In recent years, the theory of operators got the attention of many authors due to its applications in different fields of sciences and engineering. In this paper, making use of the Bernardi integral operator, we define 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.

1. Introduction

Let

be the family of holomorphic (or analytic) functions in

, and

such that

has the series representation:

(1)

Let

be the subfamily of

containing univalent functions in

. 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 |a_n| ≤ n, which was initially proven by de Branges in 1985, and as a result, they identified multiple subfamilies of the class

that are connected to various image domains. The starlike, convex, and close-to-convex functions are among those families which are defined by

(2)

Let f and g be the two analytic functions in

; then, f is subordinate to g, denoted by

(3)

if there exists a Schwarz function w(z) satisfying the conditions:

(4)

such that

(5)

Let p(z) = 1 + c₁z + c₂z² + ⋯ be an analytic and regular function in

with p(0) = 1,

, satisfying the criteria:

(6)

Then, this function is referred to as the Janowski function which is represented by

. Geometrically,

, and

is inside the domain specified by

(7)

having diameter end points:

(8)

Let

be the class of functions ϰ(z), where ϰ(0) = 0 = ϰ^′(0) − 1 are holomorphic in

and meet the following requirements:

(9)

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:

(10)

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

(11)

The following options are provided for some special choices of n and q:

(1)
For q = 2, n = 1,

(12)

is the famed Fekete-Szegő functional.

(2)
For q = 2, n = 2,

(13)

is the second Hankel determinant.

There are relatively few findings in the literature in connection with the Hankel determinant for functions belonging to the general family

. Hayman [11] established the well-known sharp inequality:

(14)

where λ is the absolute constant. Similarly for the same class

, it was obtained in [12] that

(15)

For different subclasses of the set

of univalent functions, the growth of |Δ_q,n(f)| has been estimated many times. For example, Janteng [13] investigated the sharp bounds of Δ_2,2(f) for the classes

, and

as given below:

(16)

The sharp bound of Δ_2,2(f) for the class of close-to-convex functions is unknown. On the other hand, Krishna and Reddy [14] calculated a precision estimate of Δ_2,2(f) for the Bazilevic function class.

(3)
For q = 3, n = 1,

(17)

is the third Hankel determinant.

The calculations in (17) represent that estimating |Δ_3,1(f)| is significantly more difficult than estimating the bound of |Δ_2,2(f)|. In the first paper of Babalola [15] on Δ_3,1(f), he obtained the upper bound of |Δ_3,1(f)| for the classes

, and

. Later, some more contributions have been made by different authors to calculate the bounds of |Δ_3,1(f)| for different subclasses of analytic and univalent functions. Zaprawa [16] enhanced the results of Babalola [15] and demonstrated that

(18)

He also observed that the bounds are still not sharp.

(4)
For q = 4, n = 1,

(19)

is the fourth Hankel determinant.

Since

and a₁ = 1, thus Δ_4,1(f) = a₇Λ₁ − a₆Λ₂ + a₅Λ₃ − a₄Λ₄, where

(20)

In the last few years, many articles have been published to investigate the upper bounds for the second-order Hankel determinant Δ_2,2(f), third-order Hankel determinant Δ_3,1(f), and fourth Hankel determinant Δ_4,1(f). For the functions with bounded turning, Arif et al. [17, 18] estimated the bound for the fourth- and fifth-order Hankel determinants. Khan et al. [19] also addressed this issue and derived upper bounds for the third- and fourth-order Hankel determinants for a class of functions with bounded turning that are related to sine functions. For more study about the Hankel determinant, we refer to [19–29].

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 complex-order 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 [34–40].

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

. The Bernardi integral operator

was defined by Bernardi [41], which is given by the following relation:

(21)

In the first part of the study, we extend Bernardi’s theorem to a certain class of univalent starlike functions in . Particularly, we prove that if , then . In the second part of the study, we investigate the upper bounds for the fourth-order Hankel determinant Δ_4,1(f) with respect to the function class associated with the sine function.

2. Main Results

In order to obtain our desired results, we first need the following lemmas.

