본문 바로가기
KERD
논문KCI 등재

The sharp bound of the third Hankel determinant for some classes of analytic functions

심영재|Bogumi la Kowalczyk|Adam Lecko|Millenia Lecko
대한수학회 (The Korean Mathematical Society)|2018.11|대한수학회보 (Bulletin of the KMS)|제55권 제6호|pp. 1859-1868

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

univalent functionsHankel determinantCaratheodory functions
주제분류수학

초록

In the present paper, we have proved the sharp inequality $|H_{3,1}(f)|$ $\le 4$ and $|H_{3,1}(f)|\le 1$ for analytic functions $f$ with $a_n:=f^{(n)}(0)/n!,\ n\in\mathbb{N},$ such that $$\mathrm{Re}\, \frac{f(z)}{z}> \alpha,\quad z\in\mathbb{D}:=\{z \in\mathbb{C} : |z|<1\}$$ for $\alpha=0$ and $\alpha=1/2,$ respectively, where \begin{equation*} H_{3,1}(f):= \begin{vmatrix} a_1 & a_2 & a_3 \\ a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 \end{vmatrix} \end{equation*} is the third Hankel determinant.

Abstract

In the present paper, we have proved the sharp inequality $|H_{3,1}(f)|$ $\le 4$ and $|H_{3,1}(f)|\le 1$ for analytic functions $f$ with $a_n:=f^{(n)}(0)/n!,\ n\in\mathbb{N},$ such that $$\mathrm{Re}\, \frac{f(z)}{z}> \alpha,\quad z\in\mathbb{D}:=\{z \in\mathbb{C} : |z|<1\}$$ for $\alpha=0$ and $\alpha=1/2,$ respectively, where \begin{equation*} H_{3,1}(f):= \begin{vmatrix} a_1 & a_2 & a_3 \\ a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 \end{vmatrix} \end{equation*} is the third Hankel determinant.

참고문헌 (30)

  1. K. O. Babalola, "Inequality Theory and Applications, vol. 6", pp.1~7 (2010), Nova Science Publishers
  2. Deepak Bansal, "THIRD ORDER HANKEL DETERMINANT FOR CERTAIN UNIVALENT FUNCTIONS", 대한수학회지, 52(6), pp.1139~1148 (2015), 대한수학회
  3. C. Caratheodory, "Uber den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen", Math. Ann, 64(1), pp.95~115 (1907)
  4. N. E. Cho, "Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha", J. Math. Inequal, 11(2), pp.429~439 (2017)
  5. N. E. Cho, "The bounds of some determinants for starlike functions of order alpha", Bull. Malays. Math. Sci. Soc, 41(1), pp.523~535 (2018)
  6. P. L. Duren, "Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259" (1983), Springer-Verlag
  7. M. Elin, "Linearization Models for Complex Dynamical Systems" (2010), Birkhauser Verlag
  8. A. W. Goodman, "Univalent Functions" (1983), Mariner
  9. A. Janteng, "Hankel determinant for starlike and convex functions", Int. J. Math. Anal. (Ruse), 1(13-16), pp.619~625 (2007)
  10. BOGUMIŁA KOWALCZYK, "THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS", Bulletin of the Australian Mathematical Society, 97(03), pp.435~445 (2018), Cambridge University Press (CUP)[DOI]
  11. B. Kowalczyk, "The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1=2", ~
  12. Oh Sang Kwon, "On the Fourth Coefficient of Functions in the Carathéodory Class", Computational Methods and Function Theory, 18(2), pp.307~314 (2018), Springer Nature[DOI]
  13. S. K. Lee, "Bounds for the second Hankel determinant of certain univalent functions", J. Inequal. Appl, 2013, pp.281~ (2013)
  14. R. J. Libera, "Early coefficients of the inverse of a regular convex function", Proc. Amer. Math. Soc, 85(2), pp.225~230 (1982)
  15. R. J. Libera, "Coefficient bounds for the inverse of a function with derivative in P", Proc. Amer. Math. Soc, 87(2), pp.251~257 (1983)
  16. T. H. MacGregor, "The radius of univalence of certain analytic functions", Proc. Amer. Math. Soc, 14, pp.514~520 (1963)
  17. A. Marx, "Untersuchungen uber schlichte Abbildungen", Math. Ann, 107(1), pp.40~67 (1933)
  18. S. S. Miller, "Differential Subordinations" (2000), Marcel Dekker, Inc.
  19. A. K. Mishra, "Second Hankel determinant for a class of analytic functions defined by fractional derivative", Int. J. Math. Math. Sci, 2008, pp.10~ (2008)
  20. C. Pommerenke, "Univalent Functions" (1975), Vandenhoeck & Ruprecht
  21. J. K. Prajapat, "Bounds on third Hankel determinant for close-to-convex functions", Acta Univ. Sapientiae Math, 7(2), pp.210~219 (2015)
  22. J. S. Ratti, "The radius of univalence of certain analytic functions", Math. Z, 107, pp.241~248 (1968)
  23. M. Raza, "Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli", J. Inequal. Appl, 2013, pp.412~ (2013)
  24. M. I. S. Robertson, "On the theory of univalent functions", Ann. of Math. (2), 37(2), pp.374~408 (1936)
  25. G. Shanmugam, "Third Hankel determinant for -starlike functions", Gulf J. Math, 2(2), pp.107~113 (2014)
  26. D. Shoikhet, "Rigidity and parametric embedding of semi-complete vector fields on the unit disk", Milan J. Math, 84(1), pp.159~202 (2016)
  27. E. Strohhacker, "Beitrage zur Theorie der schlichten Funktionen", Math. Z, 37(1), pp.356~380 (1933)
  28. E. Study, "Vorlesungen uber ausgewahlte Gegenstande der Geometrie, Zweites Heft;Konforme Abbildung Einfach-Zusammenhangender Bereiche" (1913), Druck und Verlag von B.G. Teubner
  29. T. V. Sudharsan, "Third Hankel determinant for a subclass of analytic functions", Malaya J. Math, 2(4), pp.438~444 (2014)
  30. P. Zaprawa, "Third Hankel determinants for subclasses of univalent functions", Mediterr. J. Math, 14(1), pp.10~ (2017)

