Research Article | Open Access

Ali Ebadian, Nafya Hameed Mohammed, Ebrahim Analouei Adegani, Teodor Bulboacă, "New Results for Some Generalizations of Starlike and Convex Functions", *Journal of Function Spaces*, vol. 2020, Article ID 7428648, 12 pages, 2020. https://doi.org/10.1155/2020/7428648

# New Results for Some Generalizations of Starlike and Convex Functions

**Academic Editor:**Konstantin M. Dyakonov

#### Abstract

The purpose of the current paper is to investigate several various problems for the categories , and other related categories such as various new outcomes for the coefficients of , together with majorization issues, the Hankel determinant, and the logarithmic coefficients with sharp inequalities and differential subordination implications.

*Dedicated to the memory of Professor Stephan Ruscheweyh (1944-2019)*

#### 1. Introduction and Preliminaries

Let denote the open unit disk in the complex plane . Let be the category of functions analytic in that has the following representationand denoted by the subclass of all functions of which are univalent in . Then, the logarithmic coefficients of 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 [1–3] and [4], Chapter 2), and note that we will use the notation instead of .

Utilizing the principle of subordination, Ma and Minda [5] introduced the classes and , where we make here the assumptions that the function is univalent in the unit disk and satisfies , 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 and play a significant role in the Geometric Function Theory because of their geometric properties.

For instance, the categories and reduce to the categories and of the popular *Janowski starlike* and *Janowski convex functions* for with , respectively. By replacing and in these function families, we get the classes and of the *starlike functions of order* and *convex functions of order*, respectively. Especially, and are the family of starlike functions and of convex functions in the unit disk , respectively.

Further, for , we get the family defined by Sok’ol and Stankiewicz [6], including functions such that stands in the region bounded by the right-half branch of the lemniscate of Bernoulli given by . Moreover, the properties of the classes consisting of functions satisfying the condition were considered by Mendiratta et al. in [7]. Raina and Sokół [8] studied the family , where maps onto the crescent-shaped region .

In addition, for , we obtain the categories and studied by Kanas et al. [9], with the property that and lies in a domain bounded by a right branch of a hyperbola where , respectively. Also, Goel and Kumar [10] introduced the classes and for where this function maps onto a domain .

Lately, Masih and Kanas [11] introduced and studied the categories and by and investigated some outcomes regarding the behavior of the functions of these classes. The function maps the unit disk onto a domain bounded by the limacon . Further, is symmetric respecting the real axis, and is starlike with respect to . Also, and in (see for more details [11]); hence, satisfies the conditions of the category of the Ma-Minda functions (see [5]). In addition, for the functionsplay as extremal functions for some problems for the classes and , respectively.

For instance, the quantity lies in a domain bounded by the lima¸con in the category while lies in a domain bounded by a right branch of a hyperbola where in the category . 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 and and obtained some characteristic properties of these classes defined by

The function maps onto the interior of the nephroid, a 2-cusped kidney-shaped curve,

Further, for , the functions play the role of extremal functions for several problems for the categories and , 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 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 , 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.

#### 2. 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 and similar classes. For this purpose, suppose represents the set of all analytic functions in , with and for , i.e., is the set of Schwarz functions. Also, we recall the following lemmas:

Lemma 1 (see [15], Theorem 1). *Let the function . If is starlike with respect to 1, then the logarithmic coefficients of f satisfy the inequality:*

*The above inequality is sharp for any , for the function given by .*

Lemma 2 (see [16], p. 172). *Assume that ω is a Schwarz function, such that .*

*Then,*

Lemma 3 (see [15], Theorem 2). *Let the function . Then, the logarithmic coefficients of satisfy the inequalities**The bounds (12) and (13) are sharp. Also, if , , and are real values, thenwhere and is given by (see [17, 18])and the sets , are stated as given below:*

For two analytic functions and in , the function is said to be majorized to , written as , if there exists an analytic function in , with , such that (see MacGregor [14]).

Cho et al. [19] studied the majorization issue for the category of starlike functions as follows:

Lemma 4 (see [19], Theorem 2). *If with , then for all in the disk, where is the smallest positive root of the equation*

Setting and in Lemma 1, since we obtain the following two results:

Theorem 5. *If the function , then**This inequality is sharp for for the function .*

Theorem 6. *If the function , then**This inequality is sharp for for the function .*

Theorem 7. *If the function , then**These bounds are sharp for and , respectively.*

*Proof. *Applying Lemma 3 with , the first two estimates follows immediately, and the results are sharp. To prove the above inequality |*γ*_{3}|, by Lemma 3 we havewhereFirst, regarding *D*_{1} it is clear that for , and the inequality holds for . Therefore, for .