Lemma 1. Let M and N be holomorphic functions in such that N maps onto many sheeted starlike regions with M(0) = N(0) = 0 and M^′(0) = N^′(0) = 1. If , then

Proof. We know that

(22)

Also, σ(z) = 1 + sinz maps |z| < r onto the disc |σ(z) − 1| < sin(1). But M^′(z)/N^′(z) takes values in the same disc, and therefore,

(23)

Choose Λ(z) so that

(24)

Then, |Λ(z)| < sin(1). Fix z₀ in . Let be the segment joining 0 and N(z₀), which lies in one sheet of the starlike image of by N. Let be the preimage of under N. Then,

(25)

That is,

(26)

This implies that

(27)

and hence,

(28)

Lemma 2 (see [12].)Let M(z) and N(z) be regular in and N(z) map onto many sheeted starlike regions:

(29)

Then,

(30)

Lemma 3. Let such that

(31)

Then, is (1 + γ)-valent starlike for γ = 1, 2, 3, ⋯, in .

Proof. The proof is analogous to the one given in [41] and hence omitted.

Lemma 4. Let and for γ = 1, 2, 3, ⋯, where is given by (31) in Lemma 3. Then, .

Proof. Let

(32)

Then,

(33)

where

(34)

By Lemma 3, is (1 + γ)-valent starlike for γ = 1, 2, 3, ⋯ in :

(35)

and since

(36)

From Lemma 1, we can get the conclusion:

(37)

Lemma 5 (see [42].)If , then there exists some x, z with |x| ≤ 1, |z| ≤ 1, such that

(38)

Lemma 6 (see [43].)Let ; then,

(39)

Lemma 7 (see [44].)Let ; then,

(40)

Now, we are in position to present our main results.

Theorem 8. If the function and is of the form (1), then

(41)

(42)

(43)

(44)

(45)

(46)

Proof. Since , according to the definition of subordination, there exists a Schwarz function w(z) with w(0) = 0 and |w(z)| < 1 such that

(47)

Now,

(48)

where β_n = (n + γ)/(1 + γ). We define a function:

(49)

It is easy to see that and

(50)

On the other hand,

(51)

When the coefficients of z, z², z³ are compared between the equations (51) and (48), then we get

(52)

(53)

(54)

(55)

Using Lemma 6, we can simply obtain

(56)

(57)

with b₁ = 1 and

(58)

If , then by comparing like powers of z, z², ⋯, zⁿ, we have

(59)

For sharpness, if we take

(60)

and thus b₂ = 1, b₃ = 0, and b₄ = −1/6, then A₂ = 1/β₂. This shows that the obtained second coefficient bound is sharp.

Again, by Lemma 6,

(61)

Let c₁ = c, with c ∈ [0, 2]; then, by Lemma 7, we can get

(62)

Now, suppose that

(63)

Then obviously,

(64)

Setting F^′(c) = 0, we can get , and hence, the maximum value of F(c) is given by

(65)

Also,

(66)

Let c₁ = c, with c ∈ [0, 2]; then, again by Lemma 7,

(67)

Assume that

(68)

Obviously, we meet the requirement:

(69)

So the function F(c) attains its maximum value at c = 0, and it is given by

(70)

Next,

(71)

Take c₁ = c, with c ∈ [0, 2]; then, according to Lemma 7, we have

(72)

Suppose that

(73)

Then obviously,

(74)

We see that F^′′(0) < 0, and we get the maximum value at c = 0:

(75)

Finally,

(76)

Again, taking c₁ = c, with c ∈ [0, 2], and using the result of Lemma 7, we can obtain

(77)

Let

(78)

Then obviously, F^′(c) ≥ 0. As a result, the function F(c) attains its maximum value at c = 2. Hence,

(79)

Theorem 9. If the function and is of the form (1), then we have

(80)

Proof. From (52), we can write

(81)

Using Lemma 5, we get

(82)

We suppose that |x| = t ∈ [0, 1], and c₁ = c ∈ [0, 2]. Also, if we apply the triangle inequality to the above equation, then we get

(83)

Assume that

(84)

Obviously, we can write

(85)

F(c, t) is increasing on [0, 1]. Therefore, at t = 1, the function F(c, t) will obtain its maximum value:

(86)

Let us take

(87)

It is clear that G(c) is decreasing on [0, 2]. So at c = 0, the function G(c) will obtain its maximum value:

(88)

This complete the proof.

