Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 6687805 | https://doi.org/10.1155/2021/6687805

Gaganpreet Kaur, Gurmeet Singh, Muhammad Arif, Ronnason Chinram, Javed Iqbal, "A Study of Third and Fourth Hankel Determinant Problem for a Particular Class of Bounded Turning Functions", Mathematical Problems in Engineering, vol. 2021, Article ID 6687805, 8 pages, 2021. https://doi.org/10.1155/2021/6687805

A Study of Third and Fourth Hankel Determinant Problem for a Particular Class of Bounded Turning Functions

Academic Editor: Erhan Set
Received29 Nov 2020
Revised15 Mar 2021
Accepted03 Apr 2021
Published19 Apr 2021

Abstract

In this present paper, a new generalized class from the family of function with bounded turning was introduced by using - derivative operator. Our aim for this class is to find out the upper bound of third- and fourth-order Hankel determinant. Moreover, the upper bounds for two-fold and three-fold symmetric functions for this class are also obtained.

1. Introduction and Motivation

In order to better explain the terminology included in our key observations, some of the essential relevant literatures on geometric function theory need to be provided and discussed here. We start with symbol which represents the class of holomorphic functions in the region of open unit disc and satisfy the relationship for . That is; if , then it has the following Taylor series form:

Also, let represent all univalent functions in . Next, we are going to define the most useful class of geometric function theory known as the class of Caratheodory functions and is defines as; a holomorphic function belongs to if it satisfies along with the series expansion:

Using the Caratheodory functions family, we consider the following basics subclasses of as

The investigation of -calculus ( stands for quantum) fascinated and inspired many scholars due its use in various areas of the quantitative sciences. Jackson [1, 2] was among the key contributors of all the scientists who introduced and developed the -calculus theory. Just like -calculus was used in other mathematical sciences, the formulations of this idea are commonly used to examine the existence of various structures of function theory. Though the link between certain geometric nature of the analytic function and the -derivative operator was established by the authors in [3], but, for the usage of -calculus in function theory, a solid and comprehensive foundation is given in [4] by Srivastava. After this development, many researchers introduced and studied some useful operators in -analog with the applications of convolution concepts. For example; Kanas and Răducanu [5] established the -differential operator and then examined the behavior of this operator in function theory. This operator was generalized further for multivalent analytic functions by Arif et al. [6]. Analogous to -differential operator, Arif et al. and Khan et al. contributed the integral operators for analytic and multivalent functions in [7, 8], respectively. Similarly, in the article [9], the authors developed and analyzed the operator in -analog for meromorphic functions. Also, see the survey type article [10] on quantum calculus and their applications. With the use of these operators, many researchers were contributed some good articles in this direction in the field of geometric function theory, see [11, 12]. The substitution of by in -calculus has given us -calculus which is the extension of -calculus. Chakrabarti and Jagannathan [13] considered -integer.

For and , the -derivative operator of the function is defined as

From the use (1) and (4), the following formulae are easily obtained:and for a constant ,

Here, we also note that when we take and .

Further by using (1) and (4), we havewith

Let us define the class , which consists all the functions , satisfying

The Hankel determinant with and for a function of the series form (1) was given by Pommerenke [14, 15], as

In particular, the following determinants are known as the first-, second-, and third-order Hankel determinants, respectively.

There are comparatively few observations in literature in relation to the Hankel determinant for the function which belongs to the general family . For the function , the best established sharp inequality is , where is the absolute constant, which is due to Hayman [16]. Further for the same class , it was obtained in [17] that

In a given family of functions, the problem of calculating the bounds, probably sharp, of Hankel determinants attracted the minds of several mathematicians. For example, the sharp bound of , for the subfamilies , , and (family of bounded turning functions) of the set , was calculated by Janteng et al. [18, 19]. These estimates are

For the following two families, of star-like functions of order and for of strongly star-like functions of order , the authors computed in [20, 21] that is bounded by and , respectively. The exact bound for the family of Ma-Minda star-like functions was measured in [22], see also [23]. For more work on , see references [2428].

