Fekete-Szegö and Hankel inequalities for certain class of analytic functions related to the sine function

Huo Tang; Gangadharan Murugusundaramoorthy; Shu-Hai Li; Li-Na Ma; Huo Tang; Gangadharan Murugusundaramoorthy; Shu-Hai Li; Li-Na Ma

doi:10.3934/math.2022354

AIMS Mathematics

2022, Volume 7, Issue 4: 6365-6380. doi: 10.3934/math.2022354

Previous Article Next Article

Research article Special Issues

Fekete-Szegö and Hankel inequalities for certain class of analytic functions related to the sine function

1.
School of Mathematics and Computer Sciences, Chifeng University, Chifeng 024000, Inner Mongolia, China
2.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, Tamilnadu, India

Received: 24 August 2021 Revised: 10 January 2022 Accepted: 12 January 2022 Published: 19 January 2022
MSC : 30C45, 30C50, 30C80

In this present investigation, the authors obtain Fekete-Szegö inequality for certain normalized analytic function $ f(\zeta) $ defined on the open unit disk for which

$ (f'(\zeta)^{\vartheta}\left( \frac{\zeta f'(\zeta )}{f(\zeta )}\right)^{1-\vartheta} \prec 1+\sin \zeta ; \qquad (0\leq \vartheta \leq 1) $

lies in a region starlike with respect to $ 1 $ and symmetric with respect to the real axis. As a special case of this result, the Fekete-Szegö inequality for a class of functions defined through Poisson distribution series is obtained. Further, we discuss the second Hankel inequality for functions in this new class.
- analytic function,
- starlike function,
- subordination,
- coefficient problem,
- Fekete-Szegöinequality,
- Hankel determinant,
- Poisson distribution series
Citation: Huo Tang, Gangadharan Murugusundaramoorthy, Shu-Hai Li, Li-Na Ma. Fekete-Szegö and Hankel inequalities for certain class of analytic functions related to the sine function[J]. AIMS Mathematics, 2022, 7(4): 6365-6380. doi: 10.3934/math.2022354

Related Papers:

Abstract

In this present investigation, the authors obtain Fekete-Szegö inequality for certain normalized analytic function $ f(\zeta) $ defined on the open unit disk for which

$ (f'(\zeta)^{\vartheta}\left( \frac{\zeta f'(\zeta )}{f(\zeta )}\right)^{1-\vartheta} \prec 1+\sin \zeta ; \qquad (0\leq \vartheta \leq 1) $

lies in a region starlike with respect to $ 1 $ and symmetric with respect to the real axis. As a special case of this result, the Fekete-Szegö inequality for a class of functions defined through Poisson distribution series is obtained. Further, we discuss the second Hankel inequality for functions in this new class.