Theorem 10. If the function and is of the form (1), then we have

(89)

Proof. From (52), we can write

(90)

From Lemma 5, we can deduce that

(91)

We suppose that |x| = t ∈ [0, 1], and c₁ = c ∈ [0, 2]. Once again, if we apply the triangle inequality to the above equation, then we get

(92)

Suppose that

(93)

Then, we get

(94)

The above expression shows that F(c, t) is a decreasing function about t on the closed interval [0, 1]. This implies that F(c, t) will attain its maximum value at t = 0, which is

(95)

Now define

(96)

Since G^′′(c) < 0, the function G(c) has maximum value at c = 0. That is,

(97)

and this completes the proof.

Theorem 11. If the function and is of the form (1), then we have

(98)

Proof. Again from (52), we can write

(99)

Using the result of Lemma 5, we can obtain

(100)

Also, by Lemma 7, we have

(101)

where

(102)

Clearly H(c) is an increasing function about c on the closed interval [0, 2]. This means that H(c) will attain its maximum value at c = 2, which is H(c) ≤ 3. Thus,

(103)

Theorem 12. If the function and is of the form (1), then we have

(104)

Proof. From (52) and (53), we have

(105)

Using the result of Lemma 7, we can write

(106)

where

(107)

Obviously,

(108)

For M^′(c) = 0, we can get c = 1.71468508801 and consequently M^′′(1.71468508801) = −2.8693. As M^′′(0) < 0, the maximum value at c = 0 is

(109)

Also,

(110)

where H(c) attains its maximum value at c = 2, so

(111)

Using the results of (109) and (111) in (106), we can get

(112)

Theorem 13. If the function and is of the form (1), then we have

(113)

Proof. From (52) and (53), we have

(114)

Letting |x| = t ∈ [0, 1] and c₁ = c ∈ [0, 2] and using the results of Lemmas 6 and 7, we get

(115)

Suppose that

(116)

where

(117)

We see that H^′(c) ≥ 0 and the maximum value of H(c) can be attained at c = 2, which is H(2) ≤ 7/9. Also,

(118)

If we set M^′(c) = 0, then we get c = 1.46416723. Consequently, M^′′(1.46416723) = −0.86. As M^′′(0) < 0, the maximum value at c = 0 is given by M(0) ≤ 0.50. Hence,

(119)

Theorem 14. If the function and is of the form (1), then we have

(120)

Proof. From (52) and (53), we have

(121)

Now, using the results of Lemmas 6 and 7, we obtain

(122)

where

(123)

It is clear that H(c) is an increasing function, so it attains its maximum value at c = 2, which is

(124)

Also, for all c ∈ [0, 2], we have

(125)

When we set M^′(c) = 0, then we get c = 1.464167. Obviously,

(126)

and it attains its maximum value at c = 1.464167, which is given by

(127)

Hence,

(128)

which completes the proof of Theorem 14.

Theorem 15. If the function and is of the form (1), then we have

(129)

Proof. We know that

(130)

which further implies that

(131)

Using the triangle inequality, we can write

(132)

By substituting the results of (41), (80), (89), (98), (104), (113), (120), and (128) in (132), we can get the desired result in (129).

3. Conclusion

In the present investigation, first, we have extended the well-known Bernardi theorem to a specific class

of univalent starlike functions in the open unit disk

. We have proven that if g is a starlike univalent function in the unit disk

and

, then

, where

(133)

Additionally, we have estimated the upper bounds of the fourth-order Hankel determinant for the functions class associated with the sine function systematically.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors equally contributed to this manuscript and approved the final version.

Open Research

Data Availability

No data were used to support this study.

