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Vladikavkaz Mathematical Journal 2020, Volume 22, Issue 1, P. 43-48
YAK 512.643.86+517.546 DOI 10.23671/VNC.2020.1.57538
HANKEL DETERMINANT OF THIRD KIND FOR CERTAIN SUBCLASS OF MULTIVALENT ANALYTIC FUNCTIONS
D. Vamshee Krishna1 and D. Shalini2
1 GITAM Institute of Science, Visakhapatnam 530045, Andhra Pradesh, India; 2 Dr. B. R. Ambedkar University, Srikakulam 532410, Andhra Pradesh, India E-mail: vamsheekrishna1972@gmail.com, shaliniraj1005@gmail.com
Abstract. The objective of this paper is to obtain an upper bound (not sharp) to the third order Hankel determinant for certain subclass of multivalent (p-valent) analytic functions, defined in the open unit disc E. Using the Toeplitz determinants, we may estimate the Hankel determinant of third kind for the normalized multivalent analytic functions belongng to this subclass. But, using the technique adopted by Zaprawa [1], i. e., grouping the suitable terms in order to apply Lemmas due to Hayami [2], Livingston [3] and Pommerenke [4], we observe that, the bound estimated by the method adopted by Zaprawa is more refined than using upon applying the Toeplitz determinants.
Key words: p-valent analytic function, upper bound, third Hankel determinant, positive real function. Mathematical Subject Classification (2010): 30C45, 30C50.
For citation: Vamshee Krishna, D. and Shalini, D. Hankel Determinant of Third Kind for Certain Subclass of Multivalent Analytic Functions, Vladikavkaz Math. J., 2019, vol. 21, no. 3, pp. 43-48. DOI: 10.23671/VNC.2020.1.57538.
1. Introduction
Let Ap (p is a fixed integer ^ 1) denotes the class of functions f of the form
f (z) = zpJ2 ap+nzn
(1.1)
n=0
in the open unit disc E = {z : |z| < 1} with p e N = {1,2,3,...}. Let S be the subclass of A\ = A, consisting of univalent functions. In 1985, Louis de Branges de Bourcia proved the Bieberbach conjecture also called as Coefficient conjecture, which states that for a univalent function its nth-Taylor's coefficient is bounded by n (see [5]). The bounds for the coefficients of these functions give information about their geometric properties. In particular, the growth and distortion properties of a normalized univalent function are determined by the bound of its second coefficient. The Hankel determinant of f given in (1.1) (when p = 1), for q,n e N was defined by Pommerenke [6] as follows and has been extensively studied by many authors:
(1.2)
(in an+1 ' O-ra+ç— 1
Hq{n) = öra+l an+2 ' O-ra+ç
an+q-1 an+q ' dn+2q-2
© 2020 Vamshee Krishna, D. and Shalini, D.
One can easily observe that the Fekete-Szego functional is H2(1). In recent years, the research on Hankel determinants has focused on the estimation of |H2(2)|, where
H2(2) =
a2
0>3 (I4
(12U4
known as the second Hankel determinant obtained for q = 2 and n = 2 in (1.2). Many authors obtained upper bound to the functional |a2a4 — a31 for various subclasses of univalent and multivalent analytic functions. The exact (sharp) estimates of |H2(2)| for the subclasses of S namely, bounded turning, starlike and convex functions denoted by R, S* and K respectively in the open unit disc E, that is, functions satisfying the conditions Re f '(z) > 0, Re |Zf(z) } > 0 and Re |l + } > 0 were proved by Janteng et al. [7, 8] and obtained the bounds as 4/9, 1, and 1/8 respectively. For the class S*(^) of Ma-Minda starlike functions, the exact bound of the second Hankel determinant was obtained by Lee et al. [9]. Choosing q = 2 and n = p + 1 in (1.2), we obtain the second Hankel determinant for the p-valent function (see [10]), namely
H2(p + 1) =
ap+1 ap+2 ap+2 ap+3
ap+1ap+3 — ap+2-
The case q = 3 appears to be much more difficult than the case q = 2. Very few papers have been devoted to the third Hankel determinant denoted by H3(1), obtained by choosing q = 3 and n = 1 in (1.2). Babalola [11] is the first one, who tried to estimate an upper bound to |H3(1)| for the classes R, S* and K. Following this paper, Raza and Malik [12] obtained an upper bound for the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli. Sudharsan et al. [13] derived an upper bound to H3(1) for a subclass of analytic functions. Bansal et al. [14] modified the upper bound for |H3(1)| for some of the classes estimated by Babalola [11] to some extent. Recently, Zaprawa [1] improved the results obtained by Babalola [11]. Further, Orhan and Zaprawa [15] obtained an upper bound for third Hankel determinant for the classes S* and K functions of order alpha. Very recently, Kowalczyk et al. [16] estimated sharp upper bound to |H3(1)| for the class of convex functions Jf and showed as |ii3(l)| ^ j^, which is far better than the bound obtained by Zaprawa [1]. Lecko et al. [17] calculated sharp bound for Hankel determinant of the third kind for starlike functions of order 1/2. For our discussion in this paper, we consider H3(p) for the values q = 3 and n = p in (1.2), called as Hankel determinant of third order for the p-valent function given in (1.1), namely
H3(p) =
ap ap+1 ap+2 ap+1 ap+2 ap+3 ap+2 ap+3 ap+4
((p = 1).