References

[1]	S. S. Miller, P. T. Mocanu, Differential subordinations: Theory and applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, No. 225, CRC Press, 2000.
[2]	L. De Branges, A proof of the Bieberbach conjecture, Acta Math., 154 (1985), 137–152. http://dx.doi.org/10.1007/BF02392821 doi: 10.1007/BF02392821
[3]	W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, In: Proceeding of the International Conference on Complex Analysis at the Nankai Institute of Mathematics, International Press, 1992,157–169.
[4]	W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Annales Polonici Mathematici, Institute of Mathematics Polish Academy of Sciences, 23 (1970), 159–177.
[5]	M. S. Robertson, Certain classes of starlike functions, Mich. Math. J., 32 (1985), 135–140.
[6]	F. Ronning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Am. Math. Soc., 118 (1993), 189–196.
[7]	J. Sokół, Radius problem in the class $\mathcal{SL} ^{\ast }$, Appl. Math. Comput., 214 (2009), 569–573. http://dx.doi.org/10.1016/j.amc.2009.04.031 doi: 10.1016/j.amc.2009.04.031
[8]	M. Mohsin, S. N. Malik, Upper bound of third hankel determinant for class of analytic functions related with lemniscate of bernoulli, J. Inequal. Appl., 2013 (2013), 412. http://dx.doi.org/10.1186/1029-242X-2013-412 doi: 10.1186/1029-242X-2013-412
[9]	R. K. Raina, J. Sokół, On coefficient estimates for a certain class of starlike functions, Hacet. J. Math. Stat., 44 (2015), 1427–1433.
[10]	K. Sharma, N. K. Jain, V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat., 27 (2016), 923–939. https://doi.org/10.1007/s13370-015-0387-7 doi: 10.1007/s13370-015-0387-7
[11]	L. Shi, I. Ali, M. Arif, N. E. Cho, S. Hussain, H. Khan, A study of third hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain, Mathematics, 7 (2019), 418. https://doi.org/10.3390/math7050418 doi: 10.3390/math7050418
[12]	R. Mendiratta, S. Nagpal, V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc., 38 (2015), 365–386. https://doi.org/10.1007/s40840-014-0026-8 doi: 10.1007/s40840-014-0026-8
[13]	L. Shi, H. M. Srivastava, M. Arif, S. Hussain, H. Khan, An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function, Symmetry, 11 (2019), 598. https://doi.org/10.3390/sym11050598 doi: 10.3390/sym11050598
[14]	A. Alotaibi, M. Arif, M. A. Alghamdi, S. Hussain, Starlikeness associated with cosine hyperbolic function, Mathematics, 8 (2020), 1118. https://doi.org/10.3390/math8071118 doi: 10.3390/math8071118
[15]	N. E. Cho, V. Kumar, S. S. Kumar, V. Ravichandran, Radius problems for starlike functions associated with the sine function, Bull. Iran. Math. Soc., 45 (2019), 213–232. https://doi.org/10.1007/s41980-018-0127-5 doi: 10.1007/s41980-018-0127-5
[16]	M. Arif, M. Raza, H. Tang, S. Hussain, H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math., 17 (2019), 1615–1630. https://doi.org/10.1515/math-2019-0132 doi: 10.1515/math-2019-0132
[17]	K. Bano, M. Raza, Starlike functions associated with cosine functions, Bull. Iran. Math. Soc., 47 (2021), 1513–1532. https://doi.org/10.1007/s41980-020-00456-9 doi: 10.1007/s41980-020-00456-9
[18]	N. E. Cho, S. Kumar, V. Kumar, V. Ravichandran, H. M. Srivasatava, Starlike functions related to the Bell numbers, Symmetry, 11 (2019), 219. https://doi.org/10.3390/sym11020219 doi: 10.3390/sym11020219
[19]	L. Shi, I. Ali, M. Arif, N. E. Cho, S. Hussain, H. Khan, A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain, Mathematics, 7 (2019), 418. https://doi.org/10.3390/math7050418 doi: 10.3390/math7050418
[20]	J. Dzoik, R. K. Raina, J. Sokół, On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers, Math. Comput. Model., 57 (2013), 1203–1211. https://doi.org/10.1016/j.mcm.2012.10.023 doi: 10.1016/j.mcm.2012.10.023
[21]	S. Kanas, D. Răducanu, Some class of analytic functions related to conic domains, Math. Slovaca, 64 (2014), 1183–1196. https://doi.org/10.2478/s12175-014-0268-9 doi: 10.2478/s12175-014-0268-9
[22]	S. Kumar, V. Ravichandran, A subclass starlike functions associated with a rational function, South East Asian Bull. Math., 40 (2016), 199–212.
[23]	J. W. Noonan, D. K. Thomas, On the second Hankel determinant of areally mean $p$-valent functions, Trans. Amer. Math. Soc., 223 (1976), 337–346. https://doi.org/10.1090/S0002-9947-1976-0422607-9 doi: 10.1090/S0002-9947-1976-0422607-9
[24]	M. Fekete, G. Szegö, Eine benberkung uber ungerada schlichte funktionen, J. London Math. Soc., s1-8 (1933), 85–89. https://doi.org/10.1112/jlms/s1-8.2.85 doi: 10.1112/jlms/s1-8.2.85
[25]	W. Koepf, On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc., 101 (1987), 89–95. https://doi.org/10.2307/2046556 doi: 10.2307/2046556
[26]	W. Koepf, On the Fekete-Szegö problem for close-to-convex functions Ⅱ, Arch. Math., 49 (1987), 420–433. https://doi.org/10.1007/BF01194100 doi: 10.1007/BF01194100
[27]	D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett., 26 (2013), 103–107. https://doi.org/10.1016/j.aml.2012.04.002 doi: 10.1016/j.aml.2012.04.002
[28]	A. Janteng, S. A. Halim, M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math., 7(2006), 50.
[29]	A. Janteng, S. A. Halim, M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., 1 (2007), 619–625.
[30]	S. K. Lee, V. Ravichandran, S. Subramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl., 2013 (2013), 281. https://doi.org/10.1186/1029-242X-2013-281 doi: 10.1186/1029-242X-2013-281
