Bernardi Integral Operator and Its Application to the Fourth Hankel Determinant

In recent years, the theory of operators got the attention of many authors due to its applications in di ﬀ erent ﬁ elds of sciences and engineering. In this paper, making use of the Bernardi integral operator, we de ﬁ ne a new class of starlike functions associated with the sine functions. For our new function class, extended Bernardi ’ s theorem is studied, and the upper bounds for the fourth Hankel determinant are determined.


Introduction
Let H be the family of holomorphic (or analytic) functions in D = fz ∈ ℂ : jzj < 1g, and A n ⊂ H such that f ∈ A n has the series representation: Let S be the subfamily of A 1 ≡ A containing univalent functions in D. Despite the fact that function theory was first proposed in 1851, it only became a viable research topic in 1916. Many academics attempted to prove or refute the conjecture ja n j ≤ n, which was initially proven by de Branges in 1985, and as a result, they identified multiple subfamilies of the class S that are connected to various image domains. The starlike, convex, and close-to-convex functions are among those families which are defined by Let f and g be the two analytic functions in D; then, f is subordinate to g, denoted by if there exists a Schwarz function wðzÞ satisfying the conditions: f z ð Þ = g w z ð Þ ð Þ, z ∈ D ð Þ: ð5Þ Let pðzÞ = 1 + c 1 z + c 2 z 2 + ⋯ be an analytic and regular function in D with pð0Þ = 1,RpðzÞ > 0, satisfying the criteria: Then, this function is referred to as the Janowski function which is represented by P ðA, BÞ. Geometrically, pðzÞ ∈ P ðA, BÞ ⟺ pð0Þ = 1, and pðU d Þ is inside the domain specified by having diameter end points: Let S * ðA, BÞ be the class of functions ϰðzÞ, where ϰð0Þ = 0 = ϰ ′ ð0Þ − 1 are holomorphic in U d and meet the following requirements: Distinct subclasses of analytic functions associated with various image domains have been introduced by many scholars. For example, Cho et al. [1] and Dziok et al. [2] discussed various properties of starlike functions related to Bell numbers and a shell-like curve connected with Fibonacci numbers, respectively. Similarly, Kumar and Ravichandran [3] and Mendiratta et al. [4] investigated subclasses of starlike functions associated with rational and exponential functions, respectively. Kanas and Raducanu [5] and Sharma et al. [6] explored some subclasses of analytic functions related to conic and cardioid domains, respectively. Raina and Sokól [7] investigated some important properties related to a certain class of starlike functions. Sokól and Stankiewicz [8] discussed radius problems of some subclasses of strongly starlike functions. Recently, Cho et al. [9] explored a family of starlike functions related to the sine function, which is defined as follows: The qth Hankel determinant for q ≥ 1 and n ≥ 1 of the functions f is introduced by Noonan and Thomas [10], which is given by The following options are provided for some special choices of n and q: (1) For q = 2, n = 1, is the famed Fekete-Szegő functional.
There are relatively few findings in the literature in connection with the Hankel determinant for functions belonging to the general family S. Hayman [11] established the well-known sharp inequality: where λ is the absolute constant. Similarly for the same class S, it was obtained in [12] that For different subclasses of the set S of univalent functions, the growth of jΔ q,n ðf Þj has been estimated many times. For example, Janteng [13] investigated the sharp bounds of Δ 2,2 ð f Þ for the classes S * , C, and R as given below: The sharp bound of Δ 2,2 ð f Þ for the class of close-toconvex functions is unknown. On the other hand, Krishna and Reddy [14] calculated a precision estimate of Δ 2,2 ð f Þ for the Bazilevic function class.

2
Journal of Function Spaces is the third Hankel determinant. The calculations in (17) represent that estimating jΔ 3,1 ð f Þj is significantly more difficult than estimating the bound of jΔ 2,2 ð f Þj. In the first paper of Babalola [15] on Δ 3,1 ð f Þ, he obtained the upper bound of jΔ 3,1 ðf Þj for the classes S * , C, and K . Later, some more contributions have been made by different authors to calculate the bounds of jΔ 3,1 ð f Þj for different subclasses of analytic and univalent functions. Zaprawa [16] enhanced the results of Babalola [15] and demonstrated that He also observed that the bounds are still not sharp.
In geometric function theory (GFT), especially in the category of univalent functions, integral and differential operators are extremely helpful and important. Convolution of certain analytic functions has been used to introduce certain differential and integral operators. This approach is developed to facilitate further exploration of geometric features of analytic and univalent functions. Libera and Bernardi were the ones who investigated the classes of starlike, convex, and close-to-convex functions by introducing the idea of integral operators. Recently, some researchers have shown a keen interest in this field and developed various features of the integral and differential operators. Srivastava et al. [30] investigated a new family of complexorder analytic functions by using the fractional q-calculus operator. Mahmood et al. [31] looked at a group of analytic functions that were defined using q-integral operators. Using the q-analogue of the Ruscheweyh-type operator, Arif et al. [32] constructed a family of multivalent functions. Srivastava [33] presented a review on basic (or q-) calculus operators, fractional q-calculus operators, and their applications in GFT and complex analysis. This review article has been proven very helpful to investigate some new subclasses from different viewpoints and perspectives [34][35][36][37][38][39][40].
Inspired from the above recent developments, in this study, we investigate the inclusion of the Bernardi integral operator in the class of starlike function associated with sine function in D. The Bernardi integral operator JðzÞ: A ⟶ A was defined by Bernardi [41], which is given by the following relation: In the first part of the study, we extend Bernardi's theorem to a certain class S * s of univalent starlike functions in D. Particularly, we prove that if g ∈ S * s , then JðzÞ ∈ S * s . In the second part of the study, we investigate the upper bounds for the fourth-order Hankel determinant Δ 4,1 ð f Þ with respect to the function class S * s associated with the sine function.

