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37Probl. Anal. Issues Anal. Vol. 5(23), No. 1, 2016, pp. 45-54
DOI: 10.15393/j3.art.2016.3010
45
UDC 517.54
D. Vamshee Krishna, T. RamReddy
COEFFICIENT INEQUALITY FOR MULTIVALENT BOUNDED TURNING FUNCTIONS OF ORDER a
Abstract. The objective of this paper is to obtain the sharp upper bound to the H2 (p + 1), second Hankel determinant for p-valent (multivalent) analytic bounded turning functions (also called functions whose derivatives have positive real parts) of order a (0 < a < 1), using Toeplitz determinants. The result presented here includes three known results as their special cases.
Key words: p-valent analytic function, bounded turning function, upper bound, Hankel determinant, positive real function, Toeplitz determinants
2010 Mathematical Subject Classification: 30C45, 30C50 1. Introduction. Let Ap denote the class of functions f of the form
f (z)= zp + ap+1zp+1 +
(1)
in the open unit disc E = {z : |z| < 1} with p G 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, i.e., for a univalent function its nth coefficient is bounded by n (see [1]). 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 for q > 1 and n > 1 was defined by Pommerenke [2] as
Hq (n) =
an an+1
an+1 an+2
l"n+q-1 an+q
an+q-1 an+q
an+2q-2
©Petrozavodsk State University, 2016
[MglHl
This determinant has been considered by several authors in the literature. For example, Noonan and Thomas [3] studied the second Hankel determinant of areally mean p-valent functions. Noor [4] determined the rate of growth of Hq (n) as n ^ to for functions in S with bounded boundary rotation. The Hankel transform of an integer sequence and some of its properties were discussed by Layman [5]. One can easily observe that the Fekete-Szego functional is H2(1). Fekete-Szego then further generalized the estimate |a3 — [j.a21 with real ^ and f E S. Ali [6] found sharp bounds on the first four coefficients and sharp estimate for the Fekete-Szego functional I73 — ¿y! |, where t is real, for the inverse function of f for p = 1, given in (1.1), defined as f-1 (w) = w + Ynwn, when f E ST (a),
the class of strongly starlike functions of order a (0 < a < 1). Further sharp bounds for the functional |a2a4 — a31, the Hankel determinant in the case of q = 2 and n = 2, known as the second Hankel determinant (functional), given by
H2(2) =
a2 a3
a3 a4
= a2a4 — a3, (3)
were obtained for various subclasses of univalent and multivalent analytic functions by several authors in the literature. Janteng et al. [7] have considered the functional |a2a4 — a§ | and found a sharp upper bound for the function f in the subclass R of S, consisting of functions whose derivative has a positive real part (also called bounded turning functions) studied by MacGregor [8]. In their work, they have shown that if f E R then a2 a4 — a23 | < 4. Motivated by this result, in this paper we consider the Hankel determinant in the case of q = 2 and n = p +1, denoted by H2 (p + 1), given by
H2 (p +1) =
ap+1 ap+2 ap+2 ap+3
ap+1ap+3 — ap+2. (4)
Further, we seek a sharp upper bound to the functional |ap+1ap+3 — ap+21 for the functions belonging to the certain subclass of p-valent analytic functions, defined as follows.
Definition 1. A function f (z) E Ap is said to be in the class Rp(a) (0 < a < 1) if it satisfies the condition
f' (z)
Re-f-(-) > a, V z E E. pzp-1
1) If p = 1, we obtain R(a) = R(a), the class of bounded turning functions of order a.
2) Choosing a = 0, we get Rp(0) = Rp, the class of p-valent bounded turning functions.
3) Selecting p =1 and a = 0, we have Ri(0) = R.
In the next section we give some preliminary Lemmas required for proving our result.
2. Preliminary Results. Let P denote the class of functions consisting of g such that
g(z) = 1 + C1 Z + C2 z2 + C3 z3 + ... = 1 + 5^
Cn Z
(6)
n=1
which are regular in the open unit disc E and satisfy Reg(z) > 0 for any z E E. Here g(z) is called a Caratheodory function [9].
Lemma 1. [10, 11] If g E P, then |ck | < 2, for each k > 1 and the inequality is sharp for the function 1+Z.
Lemma 2. [12] The power series for g(z) = 1 + 5^cnzn given in (6) converges in the open unit disc E to a function in P if and only if the Toeplitz determinants
2 C1 C2 ■ ■ cn
C-1 2 C1 ■ ■ Cn-1
Dn = C-2 C-1 2 ■ ■ Cn-2 , n = 1, 2, 3
c-n C-n+1 C-n+2 ' ■2
and c-k = ck, are all non-negative. They are strictly positive except for p(z) = pkPo (eitk z), with Pk = 1, tk real and tk = tj, for k = j,
where p0(z) = 1+|; in this case Dn > 0 for n < (m — 1) and Dn = 0 for n > m.