[31]	G. Murugusundaramoorthy, T. Bulboac$\breve{a}$, Hankel determinants for new subclasses of analytic functions related to a Shell shaped region, Mathematics, 8 (2020), 1041. https://doi.org/10.3390/math8061041 doi: 10.3390/math8061041
[32]	S. K. Lee, K. Kanika, V. Ravichandran, Radius of starlikeness for classes of analytic functions, Bull. Malays. Math. Sci. Soc., 43 (2020), 4469–4493. https://doi.org/10.1007/s40840-020-01028-0 doi: 10.1007/s40840-020-01028-0
[33]	M. G. Khan, B. Ahmad, J. Sokol, Z. Muhammad, W. K. Mashwani, R. Chinram, et al., Coefficient problems in a class of functions with bounded turning associated with sine function, Eur. J. Pure Appl. Math., 14 (2021), 53–64. https://doi.org/10.29020/nybg.ejpam.v14i1.3902 doi: 10.29020/nybg.ejpam.v14i1.3902
[34]	H. Y. Zhang, H. Tang, L. N. Ma, Upper bound of third Hankel determinant for a class of analytic functions, Pure Appl. Math., 33 (2017), 211–220.
[35]	H. Y. Zhang, H. Tang, A study of fourth-order Hankel determinants for starlike functions connected with the sine function, J. Funct. Spaces, 2021 (2021), 9991460. https://doi.org/10.1155/2021/9991460 doi: 10.1155/2021/9991460
[36]	H. Y. Zhang, R. Srivastava, H. Tang, Third-order Hankel and Toeplitz determinants for starlike functions connected with the sine function, Mathematics, 7 (2019), 404. https://doi.org/10.3390/math7050404 doi: 10.3390/math7050404
[37]	Z. Paweł, O. Milutin, T. Nikola, Third Hankel determinant for univalent starlike functions, RACSAM, 115 (2021), 1–6. https://doi.org/10.1007/s13398-020-00977-2 doi: 10.1007/s13398-020-00977-2
[38]	H. Tang, S. Khan, S. Hussain, N. Khan, Hankel and Toeplitz determinant for a subclass of multivalent $q$-starlike functions of order $\alpha$, AIMS Math., 6 (2021), 5421–5439. https://doi.org/10.3934/math.2021320 doi: 10.3934/math.2021320
[39]	U. Grenander, G. Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley, 1958.
[40]	R. J. Libera, E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in $P$, Proc. Amer. Math. Soc., 87 (1983), 251–257. https://doi.org/10.1090/S0002-9939-1983-0681830-8 doi: 10.1090/S0002-9939-1983-0681830-8
[41]	W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, In: Proceedings of the Conference on Complex Analysis, International Press, 1992.
[42]	S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal., 2014 (2014), 984135. https://doi.org/10.1155/2014/984135 doi: 10.1155/2014/984135
[43]	G. Murugusundaramoorthy, Subclasses of starlike and convex functions involving Poisson distribution series, Afr. Mat., 28 (2017), 1357–1366. https://doi.org/10.1007/s13370-017-0520-x doi: 10.1007/s13370-017-0520-x
[44]	G. Murugusundaramoorthy, K. Vijaya, S. Porwal, Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series, Hacet. J. Math. Stat., 45 (2016), 1101–1107.
[45]	S. Porwal, M. Kumar, A unified study on starlike and convex functions associated with Poisson distribution series, Afr. Mat., 27 (2016), 1021–1027. https://doi.org/10.1007/s13370-016-0398-z doi: 10.1007/s13370-016-0398-z
[46]	H. M. Srivastava, G. Kaur, G. Singh, Estimates of the fourth Hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains, J. Nonlinear Convex Anal., 22 (2021), 511–526.
[47]	H. M. Srivastava, B. Khan, N. Khan, M. Tahir, S. Ahmad, N. Khan, Upper bound of the third Hankel determinant for a subclass of $q-$ starlike functions associated with the $q-$ exponential function, Bull. Sci. Math., 167 (2021), 102942. https://doi.org/10.1016/j.bulsci.2020.102942 doi: 10.1016/j.bulsci.2020.102942
[48]	Q. X. Hu, H. M. Srivastava, B. Ahmad, N. Khan, M. G. Khan, W. K. Mashwani, et al., A subclass of multivalent Janowski type $q-$ starlike functions and its consequences, Symmetry, 13 (2021), 1275. https://doi.org/10.3390/sym13071275 doi: 10.3390/sym13071275
[49]	H. M. Srivastava, A. K. Wanas, R. Srivastava, Applications of the $q-$ Srivastava-Attiya operator involving a certain family of bi-univalent functions associated with the Horadam polynomials, Symmetry, 13 (2021), 1230. https://doi.org/10.3390/sym13071230 doi: 10.3390/sym13071230
[50]	H. M. Srivastava, N. Khan, M. Darus, S. Khan, Q. Z. Ahmad, S. Hussain, Fekete-Szegö type problems and their applications for a subclass of $q-$ starlike functions with respect to symmetrical points, Mathematics, 8 (2020), 842. https://doi.org/10.3390/math8050842 doi: 10.3390/math8050842
[51]	H. M. Srivastava, C. Kizilateş, A parametric kind of the Fubini-type polynomials, RACSAM, 113 (2019), 3253–3267. https://doi.org/10.1007/s13398-019-00687-4 doi: 10.1007/s13398-019-00687-4
[52]	H. M. Srivastava, Operators of basic (or $q-$) calculus and fractional $q-$ calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. Sci., 44 (2020), 327–344. https://doi.org/10.1007/s40995-019-00815-0 doi: 10.1007/s40995-019-00815-0
[53]	B. Khan, Z. G. Liu, H. M. Srivastava, N. Khan, M. Daru, M. Tahir, A study of some families of multivalent $q-$ starlike functions involving higher-order $q-$ derivatives, Mathematics, 8 (2020), 1470. https://doi.org/10.3390/math8091470 doi: 10.3390/math8091470
[54]	H. M. Srivastava, Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations, J. Nonlinear Convex Anal., 22 (2021), 1501–1520.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)