Expanding the determinant, we have
H3(p) = [ap(ap+2(p+4 — ap+3)
equivalently
+ (p+1 (ap+2ap+3 — ap+1(p+4) + ap+2(ap+1(p+3 — ap+2 )], (1.3)
H3(p) = H2 (p + 2) + flp+1 Jp+1 + (p+2H2(p + 1),
where Jp+1 = (ap+2ap+3 — ap+1ap+4) and H2(p + 2) = (ap+2ap+4 — a^).
Motivated by the results obtained by different authors mentioned above and who are working in this direction (see [18, 19]), in particular the result obtained by Zaprawa [1] in finding an upper bound to the Hankel determinant of third kind for the subclass R of S, consisting of functions whose derivative has a positive real part (also called as bounded turning functions), introduced by Alexander in 1915 and a systematic study of properties of these functions was conducted by MacGregor [20], who indeed referred to numerous earlier investigations involving functions whose derivative has a positive real part. In the present paper, we are making an attempt to obtain an upper bound to |H3(p)|, for the function f given in (1.1), when it belongs to certain subclass of analytic functions, defined as follows.
Definition 1.1. A function f e Ap is said to be in the class Ip(ft) (ft is real) (see [21]), if it satisfies the condition
¿€£-{0}. (1.4)
1. Choosing ft = 1 and p = 1, we obtain I1(1) = R.
2. Selecting ft = 1, we get Ip(1) = Rp, denotes the class of multivalent bounded turning functions.
In proving our result, we require a few sharp estimates in the form of Lemmas valid for functions with positive real part.
Let P denote the class of functions consisting of g, such that
<x
g(z) = 1 + ciz + C2Z2 + C3Z3 + ... = 1 + Y^ CnZn, (1.5)
n=1
which are analytic in E and Reg(z) > 0 for z e E. Here g is called the Caratheodory function [22].
Lemma 1.1 [2]. If g e P, then the sharp estimate |ck — ¡ckcn-k| ^ 2, holds for n,k e N, with n > k and i e [0,1].
Lemma 1.2 [3]. If g e P, then the sharp estimate |ck — ckcn-k| ^ 2, holds for n,k e N, where n > k.
Lemma 1.3 [4]. If g e P then |c^ | ^ 2, for each k ^ 1 and the inequality is sharp for the function g(z) = j^, z £ E.
In order to obtain our result, we referred to the classical method devised by Libera and Zlotkiewicz [23, 24], used by several authors in the literature.
2. Main Result
Theorem 2.1. If f e Ip(ß) (ß ^ 1 is real ) with p € N, then
"4p2(6p6 + 60p5ß + 227p4ß2 + 426p3ß3 + 437p2 ß4 + 252pß5 + 68ß6)
|H3(p)| <
(p + ß)2(p + 2ß)3 (p + 3ß)2(p + 4ß)
< For the function f(z) = zp + anzn e Ip(ft), by virtue of Definition 1.1, there
exists an analytic function g e P in the open unit disc E with g(0) = 1 and Re g(z) > 0 such that
(1 - + ft^Q = g(z) & [(1 - I3)pf(z) + Pf'(z) = pzpg(z)]. (2.1)
pzp
Replacing f' and g with their series expressions in (2.1), upon simplification, we obtain
pC™ RJ
aP+n = —;—q, n,pG N. p + np
(2.2)
Substituting the values of ap+1, ap+2, ap+3 and ap+4 from (2.2) in the functional given in (1.3), it simplifies to
|H3(p)| = p2
C2C4
3
pc2
c2
c3
(p + 2P)(p + 4P) (p + 2P)3 (p + 3P)2
2
pcfc4
+
2pC1C2C3
(p + p)2(p + 4P) (p + p)(p + 2P)(p + 3P) On grouping the terms in (2.3), in order to apply Lemmas, we have
(2.3)
|H3(p)1 = p2
pC4(C2 — C2)
_(p + P)2(p + 4P) (p + 3P)2
C3 C3 —
6pC1C2
(p + P)(p + 2P)
1
+
pc2(c4 - c|) _ 2p2c2(c4 - C1C3) (p + 2/?)3 (p + /?)(p + 2/?)(p + 3/?)2
+
(p6 + 6p5p + 3p4p2 — 30p3p3 — 36p2p4 + 24pP5 + 36P6)C2C4
(p + P)2(p + 2P)3(p + 3P)2(p + 4P) Applying the triangle inequality in (2.4), we obtain
|H3(p)| < p2
+
1
p|c4||c2 - c\ I (p + /?)2(p + 4/?) ' (p + 3/?)2
|C3|
C3 —
6pC1C2
(p + P)(p + 2P)
p\c2\\c4~4\
~r / /">\ Q i-
2p21C21| C4 — C1C3|
+
(p + 2P)3 (p + p)(p + 2P)(p + 3P)2
(p6 + 6P5f3 + 3p4/?2 - 30p3/?3 - 36p2/?4 + 24p(35 + 36/?6)|c2||c4| (p + /?)2(p + 2/?)3(p + 3/?)2(p + 4/?)