References

1 Cho N. E., Kumar S., Kumar V., Ravichandran V., and Srivastava H. M., Starlike functions related to the Bell numbers, Symmetry. (2019) 11, no. 2, https://doi.org/10.3390/sym11020219, 2-s2.0-85061864882.
10.3390/sym11020219
Web of Science® Google Scholar
2 Dziok J., Raina R. K., and Sokól J., On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers, Mathematical and Computer Modelling. (2013) 57, no. 5-6, 1203–1211, https://doi.org/10.1016/j.mcm.2012.10.023, 2-s2.0-84873083909.
10.1016/j.mcm.2012.10.023
Google Scholar
3 Kumar S. and Ravichandran V., A subclass of starlike functions associated with a rational function, Southeast Asian Bulletin of Mathematics. (2016) 40, 199–212.
Google Scholar
4 Mendiratta R., Nagpal S., and Ravichandran V., On a subclass of strongly starlike functions associated with exponential function, Bulletin of the Malaysian Mathematical Sciences Society. (2015) 38, no. 1, 365–386, https://doi.org/10.1007/s40840-014-0026-8, 2-s2.0-84923234028.
10.1007/s40840-014-0026-8
Web of Science® Google Scholar
5 Kanas S. and Raducanu D., Some class of analytic functions related to conic domains, Mathematica Slovaca. (2014) 64, no. 5, 1183–1196, https://doi.org/10.2478/s12175-014-0268-9, 2-s2.0-84919934166.
10.2478/s12175-014-0268-9
Web of Science® Google Scholar
6 Sharma K., Jain N. K., and Ravichandran V., Starlike functions associated with a cardioid, Afrika Matematika. (2016) 27, no. 5-6, 923–939, https://doi.org/10.1007/s13370-015-0387-7, 2-s2.0-84982283758.
10.1007/s13370-015-0387-7
Google Scholar
7 Raina R. K. and Sokól J., Some properties related to a certain class of starlike functions, Comptes Rendus Mathematique. (2015) 353, no. 11, 973–978, https://doi.org/10.1016/j.crma.2015.09.011, 2-s2.0-84947749783.
10.1016/j.crma.2015.09.011
Web of Science® Google Scholar
8 Sokól J. and Stankiewicz J., Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Naukowe Politechniki Rzeszowskiej. (1996) 19, 101–105.
Google Scholar
9 Cho N. E., Kumar V., Kumar S., and Ravichandran V., Radius problems for starlike functions associated with the sine function, Iranian Mathematical Society. (2019) 45, no. 1, 213–232, https://doi.org/10.1007/s41980-018-0127-5, 2-s2.0-85061665420.
10.1007/s41980-018-0127-5
Web of Science® Google Scholar
10 Noonan W. and Thomas D. K., On the second Hankel determinant of areally mean 𝑝-valent functions, Transactions of the American Mathematical Society. (1976) 223, 337–346, https://doi.org/10.1090/S0002-9947-1976-0422607-9, 2-s2.0-84968469748.
10.1090/S0002-9947-1976-0422607-9
Web of Science® Google Scholar
11 Hayman W. K., On the second Hankel determinant of mean univalent functions, Proceedings of the London Mathematical Society. (1968) s3-18, no. 1, 77–94, https://doi.org/10.1112/plms/s3-18.1.77, 2-s2.0-84894421256.
10.1112/plms/s3-18.1.77
Web of Science® Google Scholar
12 Libera R. J., Some classes of regular univalent functions, Proceedings of the American Mathematical Society. (1965) 16, no. 4, 755–758, https://doi.org/10.1090/S0002-9939-1965-0178131-2, 2-s2.0-84968519107.
10.1090/S0002-9939-1965-0178131-2
Web of Science® Google Scholar
13 Janteng A., Halim S. A., and Darus M., Coefficient inequality for a function whose derivative has a positive real part, Journal of Inequalities in Pure and Applied Mathematics. (2006) 7, no. 2, 1–5.
Google Scholar
14 Krishna D. V. and Reddy T. R., Second Hankel determinant for the class of Bazilevic functions, Studia Universitatis Babes-Bolyai Mathematica. (2015) 60, no. 3, 413–420.
Google Scholar
15 Babalola K. O., On H₃(1) Hankel determinant for some classes of univalent functions, Inequality Theory and Applications. (2007) 6, 1–7.
Google Scholar
16 Zaprawa P., Third Hankel determinants for subclasses of univalent functions, Mediterranean Journal of Mathematics. (2017) 14, no. 1, article 19, https://doi.org/10.1007/s00009-016-0829-y, 2-s2.0-85007595777.
10.1007/s00009-016-0829-y
Web of Science® Google Scholar
17 Arif M., Rani R., Raza M., and Zaprawa P., Fourth Hankel determinant for a family of functions with bounded turning, Bulletin of the Korean Mathematical Society. (2018) 55, no. 6, 1703–1711, https://doi.org/10.4134/BKMS.b170994, 2-s2.0-85062280421.