It is quite clear from the formulae given in (12) that the calculation of is far more challenging compared with finding the bound of . Babalola was the first mathematician who investigated the bounds of third=order Hankel determinant for the families of , , and in an article [29] published in 2010. Using the same approach, later, several authors [3034] published their articles regarding for certain subfamilies of analytic and univalent functions. After this study, Zaprawa [35] improved the findings of Babalola in 2017 by applying a new methodology. He obtained the following bounds:

He argued that such limits are indeed not the best. Later in 2018, Kwon et al. [36] strengthened the Zaprawa’s result for and showed that and this bound was further improved by Zaprawa et al. [37] in 2021. They got for . Recently in 2018, Kowalczyk et al. [38] and Lecko et al. [39] succeeded in finding the sharp bounds of for the families and , respectively, where indicate the star-like functions family of order . These results are given as

The estimation of fourth Hankel determinant for the bounded turning functions has been obtained by Arif et al. [40] and they proved the following bounds for :

After that, Kaur and Singh [41] proved fourth Hankel determinant for bounded turning function of order . For more contributions, see [4246]. Recently, Srivastava et al. [47] consider a family of normalized analytic functions with bounded turnings in the open unit disk which are connected with the cardioid domains and they obtained the estimates of fourth Hankel determinant.

2. A Set of Results

In order to investigate , we need the following results.

Lemma 1. (see [48]). If , having the form (2), then

Theorem 1. (see [40]). Let and for real ,

3. Bounds of Third Hankel Determinant

The third Hankel determinant is a polynomial of four variables, as:

In order to solve , we need to know the correspondence between and .

Thus,

By simplifying, we yield

Now, using the above coefficients in (22), we obtain

Rearranging the above terms,

Using triangular inequality and the results (18) and (19) of lemma of Section 2, we get

Remark 1. As we know by (7), if and , then the above result (28) coincides with Zaprawa [35].

4. Bounds of Fourth Hankel Determinant

Firstly, is the fourth Hankel determinant of the form (11) with six coefficients which can be written in the form,where , and are third-order determinants given as

Theorem 2. If , thenwhere

Proof. By using definition (10), we have . Thus, .Therefore,Substituting (36) in (30) and (31) and in (32) yields thatNow, rewrite the above equations as follows:Using the triangular inequality with the inequalities (18) and (19) of lemma on the above equations, we obtainNow, using the values (39)–(41) and (28) along with the inequality in (29), we get our desired result.

5. Bounds of for Two-Fold and Three-Fold Symmetric Functions

-Fold symmetric function consists all those functions which satisfy the following condition:where . The set of univalent functions with -fold symmetry (that is ) has the expansion of the form,

Furthermore, univalent function belongs to if and only ifwhereasand function if .

Now, if , then ; hence, . In the same way, if , then ; clearly, we can see that two-fold symmetric functions are odd. So, .

Theorem 3. If is three-fold symmetric bounded turning function, that is, , then

Proof. Firstly, consider that , then is a function of the form such that . Since and with the use of (43) for , it follows thatIdentifying the coefficients, we getWe already know , then function of the form such that . Thus,Therefore,Now, by rearranging (48) and (50), we getSince , this impliesWith the use of Theorem 1 in Section 2, where for , we get our theorem proved.

Theorem 4. If is two-fold symmetric bounded turning function, that is, , then

Proof. By the definition of two-fold symmetric function, the Hankel determinant can be written asSince , then is a function such that ; then, the expansion of (43) and (45) for yields thatNow, with the help of lemma in Section 2, we get our desired result as asserted by the statement.

Remark 2. For results of (29), (45) and (43) will coincide with results derived in [40].

