Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 15 (2020), 525 -- 543

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This work is licensed under a Creative Commons Attribution 4.0 International License.

UPPER BOUND ON HANKEL DETERMINANT FOR BOUNDED TURNING FUNCTION ASSOCIATED WITH SĂLĂGEAN-DIFFERENCE OPERATOR

A. Naik and T. Panigrahi

Abstract. By making use of Sălăgean-difference operator we introduce a new function class ℛλβ(α) which generalizes the class of functions of bounded turning of order alpha. We investigate upper bounds on the third Hankel determinant for the class ℛλβ(α). Our results generalize the results of earlier researchers in this direction.

2020 Mathematics Subject Classification: Primary: 30C45; Secondary 30C50
Keywords: Univalent function, Hankel determinant, Sălăgean-difference operator, Functions of bounded turning

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References

  1. K. O. Babalola, On H3(1) Hankel determinant for some classes of univalent function, Inequal. Theor. Appl., 6(2007), 1-7.

  2. D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett., 26(2013), 103-107. MR2971408. Zbl 1250.30006.

  3. D. Bansal, S. Maharana and J. K. Prajapat, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc., 52(6)(2015), 1139-1148. MR3418550. Zbl 1328.30005.

  4. B. Bhowmik, S. Ponnusamy and K. J. Wirths, On the Fekete-Szego problem for the concave univalent functions, J. Math. Anal. Appl., 373(2011), 432-438. MR2720694 Zbl 1202.30015.

  5. D. G. Cantor, Power series with the integral coefficients, Bull. Amer. Math. Soc., 69(1963), 362-366. MR0151581. Zbl 0112.29901.

  6. E. Deniz and L. Budak, Second Hankel determinant for certain analytic functions satisfying subordinate condition, Math. Slovaca, 68(2)(2018), 463-371. MR3783399. Zbl 06856356.

  7. P. Dienes, The Taylor Series, Dover, New York, 1957. MR0089895 Zbl 0078.05901.

  8. C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc., 311(1989), 164-183. MR0951883. Zbl 0652.33004.

  9. P. L. Duren, Univalent Function, Gurundlehren der Mathematicschen Wissenschaften, Vol.259, Springer-Verlag, New York, Berlin, Heidelberg and Tokoyo, 1983. MR0708494. Zbl 0514.30001.

  10. A. Edrei, Sur les determinants recurrent set les singularities dune function done por son development de Taylor, Compos. Math., 7(1939), 20-88. MR0001285. Zbl 0021.33005.

  11. R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107(2000), 557-560. MR1767065. Zbl 0985.15006.

  12. M. Fekete and G. Szego, Eine Bermerkung \ddotuber ungerade schlichte Funktionen, \textslJ. London Math. Soc., 8(1933), 85-89. MR1574865. Zbl 0006.35302.

  13. P. Gochhayat, A. Prajapat and A. K. Sahoo, Coefficient estimates of certain subclasses of analytic functions associated with Hohlov operator, \textslAsian European J. Math., (2021), DOI: 10.1142/S1793557121500212.

  14. T. Gronwall, Some remarks on conformal representation, \textslAnn. of. Math., 16(1914-15), 72-76. MR1502490. JFM 45.0672.01.

  15. W. K. Hayman, On second Hankel determinant of mean univalent functions, \textsl Proc. London Math. Soc., \bf18(3)(1968), 77-94. MR0219715, Zbl 0158.32101.

  16. R. W. Ibrahim, New classes of analytic functions determined by a modified differential-difference operator in complex domain, Karbala Int. J. Modern Sci., 3(1)(2017), 53-58.

  17. R. W. Ibrahim and M. Darus, Subordination inequalities of a new Sălăgean-difference operator, Int. J. Math. Comput. Sci., 14(3)(2019), 573-582. MR3959386. Zbl 1417.30011.

  18. R. W. Ibrahim and M. Darus, Univalent functions formulated by the Sălăgean-difference operator, Int. J. Anal. Appl., 17 (4) (2019), 652-658. Doi: 10.28924/2291-8639. Zbl 1438.30052.

  19. A. Janteng, S. A. Halim and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal Pure Appl. Math., 7(2)(2006), Art. 50, 5 pages. MR2221331. Zbl 1134.30310

  20. A. Janteng, S. A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., 1(13)(2007),~619-625. MR2370200. Zbl 1137.30308

  21. L. Jena and T. Panigrahi, Upper bounds of second Hankel determinant for generalized Sakaguchi type spiral-like functions, Bol. Soc. Paran. Mat., 35(3)(2017), 263-272. MR3573641.

  22. F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20(1969), 8-12. MR0232926. Zbl 0165.09102.

  23. W. Koepf, On the Fekete-Szeg\ddoto problem for close-to-convex functions, Proc. Amer. Math. Soc., 101(1987), 89-95. MR0897076. Zbl 0635.30019.

  24. D. V. Krishna and T. Ramreddy, Hankel determinant for starlike and convex functions of order alpha, Tbilisi Math. J., 5(1)(2012), 65-76. MR3006759. Zbl 1279.30017.

  25. S. K. Lee, V. Ravichandran and S. Supramaniam, Bounds for the second Hankel determinants of certain univalent functions, J. Inequal. Appl., 2013, 281 (2013). MR3073988. Zbl 1302.30018.

  26. R. J. Libera and E. J. Zlotkiewicz, Early coefficient of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85(2) (1982), 225-230 MR0652447. Zbl 0464.30019.

  27. R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in \mathcalP, \textslProc. Amer. Math. Soc., 87(2) (1983), 251-257. MR0681830. Zbl 0488.30010.