Next, if we select *D*_{2} then holds for . Also, the inequality is equivalent tothat holds for (see Figure 1) and the fact that . Further, holds for . Therefore, regarding these results, we conclude .

From the abovementioned two results, using (23) and Lemma 3, we conclude that .

Sinceit follows that the bounds are sharp for and , respectively.

Theorem 8. *If the function , then**These bounds are sharp for and , respectively.*

*Proof. *Applying Lemma 3 for , we obtain the first two bounds, and these results are sharp. To find the upper bound for , by Lemma 3, we havewhereTherefore, and from Lemma 3, we getSincewe conclude that the bounds are sharp for and , respectively.

Theorem 9. *If the function has the power expansion series given by (1), thenandwhere and . The first two bounds and the first and last bound for are sharp.*

*Proof. *If , by the concept of the subordination, there exists the function such thatwhere . From the above equality, it follows thatFor the estimate of , applying Lemma 2 for the first of the above equalities, we get . Next, using again Lemma 2, for , it follows thatNow, for , we obtain (see also [18])whereFirst, it is clear that for , and the inequality holds for .

Therefore, according to the definition of , it follows that for .

Also, regarding the definition of , we conclude that the inequality holds for . Also, the inequality is equivalent towhich holds for , where (see Figure 2). Further, holds for ; therefore, considering these results, we conclude for.

Next, regarding the definition of holds for , and the second inequality holds for . Therefore, for .

According to the definition of holds for , and a simple computation shows that the first and second inequality from the right side hold for and , respectively. Therefore, for .

Further, for the definition of , it is clear that for . On other the hand, the second inequality from the right side holds for , and the first inequality from the right side is equivalent toholds for (see Figure 3). Therefore, for .

Finally, according to the definition of , from the above explanations, it follows that for .

Hence, applying the mentioned outcomes from (37), we conclude thatThe extremal function for is , and for in the first and second inequalities is given by and , respectively. Also, the extremal function for in the first and fourth above inequalities is given by and , respectively.

For a function , we have if and only if . Setting , we have where for . Hence, we can get similarly the following theorem:

Theorem 10. *If the function has the power expansion series given by (1), thenwhere and . The extremal function for is , and for in the first and second inequalities is given by and , respectively. Also, the extremal function for in the first and fourth above inequalities is given by and , respectively.*

Setting in the proof of Theorem 9, hence, , , and , and we get similarly the following result:

Theorem 11. *If the function has the power expansion series given by (1), then**The inequalities are sharp for .*

For the above choice of , the analogue of Theorem 10 is the next one.

Theorem 12. *If the function has the power expansion series given by (1), then**The inequalities are sharp for .*

Our next result deals with a majorization problem for the functions of the class :

Theorem 13. *Let and , with . Then, for all in the disk , we have , where r_{2} is the smallest positive root of the equation*

*Proof. *The result follows from Lemma 4 using the fact that min that can be found in [11], Lemma 2.

#### 3. Second Hankel Determinant Problem

In this section, we investigate the problem of coefficients for the second Hankel determinant problem (see [20–24]) for the classes and similar classes. For this purpose, we need some parts of Theorems 1 and 2 of [25], as follows:

Lemma 14 (see [25], Theorem 1). *Let the function . If , , and satisfy the conditionsthen the second Hankel determinant satisfies*

Lemma 15 (see [25], Theorem 2). *Let the function .*(i)*If , , and satisfy the conditionsthen the second Hankel determinant satisfies*(ii)*If , , and satisfy the conditionsthen the second Hankel determinant satisfies*

By choosing and in Lemmas 14 and 15, then we obtain the following outcomes:

Theorem 16. *If the function , then the second Hankel determinant satisfies the inequality**The inequality is sharp for .*

Theorem 17. *If the function , then the second Hankel determinant satisfies**The inequality is sharp for .*

Theorem 18. *If the function , then the second Hankel determinant satisfies the inequality**The first inequality is sharp for .*

*Proof. *From Lemma 15 (i), a simple computation shows that the first and second inequalities of the assumption hold for and , respectively. Also, regarding Lemma 15 (ii), the first and second (which is equivalent to ) inequalities of the assumption hold for and , respectively; hence, we obtain the required result.

Theorem 19. *If the function *