Data Availability

The required data are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. D. O. Jackson, T. Fukuda, O. Dunn, and E. Majors, “On q-definite integrals,” Quarterly Journal of Pure and Applied Mathematics, vol. 14, 1910. View at: Google Scholar
  2. F. H. Jackson, “q-difference equations,” American Journal of Mathematics, vol. 32, no. 4, pp. 305–314, 1910. View at: Publisher Site | Google Scholar
  3. M. E. H. Ismail, E. Merkes, and D. Styer, “A generalization of starlike functions. Complex Variables, Theory and Application,” An International Journal, vol. 14, no. 1–4, pp. 77–84, 1990. View at: Publisher Site | Google Scholar
  4. H. M. Srivastava and S. Owa, “Univalent functions, fractional calculus, and associated generalized hypergeometric functions,” Fundamental Theory of Fractional Calculus, vol. 39, pp. 329–354, 1989. View at: Google Scholar
  5. S. Kanas and D. Răducanu, “Some class of analytic functions related to conic domains,” Mathematica Slovaca, vol. 64, no. 5, pp. 1183–1196, 2014. View at: Publisher Site | Google Scholar
  6. M. Arif, H. M. Srivastava, and S. Umar, “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, vol. 113, no. 2, pp. 1211–1221, 2019b. View at: Publisher Site | Google Scholar
  7. M. Arif, M. U. Haq, and J. L. Liu, “A subfamily of univalent functions associated with q-analog of noor integral operator,” Journal of Function Spaces, vol. 2018, Article ID 3818915, 5 pages, 2018a. View at: Publisher Site | Google Scholar
  8. Q. Khan, M. Arif, M. Raza, G. Srivastava, H. Tang, and S. u. Rehman, “Some applications of a new integral operator in q-analog for multivalent functions,” Mathematics, vol. 7, no. 12, p. 1178, 2019. View at: Publisher Site | Google Scholar
  9. M. Arif and B. Ahmad, “New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator,” Mathematica Slovaca, vol. 68, no. 5, pp. 1049–1056, 2018. View at: Publisher Site | Google Scholar
  10. H. M. Srivastava, “Operators of basic(or q-) calculas and fractional q-calculas and their applications in geometric function theory of complex analysis,” Iranian Journal of Science and Technology, Transactions A: Science, vol. 44, pp. 1–18, 2020. View at: Google Scholar
  11. M. Arif, O. Barkub, H. Srivastava, S. Abdullah, and S. Khan, “Some janowski type harmonic q-starlike functions associated with symmetrical points,” Mathematics, vol. 8, no. 4, p. 629, 2020. View at: Publisher Site | Google Scholar
  12. L. Shi, Q. Khan, G. Srivastava, J.-L. Liu, and M. Arif, “A study of multivalent q-starlike functions connected with circular domain,” Mathematics, vol. 7, no. 8, p. 670, 2019b. View at: Publisher Site | Google Scholar
  13. R. Chakrabarti and R. Jagannathan, “A (p, q)-oscillator realization of two-parameter quantum algebras,” Journal of Physics A: Mathematical and General, vol. 24, no. 13, pp. L711–L718, 1991. View at: Publisher Site | Google Scholar
  14. C. Pommerenke, “On the coefficients and hankel determinants of univalent functions,” Journal of the London Mathematical Society, vol. s1-41, no. 1, pp. 111–122, 1966. View at: Publisher Site | Google Scholar
  15. C. Pommerenke, “On the hankel determinants of univalent functions,” Mathematika, vol. 14, no. 1, pp. 108–112, 1967. View at: Publisher Site | Google Scholar
  16. W. K. Hayman, “On the second hankel determinant of mean univalent functions,” Proceedings of the London Mathematical Society, vol. s3-18, no. 1, pp. 77–94, 1968. View at: Publisher Site | Google Scholar
  17. M. Obradovic and N. Tuneski, “Hankel determinants of second and third order for the class s of univalent functions,” 1912, http://arxiv.org/abs/1912.06439. View at: Google Scholar