Main Results
In order to obtain our desired results, we first need the following lemmas. Proof. We know that Also, σðzÞ = 1 + sin z maps jzj < r onto the disc jσðzÞ − 1j < sin ð1Þ. But M ′ ðzÞ/N ′ ðzÞ takes values in the same disc, and therefore, 3 Journal of Function Spaces Choose ΛðzÞ so that Then, jΛðzÞj < sin ð1Þ: Fix z 0 in U d . Let L be the segment joining 0 and Nðz 0 Þ, which lies in one sheet of the starlike image of U d by N: Let L −1 be the preimage of L under N: Then, That is, This implies that and hence, Lemma 2 (see [12]). Let MðzÞ and NðzÞ be regular in D and NðzÞ map D onto many sheeted starlike regions: Then, Proof. The proof is analogous to the one given in [41] and hence omitted.

Theorem 8.
If the function f ðzÞ ∈ S * s and is of the form (1), then a 6 j j ≤ 67 120β 6 , ð45Þ a 7 j j ≤ 5587 10800β 7 : Proof. Since JðzÞ ∈ S * s , according to the definition of subordination, there exists a Schwarz function wðzÞ with wð0Þ = 0 and jwðzÞj < 1 such that Now, where β n = ðn + γÞ/ð1 + γÞ. We define a function: It is easy to see that pðzÞ ∈ P and On the other hand, When the coefficients of z, z 2 , z 3 are compared between the equations (51) and (48), then we get Using Lemma 6, we can simply obtain with b 1 = 1 and If JðzÞ = ∑ ∞ n=1 A n z n , then by comparing like powers of z, z 2 , ⋯, z n , we have For sharpness, if we take and thus b 2 = 1, b 3 = 0, and b 4 = −1/6, then A 2 = 1/β 2 . This shows that the obtained second coefficient bound is sharp.
Again, by Lemma 6, Let c 1 = c, with c ∈ ½0, 2; then, by Lemma 7, we can get Now, suppose that Then obviously, Setting F ′ ðcÞ = 0, we can get c = 2 ffiffi ffi 3 p /2, and hence, the maximum value of FðcÞ is given by Also, Let c 1 = c, with c ∈ ½0, 2; then, again by Lemma 7, Obviously, we meet the requirement: So the function FðcÞ attains its maximum value at c = 0, and it is given by Next, Take c 1 = c, with c ∈ ½0, 2; then, according to Lemma Then obviously, We see that F′′ð0Þ < 0, and we get the maximum value at c = 0: Finally, Again, taking c 1 = c, with c ∈ ½0, 2, and using the result of Then obviously, F ′ðcÞ ≥ 0. As a result, the function FðcÞ attains its maximum value at c = 2. Hence, Theorem 9.
If the function f ðzÞ ∈ S * s and is of the form (1), then we have Proof. From (52), we can write Using Lemma 5, we get We suppose that jxj = t ∈ ½0, 1, and c 1 = c ∈ ½0, 2. Also, if we apply the triangle inequality to the above equation, then we get

Journal of Function Spaces
Assume that Obviously, we can write Fðc, tÞ is increasing on ½0, 1. Therefore, at t = 1, the function Fðc, tÞ will obtain its maximum value: Let us take It is clear that GðcÞ is decreasing on ½0, 2. So at c = 0, the function GðcÞ will obtain its maximum value: This complete the proof.

Theorem 10.
If the function f ðzÞ ∈ S * s and is of the form (1), then we have Proof. From (52), we can write From Lemma 5, we can deduce that We suppose that jxj = t ∈ ½0, 1, and c 1 = c ∈ ½0, 2. Once again, if we apply the triangle inequality to the above equation, then we get Suppose that Then, we get The above expression shows that Fðc, tÞ is a decreasing function about t on the closed interval ½0, 1. This implies that Fðc, tÞ will attain its maximum value at t = 0, which is Now define Since G ′ ′ðcÞ < 0, the function GðcÞ has maximum value at c = 0. That is, and this completes the proof.

Theorem 11.
If the function f ðzÞ ∈ S * s and is of the form (1), then we have Proof. Again from (52), we can write Journal of Function Spaces Using the result of Lemma 5, we can obtain Also, by Lemma 7, we have where Clearly HðcÞ is an increasing function about c on the closed interval ½0, 2. This means that HðcÞ will attain its maximum value at c = 2, which is HðcÞ ≤ 3. Thus, For M ′ ðcÞ = 0, we can get c = 1:71468508801 and consequently M ′ ′ð1:71468508801Þ = −2:8693: As M ′ ′ð0Þ < 0, the maximum value at c = 0 is Also, where HðcÞ attains its maximum value at c = 2, so Using the results of (109) and (111) in (106), we can get Letting jxj = t ∈ ½0, 1 and c 1 = c ∈ ½0, 2 and using the results of Lemmas 6 and 7, we get We see that H ′ ðcÞ ≥ 0 and the maximum value of HðcÞ can be attained at c = 2, which is Hð2Þ ≤ 7/9: Also, Now, using the results of Lemmas 6 and 7, we obtain