Third Hankel determinant for the class of analytic functions defined by Mathieu-type series related to a petal-shaped domain

T. Panigrahi; E. Pattnayak; R. M. El-Ashwah

doi:10.3842/umzh.v74i4.7335

T. Panigrahi Institute of Mathematics and Applications, Andharua, Bhubaneswar, Odisha, India
E. Pattnayak Institute of Mathematics and Applications, Andharua, Bhubaneswar, Odisha, India
R. M. El-Ashwah Department of Mathematics, Faculty of Science, Damietta University, New Damietta, Egypt

DOI: https://doi.org/10.3842/umzh.v74i4.7335

Keywords: Analytic function, Subordination, Fekete-Szego functional, Hankel determinant, Petalshaped domain

Abstract

UDC 517.5

We introduce a new subclass of analytic functions based on the Mathieu-type series related to a petal-shaped domain. We investigate the bounds of the initial coefficient estimates, the Fekete–Szegö inequality, and the Hankel determinant of order two and three.

References

H. Alzer, J. L. Brenner, O. G. Rueh, On Mathieu's inequality, J. Math. Anal. and Appl., 218, 607–610 (1998).

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, 1615–1630 (2019); DOI: 10.1515/math-2019-0132.

K. O. Babalola, On $H_{3}(1)$ Hankel determinant for some classes of univalent function, Inequal. Theory and Appl., 6, 1–7 (2010).

D. Bansal, J. Sokół, Geometric properties of Mathieu-type power series inside unit disk, J. Math. Inequal., 13, 911–918 (2019).

P. Ceronre, C. T. Lenard, On integral forms of generalized Mathieu series, J. Inequal. Pure and Appl. Math., 4, № 5, 1–11 (2003).

N. E. Cho, S. Kumar, V. Kumar, V. Ravichandran, Radius problems for strarlike functions associated with the sine function, Bull. Iran. Math. Soc., 45, 213–232 (2019).

J. Choi, H. M. Srivastava, Mathieu series and associated sums involving the zeta functions, Comput. Math. Appl., 592, 861–867 (2010).

P. H. Diananda, Some inequalities related to an inequality of Mathieu, Math. Ann., 250, 95–98 (1980).

O. Emersleben, Über die Reihe $Ε_{n = 1}^{∞}n(n^{2} + c^{2})^{-2}$, Math. Ann., 125, 165–171 (1952).

F. Keough, E. Merkes, A coefficient inequality for certain subclasses of analytic functions, Proc. Amer. Math. Soc., 20, 8–12 (1969).

S. S. Kumar, K. Arora, Starlike functions associated with a petal shaped domain (2020); https://arxiv.org/abs/ 2010.10072.

W. C. Ma, D. Minda, A unified treatment of some special classes of univalent function, Proceeding of the Conference on Complex Analysis (Tianjin, 1992), Lecture Notes Anal., Int. Press Cambridge, MA (1994), p. 157–169.

E. Makai, On the inequality of Mathieu, Publ. Math. Debrecen, 5, 204–205 (1957).

E. L. Mathieu, Trait'e de Physique Mathematique VI-VII: Theory del Elasticite des corps solides (part 2), Gauthier-Villars, Paris (1890).

R. Mendiratta, S. Nagpal, V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc., 38, 365–386 (2015).

A. Naik, T. Panigrahi, Upper bound on Hankel determinant for bounded turning function associated with Salagean-difference operator, Surv. Math. and Appl., 15, 525–543 (2020).

T. K. Pog'any, H. M. Srivastava, Z. Tomovski, Some families of Mathieu $alpha$-series and alternating Mathieu $alpha$-series, Appl. Math. and Comput., 173, 69–108 (2006).

C. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, G"ottingen (1975).

R. K. Raina, J. Sokół, On coefficient estimates for a certain class of starlike functions, Hacet. J. Math. and Stat., 44, № 6, 1427–1433 (2015).

M. Shafiq, H. M. Srivastava, N. Khan, Q. Z. Ahmad, M. Darus, S. Kiran, An upper bounds of the third Hankel determinant for a subclass of $q$-starlike functions associated with $k$-Fibonacci numbers, Symmetry, 12 (2020); DOI:10.3390/sym 12061043.

K. Sharma, N. K. Jain, V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat., 27, 923–939 (2016).

H. M. Srivastava, Q. Z. Ahemad, M. Darus, B. Khan, N. Zaman, Upper bounds of the third Hankel determinant for a subclass of close-to-convex functions associated with the leminiscate of Bernoulli, Mathematics, 7 (2019); DOI:10.3390/math 7090848.

Z. Tomovski, New integral and series representations of the generalized Mathieu series, Appl. Anal. and Discrete Math., 2, 205–212 (2008).

L. A. Wani, A. Swaminathan, Starlike and convex functions associated with a Nephroid domain, Bull. Malays. Math. Sci. Soc., 44, 79–104 (2021).

P. Zaprawa, Thrid Hankel determinant for subclasses of univalent functions, Mediterr. J. Math., 14, Article № 19 (2017); https:// DOI.org/10.1007/s00009-016-0829-y.