Sharp Coefficient Bounds for Starlike Functions Associated with Cosine Function †

: Let S ∗ cos denote the class of normalized analytic functions f in the open unit disk D satisfying the subordination zf ′ ( z ) f ( z ) ≺ cos z . In the first result of this article, we find the sharp upper bounds for the initial coefficients a 3 , a 4 and a 5 and the sharp upper bound for module of the Hankel determinant | H 2,3 ( f ) | for the functions from the class S ∗ cos . The next section deals with the sharp upper bounds of the logarithmic coefficients γ 3 and γ 4 . Then, in addition, we found the sharp upper bound for (cid:12)(cid:12)(cid:12) H 2,2 (cid:16) F f /2 (cid:17)(cid:12)(cid:12)(cid:12) . To obtain these results we utilized the very useful and appropriate Lemma 2.4 of N.E. Cho et al., which gave a most accurate description for the first five coefficients of the functions from the Carathéodory’s functions class, and provided a technique for finding the maximum value of a three-variable function on a closed cuboid. All the maximum found values were checked by using MAPLE™ computer software, and we also found the extremal functions in each case. All of our most recent results are the best ones and give sharp versions of those recently published by Hacet.


Introduction and Preliminaries
Let A denote the class of all analytic and normalized functions f having the Maclaurin series expansion as follows where D := {z ∈ C : |z| < 1} represents the open unit disk in the complex plane C and we consider a subclass of S ⊂ A containing all the univalent functions of A.
The subclass of A defined by is called the class of starlike functions (with respect to the origin) in D and it is well-known that S * ⊂ S. Let B denote the family of functions w, which are analytic in D, such that w(0) = 0 and |w(z)| < 1 for z ∈ D. Such functions are referred to as Schwarz functions.
Let us consider two analytic functions, f and g in D. We say that f is subordinated to g, written as f (z) ≺ g(z) if there exists a Schwarz function w such that f (z) = g(w(z)) for z ∈ D. If f (z) ≺ g(z), then f (0) = g(0) and f (D) ⊆ g(D), and if g is univalent in D, then f (z) ≺ g(z) if and only if f (0) = g(0) and f (D) ⊆ g(D).
Based on the geometric properties of the image of D, by some analytic functions, the functions can be categorized from this point of view into different families.Thus, in 1992, Ma and Minda [1] introduced a generalized subclass of S * denoted by S * (ϕ), which is defined in terms of the subordination as follows S * (ϕ where ϕ satisfies the conditions ϕ(0) = 1 and Re ϕ ′ (z) > 0 in D and ϕ maps the unit disk D onto a star-shaped domain.Several subclasses of S * can be obtained by varying the function ϕ.For example, if we choose ϕ(z) we get the class S * [L, M] := S * 1 + Lz 1 + Mz , which is called the Janowski starlike functions class and was investigated in [2].Sokól and Stankiewicz [3] defined and studied the class S * L := S * √ 1 + z , where the function ϕ(z) = √ 1 + z maps D onto the image domain bounded by w 2 − 1 < 1 (the square functions are considered at the main branch, i.e., those branches with √ 1 = 1 or log 1 = 0).The class S * C := S * 1 + z 2 was studied by Sharma [4] and is related to the cardioid function.
In the case of ϕ(z) = cosh z, the class S * cosh was defined and thoroughly examined in [5], whereas the class S * e := S * (e z ) was presented and studied in [6] by Mendiratta et al.For ϕ(z) = 1 + sin z, the class S * sin was introduced and investigated in [7,8], while if ϕ(z) = 1 + tanh z, the class S * tanh was defined and studied in [9].Recently, the class S * cos , defined by S * cos := f ∈ A : was introduced and investigated in [10], and determining sharp results connected with those of [11] represents the main goal of the present paper.
The following part of this section is necessary to understand the motivation of this work.In 1916, Bieberbach presented the well-known Bieberbach conjecture, which was proved by de Branges [12] in 1985.Prior to de Branges proof of this conjecture, numerous mathematicians exerted considerable effort to prove it, leading to the establishment of coefficient bounds for some remarkable subclasses of the class S. They also developed several new inequalities related to the coefficient bounds for some subclasses of univalent functions, and those related to the Fekete-Szegő functional, that is, a 3 − λa 2  2 is one of those inequalities.Another coefficient problem closely related to the Fekete-Szegő functional is the Hankel determinant, as we will see below.
As we could see in Pommerenke's paper [13], for a function f ∈ A, the Hankel determinant H m,n ( f ) is as follows where m, n ≥ 1.We remark that where |H ( f ) could also be found in recent papers like [17][18][19].The determinant H 2,3 ( f ) has not been studied much in the literature: Babalola [20] studied, for the first time, non-sharp bounds for the determinant H 3,1 ( f ) for various subclasses of S, while in 2017, Zaprawa [21] improved the results of Babalola by using a new technique.We mention that the sharp bounds of the modulus of H 3,1 ( f ) for the class S * were recently obtained by Kowalczyk et al. [22], whereas for the class C and bounded turning functions, sharp bounds were obtained in [23,24], respectively.For sharp inequalities results for the determinant H 3,1 ( f ) for some subclasses of S * , we refer to [18,[25][26][27][28].
Very recently, Marimuthu et al. [11] have determined the coefficient bounds, the upper bounds for the second, third, and fourth-order Hankel determinants for the functions of the class S * cos , but most of the results presented in this paper are not sharp.Motivated by the above-mentioned study, in this paper, we have established the sharp results for the upper bounds of the coefficients and the logarithmic coefficients of the functions of the class S * cos .We have also developed the sharp upper bounds for the modulus of the second-and third-order Hankel determinants for the functions of this class.
The well-known Carathéodory class P is the family of holomorphic functions h in D, which satisfy the condition Re h(z) > 0, z ∈ D, with the power series expansion of the form The study of some coefficient problems in different classes of analytic functions revolves around the idea of expressing function coefficients in a given class by function coefficients that have a positive real part.Thus, the known inequalities for the class P can be used to study coefficient functionals.We require the following results on the class P for our next proofs.
Note that the extension of this lemma for the coefficients c 4 and c 5 may also be found in Lemma 2.1 of [33].
The next lemma represents the first part of the result from [1] (Remark p. 162).

