Upper Bound of the Third Hankel Determinant for a Subclass of Multivalent Functions Associated with the Bernoulli Lemniscate

Authors

  • Khalil Ullah Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan
  • Jihad Younis Government Post Graduate Collage Dargai, Malakand, Pakistan
  • Khurshid Ahmad Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan
  • A Manickam School of Advanced Science and Languages (SASL), Department of Mathematics, VIT Bhopal University Kothri Kalan, Bhopal-Indore Highway, Sehore Dt. Madhya Pradesh - 466114 .
  • Bilal Khan 4 School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, Peoples Republic of China
  • Mirajul Haq Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan

DOI:

https://doi.org/10.21015/vtm.v9i1.1022

Abstract

Let RL; SL and CL represent the families of multivalent bounded turning, multivalent starlike, and multivalent convex functions that are subordinated with Bernoulli lemniscate in the open unit disk E = {z : |z|<1}. In this particular paper, our goal is to find the upper bounds of Hankel’s third order determinant for the above-mentioned families.

References

Ali, R. M., Cho, N. E., Ravichandran, V., & Kumar, S. S. (2012). First order differential subordination for functions associated with the lemniscate of Bernoulli. Taiwanese J. Math., 16, 1017–1026.

Altinkaya, Ş., & Yalçin, S. (2016). Third Hankel determinant for Bazilevic functions. Adv. Math., 5, 91–96.

Altinkaya, Ş., & Yalçin, S. (2016). Upper bound of second Hankel determinant for biBazilevic functions. Mediterr. J. Math., 13, 4081–4090.

Arif, M. (2013). Sufficiency criteria for a class of p-valent analytic functions of complex order. Abstract and Applied Analysis, Volume 2013, Article ID 517296, 4 pages.

Arif, M., Ayaz, M., Iqbal, J., & Haq, W. (2013). Sufficient conditions for functions to be in a class of p-valent analytic functions. Journal of Computational Analysis and Applications, 16(1), 159–164.

Arif, M., Dziok, J., Raza, M., & Sokół, J. (2015). On products of multivalent close-to-star functions. Journal of Inequality and Applications, Vol. 2015, 2015:5, 14 Pages.

Arif, M., Noor, K. I., & Raza, M. (2012). Hankel determinant problem of a subclass of analytic functions. J. Ineq. Appl., 2012(1), Art. 22, 7 pages.

Arif, M., Raza, M., Tang, H., Hussain, S., & Khan, H. (2019). Hankel determinant of order three for familiar subsets of analytic functions related with sine function. Open Mathematics, 2019, 17(1), 1615–1630.

Arif, M., Rani, L., Raza, M., & Zaprawa, P. (2018). Fourth Hankel determinant for the family of functions with bounded turning. Bull. Korean Math. Soc., 2018, 55(6), 1703–1711.

Arif, M., Raza, M., Ullah, I., & Zaprawa, P. (2020). Investigation of the fifth Hankel determinant for a family of functions with bounded turnings. Math. Slovaca, 2020, 70(2), 319–328.

Arif, M., Sokół, J., Ayaz, M., & Khan, H. (2014). Sufficient condition for functions to be in a class of meromorphic multivalent Sakaguchi type spiral-like functions. Acta Mathematica Scientia, 2014, 34B(2), 1–4.

Arif, M., Umar, S., Raza, M., Bulboaca, T., Farooq, M., & Khan, H. (In press). On fourth Hankel determinant for functions associated with Bernoulli’s lemniscate. Hacet. J. Math. Stat.

Babalola, K. O. (2010). On H(1) Hankel determinant for some classes of univalent functions. Inequal. Theory Appl., 2010, 6, 1–7.

Bansal, D. (2013). Upper bound of second Hankel determinant for a new class of analytic functions. Appl. Math. Lett., 2013, 23, 103–107.

Bansal, D., Maharana, S., & Prajapat, J. K. (2015). Third order Hankel Determinant for certain univalent functions. J. Korean Math. Soc., 2015, 32, 1139–1148.