10.4134/BKMS.b170994
Web of Science® Google Scholar
18 Arif M., Ullah I., Raza M., and Zaprawa P., Investigation of the fifth Hankel determinant for a family of functions with bounded turnings, Mathematica Slovaca. (2020) 70, no. 2, 319–328, https://doi.org/10.1515/ms-2017-0354.
10.1515/ms-2017-0354
Web of Science® Google Scholar
19 Khan M. G., Ahmad B., Sokol J., Muhammad Z., Mashwani W. K., Chinram R., and Petchkaew P., Coefficient problems in a class of functions with bounded turning associated with sine function, European Journal of Pure and Applied Mathematics. (2021) 14, no. 1, 53–64, https://doi.org/10.29020/nybg.ejpam.v14i1.3902.
10.29020/nybg.ejpam.v14i1.3902
Web of Science® Google Scholar
20 Islam S., Khan M. G., Ahmad B., Arif M., and Chinram R., Q-extension of starlike functions subordinated with a trigonometric sine function, Mathematics. (2020) 8, no. 10, https://doi.org/10.3390/math8101676.
10.3390/math8101676
Web of Science® Google Scholar
21 Lee S. K., Khatter K., and Ravichandran V., Radius of starlikeness for classes of analytic functions, Bulletin of the Malaysian Mathematical Sciences Society. (2020) 43, no. 6, 4469–4493, https://doi.org/10.1007/s40840-020-01028-0.
10.1007/s40840-020-01028-0
Web of Science® Google Scholar
22 Lee S. K., Ravichandran V., and Supramaniam S., Bounds for the second Hankel determinant of certain univalent functions, Journal of Inequalities and Applications. (2013) 2013, 281, https://doi.org/10.1186/1029-242X-2013-281, 2-s2.0-84894487423.
10.1186/1029-242X-2013-281
Google Scholar
23 Mahmood S., Srivastava H. M., Khan N., Ahmad Q. Z., Khan B., and Ali I., Upper bound of the third Hankel determinant for a subclass of q-starlike functions, Symmetry. (2019) 11, no. 3, https://doi.org/10.3390/sym11030347, 2-s2.0-85066307501.
10.3390/sym11030347
Web of Science® Google Scholar
24 Murugusundaramoorthy G. and Bulboacă T., Hankel determinants for new subclasses of analytic functions related to a shell shaped region, Mathematics. (2020) 8, no. 6, https://doi.org/10.3390/math8061041.
10.3390/math8061041
Web of Science® Google Scholar
25 Raza M. and Malik S. N., Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, Journal of Inequalities and Applications. (2013) 2013, 8, 2378, https://doi.org/10.1186/1029-242X-2013-412, 2-s2.0-84897599158.
10.1186/1029-242X-2013-412
Google Scholar
26 Shi L., Ali I., Arif M., Cho N. E., Hussain S., and Khan H., A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain, Mathematics. (2019) 7, no. 5, https://doi.org/10.3390/math7050418, 2-s2.0-85073744955.
10.3390/math7050418
Web of Science® Google Scholar
27 Shi L., Srivastava H. M., Arif M., Hussain S., and Khan H., An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function, Symmetry. (2019) 11, no. 5, https://doi.org/10.3390/sym11050598, 2-s2.0-85066340330.
10.3390/sym11050598
Web of Science® Google Scholar
28 Zaprawa P., Obradović M., and Tuneski N., Third Hankel determinant for univalent starlike functions, Fsicas y Naturales. Serie A. Matemáticas. (2021) 115, no. 2, 1–6.
Google Scholar
29 Zhang H. Y., Tang H., and Ma L. N., Upper bound of third Hankel determinant for a class of analytic functions, Pure and Applied Mathematics. (2017) 33, no. 2, 211–220.
Google Scholar
30 Srivastava H. M., Aouf M. K., and Mostafa A. O., Some properties of analytic functions associated with fractional q-calculus operators, Miskolc Mathematical Notes. (2019) 20, 1245–1260, https://doi.org/10.18514/MMN.2019.3046.
10.18514/MMN.2019.3046
Web of Science® Google Scholar
31 Mahmood S., Raza N., AbuJarad E. S. A., Srivastava G., Srivastava H. M., and Malik S. N., Geometric properties of certain classes of analytic functions associated with a q-integral operator, Symmetry. (2019) 11, https://doi.org/10.3390/sym11050719, 2-s2.0-85066330560.
10.3390/sym11050719
Web of Science® Google Scholar
