Starlike functions associated with $ \tanh z $ and Bernardi integral operator

Pratima Rai; Asena Çetinkaya; Sushil Kumar

doi:10.3934/mfc.2022032

Article Contents

2023, Volume 6, Issue 3: 573-585. Doi: 10.3934/mfc.2022032

This issue Previous Article Fuzzy approximation based on $ \tau- \mathfrak{K} $ fuzzy open (closed) sets Next Article Linear combinations of two Bernstein polynomials

Starlike functions associated with $ \tanh z $ and Bernardi integral operator

1.
Department of Mathematics, University of Delhi, Delhi, India, 110007
2.
Department of Mathematics and Computer Science, Istanbul Kültür University, İstanbul, Turkey
3.
Bharati Vidyapeeth's College of Engineering, Delhi-110063, India, 110063

^*Corresponding author: Sushil Kumar
^*Corresponding author: Sushil Kumar

Dedicated to Prof. Vijay Gupta on the occasion of his 60th birthday

Received: June 2022

Revised: September 2022

Early access: September 2022

Published: August 2023

The first and the corresponding authors are supported by the Institute of Eminence, University of Delhi, Delhi, India–110007 for providing financial support for this research under grant number /IoE/2021/12/FRP.

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Abstract

We determine the necessary and sufficient convolution conditions for the starlike functions on the open unit disk and related to some geometric aspects of the function $ \tanh z $. We also determine sharp bounds on second and third order Hermitian-Toeplitz determinants for such functions. Further, we compute estimates on some initial coefficients and the Hankel determinants of third and fourth order. In addition, using the concept of Briot-Bouquet type differential subordination, we establish a subordination inclusion involving Bernardi integral operator.

Keywords:

Starlike functions,
hyperbolic tangent function,
convolution conditions,
Hermitian–Toeplitz and Hankel determinants,
subordination inclusions,
Bernardi integral operator.

Mathematics Subject Classification: Primary: 30C45; Secondary: 30C50.

Citation:

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Figure 1. Image of unit disk $ \mathbb{D} $ under the function $ 1+\tanh z $