Caratheodory, C. (1911). Über den variabilitatsbereich der fourier’schen konstanten von positiven harmonischen funktionen. Rend. Circ. Mat. Palermo, 1911, 32, 193–127.

Çaglar, M., Deniz, E., & Srivastava, H. M. (2017). Second Hankel determinant for certain subclasses of bi-univalent functions. Turk. J. Math., 2017, 41, 694–706.

Cho, N. E., Kowalczyk, B., Kwon, O. S., Lecko, A., & Sim, Y. J. (2017). Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha. J. Math. Inequal., 2017, 11, 429–439.

Grenander, U., & Sezego, G. (1958). Toeplitz Forms and Their Applications. University of California Press, Berkeley.

Hayman, W. K. (1968). On second Hankel determinant of mean univalent functions. Proc. Lond. Math. Soc., 1968, 3, 77–94.

Janowski, W. (1971). Extremal problems for a family of functions with positive real part and for some related families. Ann. Pol. Math., 1971, 23, 159–177.

Jangteng, A., Halim, S. A., & Darus, M. (2006). Coefficient inequality for a function whose derivative has a positive real part. J. Ineq. Pure Appl. Math., 2006, 7, 1–5.

Jangteng, A., Halim, S. A., & Darus, M. (2007). Coefficient inequality for starlike and convex functions. Int. J. Ineq. Math. Anal., 2007, 1, 619–625.

Kargar, R., Ebadian, A., & Sokół, J. (2019). On Booth lemniscate of starlike functions. Anal. Math. Phys., 2019, 9, 143–154.

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

Khan, Q., Arif, M., Ahmad, B., Tang, H. (2020). On analytic multivalent functions associated with lemniscate of Bernoulli. AIMS Mathematics, 2020, 5(3), 2261–2271.

Kowalczyk, B., Lecko, A., & Sim, Y. J. (2018). The sharp bound of the Hankel determinant of the third kind for convex functions. Bull. Aust. Math. Soc., 2018, 97, 435–445.

Krishna, D. V., & RamReddy, T. (2012). Hankel determinant for starlike and convex functions of order alpha. Tbil. Math. J., 2012, 5, 65–76.

Krishna, D. V., & RamReddy, T. (2015). Second Hankel determinant for the class of Bazilevic functions. Stud. Univ. Babe’s-Bolyai Math., 2015, 60, 413–420.

Krishna, D. V., Venkateswarlu, B., & RamReddy, T. (2015). Third Hankel determinant for bounded turning functions of order alpha. J. Niger. Math. Soc., 2015, 34, 121–127.

Kumar, S. S., Kumar, V., Ravichandran, V., & Cho, N. E. (2013). Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli. J. Inequal. Appl., 2013, 176(2013).

Kwon, O. S., Lecko, A., & Sim, Y. J. (2019). The bound of the Hankel determinant of the third kind for starlike functions. Bull. Malays. Math. Sci. Soc., 2019, 42, 767–780.

Lecko, A., Sim, Y. J., & Smiarowska, B. (2018). The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2. Complex anal. Oper. theory., 2018, 1–8.

Lee, S. K., Ravichandran, V., & Supramaniam, S. (2013). Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl., 2013, 2013, 281.

Liu, M. S., Xu, J. F., & Yang, M. (2014). Upper bound of second Hankel determinant for certain subclasses of analytic functions. Abstr. Appl. Anal., 2014, 2014, 603180.

Ma, W., & Minda, D. (1992). A unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis; Li, Z., Ren, F., Yang, L., Zhang, S. Eds.; Int. Press: Cambridge, MA, USA, 1992; pp:157169.

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

Noonan, J. W., & Thomas, D. K. (1976). On the second Hankel determinant of areally mean p-valent functions. Trans. Am. Math. Soc., 1976, 233, 337–346.

Noor, K. I., Bukhari, S. Z. H., Arif, M., & Nazir, M. (2013). Some properties of p-valent analytic functions involving Cho-Kwon-Srivastava integral operator. Journal of classical analysis, 2013, 3(1), 35–43.