32 Arif M., Srivastava H. M., and Umar S., Some applications of a q-analogue of the Ruscheweyh type operator for multivalent functions, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. (2019) 113, no. 2, 1211–1221, https://doi.org/10.1007/s13398-018-0539-3, 2-s2.0-85064173442.
10.1007/s13398-018-0539-3
Web of Science® Google Scholar
33 Srivastava H. M., Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iranian Journal of Science and Technology, Transaction A: Science. (2020) 44, 327–344, https://doi.org/10.1007/s40995-019-00815-0.
10.1007/s40995-019-00815-0
Web of Science® Google Scholar
34 Attiya A. A., Ali E. E., Hassan T. S., and Albalahi A. M., On some relationships of certain uniformly analytic functions associated with Mittag-Leffler function, Journal of Function Spaces. (2021) 2021, 7, 6739237, https://doi.org/10.1155/2021/6739237.
10.1155/2021/6739237
Web of Science® Google Scholar
35 Tang H., Arif M., Ullah K., Khan N., Haq M., and Khan B., Starlikeness associated with tangent hyperbolic function, Journal of Function Spaces. (2022) 2022, 14, 8379847, https://doi.org/10.1155/2022/8379847.
10.1155/2022/8379847
Web of Science® Google Scholar
36 Khan B., Liu Z. G., Shaba T. G., Araci S., Khan N., and Khan M. G., Applications of q-derivative operator to the subclass of bi-univalent functions involving q-Chebyshev polynomials, Journal of Mathematics. (2022) 2022, 7, 8162182, https://doi.org/10.1155/2022/8162182.
10.1155/2022/8162182
Web of Science® Google Scholar
37 Yassen M. F., Attiya A. A., and Agarwal P., Subordination and superordination properties for certain family of analytic functions associated with Mittag–Leffler function, Symmetry. (2020) 12, no. 10, https://doi.org/10.3390/sym12101724.
10.3390/sym12101724
Web of Science® Google Scholar
38 Khan B., Liu Z. G., Srivastava M. M., Khan N., and Tahir M., Applications of higher-order derivatives to subclasses of multivalent q-starlike functions, Maejo International Journal of Science and Technology. (2021) 15, no. 1, 61–72, https://doi.org/10.1186/s13662-021-03611-6.
10.1186/s13662-021-03611-6
Web of Science® Google Scholar
39 Rehman M. S., Ahmad Q. Z., Khan B., Tahir M., and Khan N., Generalisation of certain subclasses of analytic and univalent functions, Maejo International Journal of Science and Technology. (2019) 13, no. 1, 1–9.
CAS Web of Science® Google Scholar
40 Ahmad Q. Z., Khan N., Raza M., Tahir M., and Khan B., Certain q-difference operators and their applications to the subclass of meromorphic q-starlike functions, Univerzitet u Nišu. (2019) 33, 3385–3397.
Google Scholar
41 Bernardi S. D., Convex and starlike univalent functions, Transactions of the American Mathematical Society. (1969) 135, 429–446, https://doi.org/10.1090/S0002-9947-1969-0232920-2, 2-s2.0-84968484535.
10.1090/S0002-9947-1969-0232920-2
Web of Science® Google Scholar
42 Libera R. J. and Zlotkiewicz E. J., Coefficient bounds for the inverse of a function with derivative in , Proceedings of the American Mathematical Society. (1983) 87, no. 2, 251–257, https://doi.org/10.1090/S0002-9939-1983-0681830-8, 2-s2.0-84966212576.
10.1090/S0002-9939-1983-0681830-8
Web of Science® Google Scholar
43 Ravichandran V. and Verma S., Borne pour le cinquieme coefficient des fonctions etoilees, Comptes Rendus Mathematique. (2015) 353, no. 6, 505–510, https://doi.org/10.1016/j.crma.2015.03.003, 2-s2.0-84929606732.
10.1016/j.crma.2015.03.003
Google Scholar
44 Pommerenke C., Univalent functions, Studia Mathematica Mathematische Lehrbucher, 1975, Vandenhoeck and Ruprecht.
Google Scholar

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Bernardi Integral Operator and Its Application to the Fourth Hankel Determinant

Abstract

1. Introduction

2. Main Results

3. Conclusion

Conflicts of Interest

Authors’ Contributions

Open Research

Data Availability

References

References

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Bernardi Integral Operator and Its Application to the Fourth Hankel Determinant

Abstract

1. Introduction

2. Main Results

3. Conclusion

Conflicts of Interest

Authors’ Contributions

Open Research

Data Availability

References

References

Related

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