Lemma 3.
If h ∈ P is given by (4), then The main novelty is found in the tools we used, that is, Lemma 2, which offers the best known estimate regarding the first five coefficients of the functions of the class P. The main strong point is the fact that all the results are the best possible, and we gave the extremal function where the equalities are obtained.Since Lemma 2 deals only with the first five coefficients of the class P, we believe that the importance of this lemma is remarkable for results involving these coefficients.Unfortunately, it is our opinion that it is hard to find similar results for higher index coefficients because of the computational difficulties, at least in the first steps of the proofs.Thus, the limitations of the below methods include the fact that the order of the Hankel determinant and the indexes of the coefficients or logarithmical coefficients cannot be higher than the present ones.Definitively, our results do not represent general investigation methods for these types of problems.

Initial Coefficients Sharp Upper Bounds
The next main results give us the sharp upper bounds for the initial coefficients of the functions from the class S * cos .
Theorem 1.Let f ∈ S * cos be given by (1).Then, and these bounds are sharp.
Proof.If f ∈ S * cos , then by the definition of subordination there exists a Schwarz function w that is analytic in D and satisfies the condition w(0) = 0 and |w(z Therefore, the function h defined by which has the property h ∈ P. Using the relations ( 6) and (7) and by equating the first four coefficients, we get (i) From the second relation of ( 8), according to the inequality (5), it follows that and this inequality is attained for the function f * from Remark 2.2 of [11].Thus, the above upper bound is sharp, which is the best possible.
(ii) To find the upper bound of |a 4 |, we see that the third equality of (8) could be written in the form and using Lemma 3 for ν = 1 2 we obtain Denoting c := |c 1 |, from Lemma 1 we have 0 ≤ c ≤ 2, hence It is easy to check that the function F attained the maximum value at c = 2 √ 3 ; then, according to the above inequality, we get To prove the sharpness of this upper bound thus, w 1 (0) = 0. To show that |w 1 (z)| < 1 in D, we remark that the function w 1 could be written in the form w 1 (z) = zφ(z), where Using the fact that every circular transform maps the circles (in the large sense, circles or lines) of Therefore, the function Setting , from the triangle inequality we obtain where Denoting by the closed unit cuboid, we will find the maximum value of F in Ω.
I. First, let us consider that (x, y, u) belongs to the interior of Ω, denoted by int Ω. Differentiating (13) with respect to u, we obtain therefore, the function F has no extremal values in int Ω. II.Next, we will discuss the existence of the maximum value for F on the open six faces of Ω, as follows.
(i) On the face x = 0, the next inequality holds (ii) On the face x = 1, we have the equality (iii) On the plane y = 0, let us denote and because it follows that the function κ 1 has no extremal values on (0, 1) × (0, 1).