Orhan, H., Magesh, N., & Yamini, J. (2016). Bounds for the second Hankel determinant of certain bi-univalent functions. Turk. J. Math., 2016, 40, 679–687.

Pommerenke, C. (1966). On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc., 1966, 41, 111–122.

Pommerenke, C. (1967). On the Hankel determinants of univalent functions. Mathematika., 1967, 14108–112.

Pommerenke, C. (1975). Univalent Functions. Vandenhoeck and Ruprecht: Gottingen, Germany, 1975.

Raza, M., & Arif, M. (2013). Fekete-Szego inequality for a subclass of p-valent analytic functions. Journal of Applied Mathematics, Volume 2013, Article ID 127615, 7 pages.

Raza, M., & Malik, S. N. (2013). Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli. J. Inequal. Appl., 2013, 2013, 412.

Ravichandran, V., & Sharma, K. (2015). Sufficient conditions for starlikeness. J. Korean Math. Soc., 2015, 52, 727–749.

Raina, R. K., & Sokol, J. (2015). On coefficient estimates for a certain class of starlike functions. Hacet. J. Math. Stat., 2015, 44, 1427–1433.

Ravichandran, V., & Verma, S. (2015). Bound for fifth coefficient of certain starlike functions. C. R. Math., 2015, 353, 505–510.

Răducanu, D., & Zaprawa, P. (2017). Second Hankel determinant for close-to-convex functions. Compt. Rendus., 2017, 355, 1063–1071.

Shi, L., Ali, I., Arif, M., Cho, N. E., Hussain, S., & Khan, H. (2019). A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain. Mathematics, 2019, 7(5), 418; 15 pages.

Shi, L., Khan, Q., Srivastava, G., Liu, J. L., Arif, M. (2019). A study of multivalent q-starlike functions connected with the circular domain. Mathematics, 7, 670. doi:10.3390/math7080670.

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

Sharma, K., Ravichandran, V. (2016). Application of subordination theory to starlike functions. Bulletin of the Iranian Mathematical Society, 42, 761–777.

Shanmugam, T. N. (1989). Convolution and differential subordination. International Journal of Mathematics and Mathematical Sciences, 12, 333–340.

Shanmugam, G., Stephen, B. A., K. O. Babalola, K.O. (2014). Third Hankel determinant for q-starlike functions. Gulf Journal of Mathematics, 2, 107–113.

Shi, L., Srivastava, H. M., Arif, M., Hussain, S., Khan H. (2019). An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function. Symmetry, 11(5), 14 pages. https://doi.org/10.3390/sym11050598.

Sokól, J. (2009). Coefficient estimates in a class of strongly starlike functions. Kyungpook Mathematical Journal, 49(2), 349–353.

Soko÷, J., Stankiewicz, J. (1996). Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka, 19, 101–105.

Sokól, J. (2009). Radius problem in the class $S_L$: Applied Mathematics and Computation, 214, 569–573.

Srivastava, H. M., Alt‹nkaya, S., Yalc‹n, S. (2018). Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator. Filomat, 32, 503–516.

Srivastava, H. M., Ahmad, Q. Z., Khan, N., Khan, B. (2019). Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics, 7, 1–15.

Wang, Z. G., Raza, M., Ayaz, M., Arif, M. (2016). On certain multivalent functions involving the generalized Srivastava-Attiya operator. Journal of Non-linear Sciences and Applications, 9, 6067–6076.

Zaprawa, P. (2017). Third Hankel determinants for subclasses of univalent functions. Mediterranean Journal of Mathematics, 14, 19.

Downloads

Published

2021-12-31

How to Cite

Ullah, K., Younis, J., Ahmad, K., Manickam, A., Khan, B., & Haq, M. (2021). Upper Bound of the Third Hankel Determinant for a Subclass of Multivalent Functions Associated with the Bernoulli Lemniscate. VFAST Transactions on Mathematics, 9(1), 37–58. https://doi.org/10.21015/vtm.v9i1.1022