  18. A. Janteng, S. A. Halim, and M. Darus, “Coefficient inequality for a function whose derivative has a positive real part,” Journal of Inequalties in Pureand Applied Mathematics, vol. 7, no. 2, pp. 1–5, 2006. View at: Google Scholar
  19. A. Janteng, S. A. Halim, and M. Darus, “Hankel determinant for starlike and convex functions,” International Journal of Mathematical Analysis, vol. 1, no. 13, pp. 619–625, 2007. View at: Google Scholar
  20. N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko, and Y. J. Sim, “Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha,” Journal of Mathematical Inequalities, vol. 11, no. 2, pp. 429–439, 2017. View at: Publisher Site | Google Scholar
  21. N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko, and Y. J. Sim, “The bounds of some determinants for starlike functions of order alpha,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 41, no. 1, pp. 523–535, 2018. View at: Publisher Site | Google Scholar
  22. S. K. Lee, V. Ravichandran, and S. Supramaniam, “Bounds for the second hankel determinant of certain univalent functions,” Journal of Inequalities and Applications, vol. 2013, no. 1, p. 281, 2013. View at: Publisher Site | Google Scholar
  23. A. Ebadian, T. Bulboacă, N. E. Cho, and E. A. Adegani, “Coefficient bounds and differential subordinations for analytic functions associated with starlike functions,” RACSAM, vol. 114, 2020. View at: Publisher Site | Google Scholar
  24. Ş. Altınkaya and S. Yalçın, “Upper bound of second hankel determinant for bi-bazilevic functions,” Mediterranean Journal of Mathematics, vol. 13, no. 6, pp. 4081–4090, 2016. View at: Google Scholar
  25. D. Bansal, “Upper bound of second hankel determinant for a new class of analytic functions,” Applied Mathematics Letters, vol. 26, no. 1, pp. 103–107, 2013. View at: Publisher Site | Google Scholar
  26. M. Çağlar, E. Deniz, and H. M. Srivastava, “Second hankel determinant for certain subclasses of bi-univalent functions,” Turkish Journal of Mathematics, vol. 41, no. 3, pp. 694–706, 2017. View at: Google Scholar
  27. S. Kanas, E. A. Adegani, and A. Zireh, “An unified approach to second hankel determinant of bi-subordinate functions,” Mediterranean Journal of Mathematics, vol. 14, no. 6, pp. 1–12, 2017. View at: Publisher Site | Google Scholar
  28. M.-S. Liu, J.-F. Xu, and M. Yang, “Upper bound of second hankel determinant for certain subclasses of analytic functions,” Abstract and Applied Analysis, vol. 2014, Article ID 603180, 10 pages, 2014. View at: Publisher Site | Google Scholar
  29. K. O. Babalola, “On h_3 (1) hankel determinant for some classes of univalent functions,” Inequality Theory and Applications, vol. 6, pp. 1–7, 2012. View at: Google Scholar
  30. S. Altinkaya and S. Yalçin, “Third hankel determinant for bazilevic functions,” Advances in Math, vol. 5, pp. 91–96, 2016. View at: Google Scholar
  31. D. Bansal, S. Maharana, and J. K. Prajapat, “Third order hankel determinant for certain univalent functions,” Journal of the Korean Mathematical Society, vol. 52, no. 6, pp. 1139–1148, 2015. View at: Publisher Site | Google Scholar
  32. D. V. Krishna, B. Venkateswarlu, and T. RamReddy, “Third hankel determinant for bounded turning functions of order alpha,” Journal of the Nigerian Mathematical Society, vol. 34, no. 2, pp. 121–127, 2015. View at: Google Scholar
  33. M. Raza and S. N. Malik, “Upper bound of the third hankel determinant for a class of analytic functions related with lemniscate of Bernoulli,” Journal of Inequalities and Applications, vol. 2013, no. 1, pp. 1–8, 2013. View at: Publisher Site | Google Scholar