(vi)
On the open face u = 1, the function F becomes and it is easy to check that the system of equations ∂κ 3 (x, y) ∂x = 0 and ∂κ 3 (x, y) ∂y = 0 has no solutions in (0, 1) × (0, 1).III.Now we will investigate the existence of the maximum of F on the edges of Ω.
(iii) Putting x = 0 in (16) we have (iv) Since ( 17) is independent of u, similar to the inequality (18), we deduce and putting x = 0 in (17), we obtain The function given by ( 15) is independent of the variables y and u, thus, (vi) Since the function defined by ( 14) is independent of the variable u, we have For all the above reasons, we conclude that max{F(x, y, u) : (x, y, u) ∈ Ω} = F(0, 1, u) = 3, and, according to (12), we finally obtain that |a 5 | ≤ 1 8 .This upper bound for |a 5 | is sharp for the function which completes our proof.
Remark 1. 1.In [11] it was proved that |a 4 | ≤ Proof.Using the relations ( 8) and ( 9) in (3), we obtain and replacing all the variables of the above relation with those of Lemma 2, it follows that using the above relation and the triangle inequality, we get where With the same notations and method as in the proof of Theorem 1, next we will find the maximum value of F in Ω.
I. If we consider that (x, y, u) ∈ int Ω, differentiating (21) with respect to u, we obtain hence, it follows that the function F has no maximum value in int Ω. II.In the sequel, we will study the existence of the maximum value of F in the interior of six faces of Ω.
(i) On the face x = 0, we have (ii) On the face x = 1, we get (iii) On the plane y = 0, the function F can be written as Since it implies that the function κ 1 has no maximum points in the face of Ω.
(vi) On u = 1 the function F takes the form Similarly, the system of equations ∂κ 3 (x, y) ∂x = 0 and ∂κ 3 (x, y) ∂y = 0 has no solutions in (0, 1) × (0, 1).III.Now we will investigate the existence of the maximum of F on the edges of Ω.
(i) From ( 24) we obtain (ii) The relation (24) at u = 1 becomes s 2 (x) := F(x, 0, 1) (iii) Putting x = 0 in (24), we have (iv) Since ( 25) is independent of u, similarly as above, we obtain while if we take x = 0 in (25) it follows (v) The relation ( 23) is independent of the variables y, u ∈ [0, 1], hence (vi) Finally, since ( 22) is independent of the variable u ∈ [0, 1], we have All the inequalities we obtained above show that max{F(x, y, u) : (x, We can see that w 3 (0) = 0, and let us write the function w 3 as w 3 (z) = zψ(z), where For the same reasons regarding the circular transforms as in the proof of the sharpness of Theorem 1 item (ii), we will show that |ψ(z)| < 1 in D by proving that Re H(z) > 0, z ∈ D, where Thus, the function cos , obtained here with those of [11], the result of Theorem 2 is a significant improvement of the previous one.Moreover, the inequality obtained in above theorem is sharp, thus the found upper bound for cos cannot be improved.Proof.Replacing in (2) the values of a 2 , a 3 , a 2 , and a 5 given by ( 8) and ( 9), we obtain According to Lemma 2, from the above relation we deduce that , then, using the triangle's inequality, the above relation leads us to where With the same notations as those in the proofs of the two previous theorems, we have to find the maximum value of F on int Ω, on the six faces, and on the twelve edges of Ω.
I. First, we consider the arbitrary interior point (x, y, u) ∈ int Ω. Differentiating (28) with respect to u, we obtain therefore, we have no maximum value of F in int Ω. II.Next, we will study the existence of the maximum value of the function F in the interior of six faces of Ω.
(ii) On the face x = 1, it takes the form (iii) On y = 0, the function F can be written as Since it follows that κ 1 has no maximum point in this face of Ω.
(i) From ( 31), we obtain (ii) Also, from (31) at u = 1 we get s 2 (x) := F(x, 0, 1) = 3x 6 + 18x 3 1 − x 2 .The solution in [0, 1] of the equation s (31) we have (iv) Since ( 32) is independent of u, according to (33), we obtain (v) If we take x = 0 in (32), we get Using the fact that the relation ( 30) is independent of the variables y and u, we deduce (vi) Since the relation ( 29) is independent of the variable u, we have Consequently, for the above reasons, we conclude that max{F(x, y, u) : (x, First, w 4 (0) = 0 and we will denote w 4 (z) = zχ(z), where Using the same property of the circular transforms as in the proof of the sharpness of Theorem 1 item (ii) and Theorem 2, we will show that |χ(z)| < 1 in D by proving that Re H(z) > 0, z ∈ D, where , is a circular transform.Computing the below values for similar reasons as in the above-mentioned proofs, these values of Consequently, the function belongs to the class S * cos .In the above power series expansion, we have a 3 = − Remark 3. The maximum values of the functions F defined by ( 21) and (28) could be also found by using the MAPLE™ computer software codes like in the Remark 1 item 2, and we obtain the same values as in both of the above two theorems.

