f ∈ CV L ðsÞ if and only if zf ′ ðzÞ ∈ ST L ðsÞ. Setting gðzÞ ≔ zf ′ ðzÞ, we have gðzÞ = z + ∑ ∞ n=2 b n z n where b n = na n for n ≥ 2. Hence, we can get similarly the following theorem: If the function f ∈ CV L ðsÞ has the power expansion series given by (1), then where q 1 ≕ 4s and q 2 ≕ 7s 2 /2. The extremal function for |a 2 | is f = K s,1 , and for |a 3 | in the first and second inequalities is given by f = K s,2 and f = K s,1 , respectively. Also, the extremal function for |a 4 | in the first and fourth above inequalities is given by f = K s,3 and f = K s,1 , respectively.
Setting φ ≕ φ Ne in the proof of Theorem 9, hence, L 1 = 1, L 2 = 0, and L 3 = −1/3, and we get similarly the following result: Theorem 11. If the function f ∈ S * Ne has the power expansion series given by (1), then The inequalities are sharp for f = Ω 2 .
For the above choice of φ, the analogue of Theorem 10 is the next one.

Theorem 12.
If the function f ∈ C Ne has the power expansion series given by (1), then The inequalities are sharp for f = Λ 2 .
Our next result deals with a majorization problem for the functions of the class ST L ðsÞ: Theorem 13. Let g ∈ A and f ∈ ST L ðsÞ, with gðzÞ ≪ f ðzÞ. Then, for all z in the disk jzj ≤ r 2 , we have jg ′ ðzÞj ≤ jf ′ ðzÞj, where r 2 is the smallest positive root of the equation Proof. The result follows from Lemma 4 using the fact that min min jzj=r jL s ðzÞj = L s ð−rÞ that can be found in [11], Lemma 2.
(i) If L 1 , L 2 , and L 3 satisfy the conditions (ii) If L 1 , L 2 , and L 3 satisfy the conditions The inequality is sharp for f = Φ s,2 .

Theorem 17.
If the function f ∈ S * Ne , then the second Hankel determinant satisfies The inequality is sharp for f = Ω 3 .

Theorem 18.
If the function f ∈ CV L ðsÞ, then the second Hankel determinant satisfies the inequality The first inequality is sharp for f = K s,2 .

Theorem 19.
If the function f ∈ C Ne , then the second Hankel determinant satisfies a 2 a 4 − a 2 The inequality is sharp for f = Λ 3 .
Proof. The result follows immediately from Lemma 15 (i).

Differential Subordinations
The principle of differential subordination has important usages in the theory of analytic functions (for details see [26]). The significant importance of the Briot-Bouquet differential subordination inspired many authors to study these types of subordinations, and recently, many generalizations and extensions of the Briot-Bouquet differential subordination have been obtained; for example, see ð51Þ Figure 5, it follows that this subordination holds