H3(p)
< 4p2
p
(p + P)2(p + 4P) 1
+
+
p
+
2p2
(p + 3P)2 (p + 2P)3 (p + P)(p + 2P)(p + 3P)2
+
(p + dp (3 + 3p /? - 30p3/?3 - 36p /? + 2Ap/3 + 36/?b)c2c4 (p + /?)2(p + 2/?)3(p + 3/?)2(p + 4/?)
Further simplification, we obtain
I" 4p2 (6p6 + 60p5f3 + 227p4/?2 + 426p3/?3 + 437p2/?4 + 252p/?5 + 68 f36) 1 3(P)I ^ [ (p + /?)2(p + 2/?)3(p + 3/?)2(p + 4/?)
This completes the proof of our Theorem. >
(2.4)
(2.5)
Upon using the Lemmas given in 1.2, 1.3 and 1.4 in the inequality (2.5), it reduces to
(2.6)
(2.7)
Remark 2.1. Choosing p = 1 and ft = 1 in the inequality (2.7), it coincides with the result obtained by Zaprawa [1].
Acknowledgement. The authors are extremely grateful to the esteemed Reviewers for a careful reading of the manuscript and making valuable suggestions leading to a better presentation of the paper.
References
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Received 26 November, 2018
Deekonda Vamshee Krishna GITAM Institute of Science, Visakhapatnam 530045, Andhra Pradesh, India, Assistant Professor
E-mail: vamsheekrishna1972@gmail. com
https://orcid.org/0000-0002-3334-9079;
D. Shalini
Dr. B. R. Ambedkar University, Srikakulam 532410, Andhra Pradesh, India, Teaching Assistant E-mail: shaliniraj 1005@gmail. com https://orcid.org/0000-0003-4059-8900
Владикавказский математический журнал 2020, Том 22, Выпуск 1, С. 43-48
ОПРЕДЕЛИТЕЛЬ ГАНКЕЛЯ ТРЕТЬЕГО РОДА ДЛЯ НЕКОТОРОГО ПОДКЛАССА МНОГОВАЛЕНТНЫХ АНАЛИТИЧЕСКИХ ФУНКЦИЙ
Вамши Кришна Д.1, Шалини Д.2
1 Университет ГИТАМ, Вишакхапатнам 530045, Андхра-Прадеш, Индия; 2 Университет доктора Б. Р. Амбедкэра, Шрикакулам, 532410, Андхра-Прадеш, Индия E-mail: vamsheekriskna1972@gmail.com, shaliniraj1005@gmail.com
Аннотация. Целью данной статьи является получение (не точной) верхней границы для определителя Ганкеля третьего порядка для некоторого подкласса многовалентных (p-валентных) аналитических функций, определенных на открытом единичном диске E. Используя определители Теплица, мы можем оценить определитель Ганкеля третьего рода для нормированных многовалентных аналитических функций, принадлежащих этому подклассу. Однако, используя технику, принятую Саправой [1], т. е. группируя подходящие члены для применения лемм Хаями [2], Ливингстона [3] и Померенке [4], мы видим, что оценка методом Саправы точнее, чем при применении определителей Теплица.
Ключевые слова: p-валентная аналитическая функция, верхняя граница, третий определитель Ганкеля, положительная вещественная функция.
Mathematical Subject Classification (2010): 30C45, 30C50.
Образец цитирования: Vamshee Krishna, D. and Shalini, D. Hankel Determinant of Third Kind for Certain Subclass of Multivalent Analytic Functions // Владикавк. мат. журн.—2019.—Т. 21, № 3.—C. 43-48 (in English). DOI: 10.23671/VNC.2020.1.57538.