관련 논문AI추천

Sharp bounds for initial coefficients and the second Hankel determinantMilutin Obradovic, Rosihan M. Ali, See Keong Lee대한수학회 · 대한수학회보2020 · 제57권 제4호Some Coefficient Inequalities Related to the Hankel Determinant for a Certain Class of Close-to-convex Functionsyong sun, Zhi-Gang Wang경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2019 · 제59권 제3호Upper bound on the third Hankel determinant for functions defined by Ruscheweyh derivative operatorTugba Yavuz대한수학회 · 대한수학회논문집2018 · 제33권 제2호Third Hankel determinants for starlike and convex functions of order alphaHalit Orhan, Paweł Zaprawa대한수학회 · 대한수학회보2018 · 제55권 제1호HANKEL DETERMINANTS FOR STARLIKE FUNCTIONS WITH RESPECT TO SYMMETRICAL POINTS심영재, 조낙은, Derek K. Thomas대한수학회 · 대한수학회보2023 · 제60권 제2호Estimate of Third Order Hankel Determinant for a Certain Subclass of Analytic Functions associated with Cardioid DomainGagandeep Singh, Gurcharanjit Singh한국수학교육학회 · 이론수학과 교직수학2022 · 제29권 제4호

이 간행물의 다른 논문 (21건)

Skew cyclic codes over $\mathbb{F}_p+v\mathbb{F}_p+v^2\mathbb{F}_p$Ahmad Moussavi, Hamed Mousavi, Saeed Rahimipp. 1627-1638Nonlocal boundary value problems for Hilfer fractional differential equationsSuphawat Asawasamrit, Atthapol Kijjathanakorn, Sotiris K. Ntouyas, Jessada Tariboonpp. 1639-1657On a Waring-Goldbach problem involving squares, cubes and biquadratesYuhui Liupp. 1659-1666On the minimum order of 4-lazy cops-win graphsKai An Sim, Kok Bin Wong, Ta Sheng Tanpp. 1667-1690On the location of eigenvalues of real constant row-sum matricesRachid Marsli, Frank J. Hallpp. 1691-1701
원문 보기