  28. A. K. Mishra and P. Gochhayat, Second Hankel determinant for a class of analytic functions defined by fractional derivative, Int. J. Math. Math. Sci., 2008(2008), Art. ID:153280; 10 Pages. MR2392999. Zbl 1158.30308.

  29. A. K. Mishra and T. Panigrahi, The Fekete-Szeg\ddoto problem for a class defined by Hohlov operator, Acta Univ. Apul., 29(2012), 241-254. MR3015070. Zbl 1289.30083.

  30. R. N. Mohapatra and T. Panigrahi, Second Hankel determinant for a class of analytic functions defined by Komantu integral operator, \textslRend. Mat. Appl., 41(1)(2020), 51-58. Zbl 07195400.

  31. A. K. Mishra, J .K. Prajapat and S. Maharana, Bounds on Hankel determinant for starlike and convex functions with respect to symmetric points, Cogent Math., 3:1(2016), 1160557, DOI: 10.1080/23311835.2016.110557. MR3625325. Zbl 1426.30009.

  32. J. W. Noonan and D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc., 223(2)(1976), 337-346. MR0422607. Zbl 0346.30012.

  33. S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39(5)(1987), 1057-1077. MR0918587. Zbl 0611.33007.

  34. T. Panigrahi and G. Murugusundarmoorthy, The Fekete-Szeg\ddoto inequality for subclass of analytic functions of complex order, Adv. Studies Contemp. Math., 24(1)(2014), 67-75. MR3157409 Zbl 1297.30029.

  35. T. Panigrahi and R. K. Raina, Fekete-Szeg\ddoto coefficient functional for quasi-subordination class, Afr. Mat. 28, No. 5-6 (2017), 707-716. DOI: 10.1007/s13370-016-0477-1. MR3687385. Zbl 1370.30008.

  36. T. Panigrah and R. K. Raina, Fekete-Szeg\ddoto problem for generalized Sakaguchi type functions associated with quasi-subordination, Stud. Univ. Babes-Bolyai Math., 63(3)(2018), 329-340. MR3864364 Zbl 1438.30082.

  37. Ch. Pommerenke, On the Hankel determinants of univalent functions, Mathematika, 14(1967), 108-112. MR0215976. Zbl 0165.09602.

  38. Ch. Pommerenke, On the coefficient and Hankel determinants of univalent functions, J. London Math. Soc., 41(1)(1996), 111-122. MR0185105. Zbl 0138.29801.

  39. M. H. Priya and R. B. Sharma, On a class of bounded turning functions subordinate to a leaf-like domain, J. Phys. Conf. Ser. (2018), DOI: 10.1088/1742-6596/1000/1/012056.

  40. T. Ramreddy and D. V. Krishna, Hankel determinant for starlike and convex functions with respect to symmetric points, J. Indian Math. Soc., 79(2012), 161-171. MR2976034.

  41. M. Raza and S. N. Malik, Upper bound of the third Hankel determinant for a class of analytic functions related with lemiscate of Bernoulli, Int. J. Inequal. Appl., 2013(2013), Art. 412. MR3339521. Zbl 1291.30106.

  42. O. Al-Refai and M. Darus, Second Hankel determinant for a class of analytic function defined by a fractional operator, European J. Sci. Resources, 28(2) (2009), 234-241.

  43. A. K. Sahoo and J. Patel, Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, Int. J. Anal. Appl., 6(2)(2014), 170-177. Zbl 1399.30071.

  44. G. S. Sălăgean, Subclasses of univalent functions, Complex Analysis-Fifth Rommanian-Finish Seminar, Part-I, Bucharest, 1981, Lecture Notes in Math., Vol. 1013, Springer, Berlin (1983), 362-372. MR0738107. Zbl 0531.30009.

  45. P. Sumalatha, R. B. Sharma and M. Hari Priya, The third Hankel determinant for starlike functions with respect to symmetric points subordinate to k-Fibonacci sequence, AIP Conf. Proced., 2112, 020069 (2019). DOI: 10.1063/1.5112254.

  46. V. S. Kumar, R. B. Sharma, and M. Haripriya, Third Hankel determinant for Bazilevic functions related to a leaf like domain, AIP Conf. Proc., 2112. DOI: 10.1063/1.5112273.

  47. D. V. Krishna and T. Ramreddy, Coefficient inequality for a function whose derivative has a positive real part of order α, Mathematica Bohemica, 140(1)(2015), 43-52. MR3324418. Zbl 1349.30056.

  48. D. V. Krishna, B. Venkateswarlu and T. Ramreddy, Third Hankel determinant for bounded turning function of order alpha, J. Nigerian Math. Soc., 34(3)(2015), 121-127. MR3512016. Zbl 1353.30013.

  49. T. Yavuz, Second Hankel determinant problem for a certain subclass of univalent functions, Int. J. Math. Anal., 9(10)(2015), 493-498.

  50. P. Zaprawa, Second Hankel determinants for the class of typically real functions, Abstr. Appl. Anal., \bf2016(2016), Art. ID: 3792367. MR3459636. Zbl 06929363.



A. Naik
Department of Mathematics,
School of Applied Sciences, KIIT Deemed to be University,
Bhubaneswar-751024, Odisha, India.
e-mail: avayanaik@gmail.com



T. Panigrahi
Department of Mathematics,
School of Applied Sciences, KIIT Deemed to be University,
Bhubaneswar-751024, Odisha, India.
e-mail: trailokyap6@gmail.com

http://www.utgjiu.ro/math/sma