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References

[1]	T. Acar, Asymptotic formulas for generalized Szász-Mirakyan operators, Appl. Math. Comput., 263 (2015), 233-239. doi: 10.1016/j.amc.2015.04.060.
[2]	T. Acar, A. Aral and I. Raşa, Iterated Boolean sums of Bernstein type operators, Numer. Funct. Anal. Optim., 41 (2020), 1515-1527. doi: 10.1080/01630563.2020.1777160.
[3]	R. M. Ali, V. Ravichandran and N. Seenivasagan, On Bernardi's integral operator and the Briot-Bouquet differential subordination, J. Math. Anal. Appl., 324 (2006), 663-668. doi: 10.1016/j.jmaa.2005.12.033.
[4]	K. O. Babalola, On ${H}_3(1)$ Hankel determinant for some classes of univalent functions, Ineq. Thoery and Appl., 6 (2010), 1-7.
[5]	S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135 (1969), 429-446. doi: 10.1090/S0002-9947-1969-0232920-2.
[6]	N. Bohra, S. Kumar and V. Ravichandran, Some special differential subordinations, Hacet. J. Math. Stat., 48 (2019), 1017-1034. doi: 10.15672/hjms.2018.570.
[7]	N. Bohra, S. Kumar and V. Ravichandran, Applications of Briot-Bouquet differential subordination, J. Class. Anal., 18 (2021), 17-28. doi: 10.7153/jca-2021-18-02.
[8]	T. Bulboacǎ, M. K. Aouf and R. M. El-Ashwah, Convolution properties for subclasses of meromorphic univalent functions of complex order, Filomat, 26 (2012), 153-163. doi: 10.2298/FIL1201153B.
[9]	N. E. Cho, S. Kumar and V. Kumar, Hermitian-Toeplitz and Hankel determinants for certain starlike functions, Asian-Eur. J. Math., 15 (2022), Paper No. 2250042, 16 pp. doi: 10.1142/S1793557122500425.
[10]	K. Cudna, O. S. Kwon, A. Lecko, Y. J. Sim and B. Smiarowska, The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of order $\alpha$, Bol. Soc. Mat. Mex., 26 (2020), 361-375. doi: 10.1007/s40590-019-00271-1.
[11]	P. L. Duren, Univalent Functions, 259, Springer, New York, 1983.
[12]	P. Eenigenburg, S. S. Miller, P. T. Mocanu and M. O. Reade, On a Briot-Bouquet differential subordination, General Inequalities 3. International Series of Numerical Mathematics, 64 (1983), 339-348.
[13]	R. M. El-Ashwah, Some convolution and inclusion properties for subclasses of bounded univalent functions of complex order, Thai J. Math., 12 (2014), 373-384.
[14]	V. Gupta, A generalized class of integral operators, Carpathian J. Math., 36 (2020), 423-431. doi: 10.37193/CJM.2020.03.10.
[15]	V. Gupta, A. Aral and C.-V. Muraru, Modification of exponential type operators preserving exponential functions connected with $x^3$, Mediterr. J. Math., 18 (2021), Paper No. 222, 14 pp. doi: 10.1007/s00009-021-01851-0.
[16]	D. J. Hallenbeck and S. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc., 52 (1975), 191-195. doi: 10.1090/S0002-9939-1975-0374403-3.
[17]	W. Janowski, Some extremal problems for certain families of analytic functions. I, Ann. Polon. Math., 28 (1973), 297-326. doi: 10.4064/ap-28-3-297-326.
[18]	P. Jastrzȩbski, B. Kowalczyk, O. S. Kwon, A. Lecko and Y. J. Sim, Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), 166. doi: 10.1007/s13398-020-00895-3.
[19]	B. Kowalczyk, A. Lecko and Y. J. Sim, The sharp bound for the Hankel determinant of the third kind for convex functions, Bull. Aust. Math. Soc., 97 (2018), 435-445. doi: 10.1017/S0004972717001125.
[20]	D. Kucerovsky, K. Mousavand and A. Sarraf, On some properties of Toeplitz matrices, Cogent Math., 3 (2016), Art. ID 1154705, 12 pp. doi: 10.1080/23311835.2016.1154705.
[21]	S. Kumar and A. Cetinkaya, Coefficient inequalities for certain starlike and convex functions, Hacet. J. Math. Stat., $\texttt{51}$ (2022), 156-171. doi: 10.15672/hujms.778148.
[22]	S. Kumar and V. Kumar, Sharp estimates on Hermitian–Toeplitz Determinant for Sakaguchi classes, Commun. Korean Math. Soc., https://ckms.kms.or.kr/.
[23]	S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math., 40 (2016), 199-212.
[24]	S. Kumar and V. Ravichandran, Subordinations for functions with positive real part, Complex Anal. Oper. Theory, 12 (2018), 1179-1191. doi: 10.1007/s11785-017-0690-4.
[25]	V. Kumar, N. E. Cho, V. Ravichandran and H. M. Srivastava, Sharp coefficient bounds for starlike functions associated with the Bell numbers, Math. Slovaca, 69 (2019), 1053-1064. doi: 10.1515/ms-2017-0289.
[26]	V. Kumar and S. Kumar, Bounds on Hermitian-Toeplitz and Hankel determinants for strongly starlike functions, Bol. Soc. Mat. Mex., 27 (2021), Paper No. 55. doi: 10.1007/s40590-021-00362-y.
[27]	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 Anal. Oper. Theory, 13 (2019), 2231-2238. doi: 10.1007/s11785-018-0819-0.
[28]	R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in $\mathcal{ P}$, Proc. Amer. Math. Soc., 87 (1983), 251-257. doi: 10.2307/2043698.
[29]	S. S. Miller and P. T. Mocanu, Univalent solutions of Briot-Bouquet diferential equations, J. Differ Equ., 56 (1985), 297-309. doi: 10.1016/0022-0396(85)90082-8.
[30]	R. Parvatham, On Bernardi's integral operators of certain classes of functions, Kyungpook Math. J., 42 (2002), 437-441.
[31]	Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc., 41 (1966), 111-122. doi: 10.1112/jlms/s1-41.1.111.
[32]	V. Ravichandran, Starlike and convex functions with respect to conjugate points, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 20 (2004), 31-37.
[33]	V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris, 353 (2015), 505-510. doi: 10.1016/j.crma.2015.03.003.
[34]	K. Ullah, H. M. Srivastava, A. Rafiq, M. Arif and S. Arjika, A study of sharp coefficient bounds for a new subfamily of starlike functions, J. Inequal. Appl., 2021, Paper No. 194, 20 pp. doi: 10.1186/s13660-021-02729-1.
[35]	K. Ullah, S. Zainab, M. Arif, M. Darus and M. Shutaywi, Radius problems for starlike functions associated with the tan hyperbolic function, J. Funct. Spaces, 2021, Art. ID 9967640, 15 pp. doi: 10.1155/2021/9967640.
[36]	Z.-G. Wang, M. Raza, M. Arif and K. Ahmad, On the third and fourth Hankel determinants for a subclass of analytic functions, Bull. Malays. Math. Sci. Soc., 45 (2022), 323-359. doi: 10.1007/s40840-021-01195-8.
[37]	K. Ye and L.-H. Lim, Every matrix is a product of Toeplitz matrices, Found. Comput. Math., 16 (2016), 577-598. doi: 10.1007/s10208-015-9254-z.