  34. G. Shanmugam, B. A. Stephen, and K. O. Babalola, “Third hankel determinant for -starlike functions,” Gulf Journal of Mathematics, vol. 2, no. 2, pp. 107–113, 2014. View at: Google Scholar
  35. P. Zaprawa, “Third hankel determinants for subclasses of univalent functions,” Mediterranean Journal of Mathematics, vol. 14, no. 1, p. 19, 2017. View at: Publisher Site | Google Scholar
  36. O. S. Kwon, A. Lecko, and Y. J. Sim, “The bound of the hankel determinant of the third kind for starlike functions,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 42, no. 2, pp. 767–780, 2019. View at: Publisher Site | Google Scholar
  37. P. Zaprawa, M. Obradović, and N. Tuneski, “Third hankel determinant for univalent starlike functions. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales,” Serie A. Matemáticas, vol. 115, no. 2, pp. 1–6, 2021. View at: Publisher Site | Google Scholar
  38. B. Kowalczyk, A. Lecko, M. Lecko, and Y. J. Sim, “The sharp bound of the third hankel determinant for some classes of analytic functions,” Bulletin of the Korean Mathematical Society, vol. 55, no. 6, pp. 1859–1868, 2018. View at: Google Scholar
  39. A. Lecko, Y. J. Sim, and B. Śmiarowska, “The sharp bound of the hankel determinant of the third kind for starlike functions of order 1/2,” Complex Analysis and Operator Theory, vol. 13, no. 5, pp. 2231–2238, 2019. View at: Publisher Site | Google Scholar
  40. M. Arif, L. Rani, M. Raza, and P. Zaprawa, “Fourth hankel determinant for the family of functions with bounded turning,” Bulletin of the Korean Mathematical Society, vol. 55, no. 6, pp. 1703–1711, 2018b. View at: Google Scholar
  41. G. Kaur and G. Singh, “4\{Th} hankel determinant for ${\Alpha}$ bounded turning function,” Advances in Mathematics: Scientific Journal, vol. 9, no. 12, pp. 10563–10567, 2020. View at: Publisher Site | Google Scholar
  42. M. Arif, M. Raza, H. Tang, S. Hussain, and H. Khan, “Hankel determinant of order three for familiar subsets of analytic functions related with sine function,” Open Mathematics, vol. 17, no. 1, pp. 1615–1630, 2019a. View at: Publisher Site | Google Scholar
  43. M. Shafiq, H. M. Srivastava, N. Khan, Q. Z. Ahmad, M. Darus, and S. Kiran, “An upper bound of the third hankel determinant for a subclass of -starlike functions associated with -fibonacci numbers,” Symmetry, vol. 12, no. 6, 2020. View at: Publisher Site | Google Scholar
  44. L. Shi, I. Ali, M. Arif, N. E. Cho, S. Hussain, and H. Khan, “A study of third hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain,” Mathematics, vol. 7, no. 5, p. 418, 2019a. View at: Publisher Site | Google Scholar
  45. H. M. Srivastava, Ş. Altınkaya, and S. Yalçın, “Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator,” Filomat, vol. 32, no. 2, pp. 503–516, 2018. View at: Publisher Site | Google Scholar
  46. H. M. Srivastava, B. Khan, N. Khan, and Q. Z. Ahmad, “Coeffcient inequalities for -starlike functions associated with the janowski functions,” Hokkaido Mathematical Journal, vol. 48, pp. 407–425, 2019. View at: Publisher Site | Google Scholar
  47. H. M. Srivastava, G. Kaur, and G. Singh, “Estimates of the fourth hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains,” Journal of Nonlinear and Convex Analysis, vol. 22, no. 3, pp. 511–526, 2021. View at: Google Scholar
  48. C. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, Germany, 1975.

Copyright © 2021 Gaganpreet Kaur et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views441
Downloads386
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.