Logarithmic Coefficients Sharp Upper Bounds
The logarithmic coefficients γ n := γ n ( f ), n ∈ N for the function f ∈ S are defined by Since the function ϕ(z) := cos z has a positive real part in D, and moreover it follows that S * cos ⊂ S * ⊂ S (see [11], p. 610).Therefore, it is possible to define the logarithmic coefficients for the functions f ∈ S * cos .In this section, we give the sharp upper bounds estimates for the third and fourth logarithmic coefficients of the functions that belong to the class S * cos .
Theorem 4. If f ∈ S * cos is given by (1), then These bounds are sharp.
And, equating the first four coefficients of (34) with those of (35), we get With the same notation as in the proof of Theorem 1, replacing in (36) and (37) the values of a 2 , a 3 , a 4 and a 5 from the relations ( 8) and ( 9), we obtain For the upper bound of |γ 3 |, using (38), we write and according to Lemma 3 for ν = 1 2 , we obtain Denoting c := |c 1 |, from Lemma 1, we have Using the result we got for the computation of the maximum F given by ( 10) we get To prove the sharpness of this bound, let us consider the function f 1 given by ( 11), were and a 4 = 2 27 √ 3. Therefore, for this function, by using the last of the relations from (36), we obtain γ 3 = √ 3 27 .
To find the upper bound of |γ 4 |, from (37) combined with Lemma 2, we can write where Using the notations and the technique from the proofs of the previous theorems, we will determine the maximum of F on Ω as follows.
I. In the points (x, y, u) ∈ int Ω, differentiating (41) with respect to u, we obtain therefore, the function F does not attain its maximum value in int Ω.

II.
In the next items, we will discuss the existence of the maximum value of F in the interior of six faces of Ω.
(vi) On u = 1, the function F will be Similarly, it is easy to check that the system of equations ∂κ 3 (x, y) ∂x = 0 and ∂κ 3 (x, y) ∂y = 0 has no solutions in (0, 1) × (0, 1).III.Now we will investigate the existence of the maximum of F on the edges of Ω.
(v) The relation ( 43) is independent with respect to the variables y and u, thus (vi) Finally, since (42) is independent of the variable u, we have The above computations lead to max{F(x, y, u) : (x, y, u) and from (40) we conclude that |γ 4 | ≤ 1 16 .
For proving the sharpness of the above inequality, we consider the function f 2 given by (19).In this case, a 1 = 1, a 2 = a 3 = a 4 = 0 and a 5 = − Using the Lemma 2, we obtain and denoting , by using the triangle's inequality, the above relation leads us to where With the notations used in the proofs of the previous three theorems, we will determine the maximum value of F in Ω.
I. For all the interior points (x, y, u) ∈ int Ω, differentiating (48) with respect to u, we obtain ∂F(x, y, u) hence, the function does not get its maximum value in int Ω. II.Next, we will study whether it is possible to obtain the maximum value of F in the interior of six faces of Ω.
(i) On the face x = 0, we have (ii) On the face x = 1, the function F takes the form (iii) On y = 0, the function F can be written as it follows that the function κ 1 has no maximum value in (0, 1) × (0, 1).
For the above reasons, it follows that max{F(x, y, u) : (x, y, u) ∈ Ω} = F(x 0 , which proves the sharpness of our estimation. Remark 5.As we already mentioned in Remark 3, the same maximum values of the functions F defined by (41) and (48) could also be found by using the MAPLE™ computer software codes as in Remark 1 item 2.
Combining with the technique for determining the extremal values of a three-variable function on a compact cuboid, we found the best upper bounds of |a 3 |, |a 4 |, |a 5 | and of the Hankel determinants |H 3,1 ( f )|, |H 2,3 ( f )|.In addition, the sharp upper bounds of the logarithmic coefficients |γ 3 | and |γ 4 |, combined with those of H 2,2 F f /2 , are presented in Section 3.

27 √ 3
belongs to the class S * cos ; hence, |a 4 | = 2 for the above function f 1 , which proves the sharpness of the second inequality of this theorem.(iii) To determine the upper bound of |a 5 |, by using the relation (9) combined with Lemma 2, we obtain
cos , for which |a 4 | = 0 and |a 5 | = In the following two theorems, we determined the sharp upper bounds for the Hankel determinants |H 3,1 | and |H 2,3 |, respectively, over the class S * cos .