This necessary and sufficient condition found in [12] is due to Caratheodory and Toeplitz. We may assume without restriction that c1 > 0. From Lemma 2, for n = 2 we have
D2 =
2 Ci C2 C1 2 c1 C2 C1 2
On expanding the determinant, we get
D2 = 8 + 2Re{cl02} - 2|c212 - 4|ci|2 > 0.
Applying the fundamental principles of complex numbers, the above expression is equivalent to
In the same way,
202 = cl + y(4 - cl).
D3 =
(7)
2 ci c2 c3
ci 2 ci c2
c2 ci 2 ci
c3 c2 ci 2
Then D3 > 0 is equivalent to |(4c3 - 4cic2 + c1)(4 - c2)+ ci (202 - c2 )2| < 2(4 - cl)2 - 2|(2c2 - c2)|2. (8) Simplifying relations (7) and (8), we obtain
4c3 = {cl + 2ci(4 - cl)y - ci(4 - c2)y2 + 2(4 - c?)(1 - |y|2)Z} (9)
for some complex valued y with |y| < 1 and for some complex valued Z with |Z| < 1. To obtain our result, we refer to the classical method devised by Libera and Zlotkiewicz [13], used by several authors in literature.
3. Main Result Theorem 1. If f (z) e Rp(a) (0 < a < 1) with p e N then
2p(1 - a) p + 2
|ap+iap+3 - ap+21 <
and the inequality is sharp.
For the function f (z) = zp + ^Lp+l anzn e Rp(a), by virtue of Definition 1, there exists an analytic function g eP in the open unit disc E with g(0) = 1 and Reg(z) > 0 such that
^a^ = g(z) ^ f (z) - pazp-i = p(1 - a)zp-ig(z). (10)
2
Replacing f'(z) and g(z) with their equivalent series expressions in (10), we have
œ œ X
Pzp-1 + ^ nanzn-1 - pazp-1 = p(1 - a)zp-1i 1 + ^ enzn I.
n=p+1 ^ n=1 J
Upon simplification, we obtain
p(1 - a)zp-1 + (p + 1)op+1 zp + (p + 2)ap+2 zp+1 + (p + 3)ap+azp+2 +... =
= p(1 - a)zp-1 [1 + C1 z + C2z2 + esz3 + ...]. (11)
Equating the coefficients of same powers of zp, zp+1 and zp+2 in (11), we have
p(1 - a)c1 p(1 - p(1 - a)e3
ap+1 = -—,-, ap+2 = -- and ap+3 = -—. (12)
p + 1 p + 2 p + 3
Substituting the values of ap+1, ap+2, and ap+3 from (12) in the functional |ap+1 ap+3 - ap+21, after simplifying we get
|aP+1ap+3 - ap+21 =
p2(1 - a)2
(p + 1)(p + 2)2 (p + 3) The above expression is equivalent to
(p + 2)2C1C3 - (p +1)(p + 3)c21
|ap+iap+4 - ap+2| = t dicic3 + d2c2 , (13)
where
p2 (1 _ a)2
t = (p +1^(p + 2)'(p + 3) ,dl = (P + 2)2 and d2 = -(P + 1)(P + 3)- (14)
Substituting the values of c2 and c3 from (7) and (9) respectively from
Lemma 2 in the expression of (13), we have
d1C1C3+d2c
, which is on the right-hand side
d1C1C3 + d2c2 = d1 C1 x 4(c? + 2c1 (4 - c?)y - d(4 - c?)y2 +
+ 2(4 - c2)(1 - |y|2)Z} + d2 x 4(c2 + y(4 - c2)}2
4
¿1 C1C3 + d2 c2
= | (d1 + d2)c1 + 2d1 C1 (4 — c1 )Z + 2(d1 + ¿2 )c2 (4 — c2)y— — {d1c2y2 + 2d1C1 |y|2Z — d2(4 — c2)y2} (4 — c2)|;
d1 C1C3 + d2 c2 = |(d1 + d2)c1 + 2d1 d(4 — c2 )Z + 2(d. + ¿2 )c1(4 — c2)y— — {(d1 + d2)c1 y2 + 2d1C1 |y|2Z — 4d2y2 } (4 — c1) |. Applying the triangle inequality, we get
¿1 C1C3 + d2c2
< |(d1 + d2)c4 + 2d1C1 (4 — c2)|Z| + 2(d1 + d2)c1 (4 —c1 )|y| + + {(d1 + d2)c1|y|2 + 2d1 C1|y|2|Z| — 4d2|y|2} (4 — c1 )|.
Using the fact that |Z| < 1 in the above iequality, we obtain
4
¿1 C1C3 + d2c2
< | (d1 + d2)c4 + 2d1 C1(4 — c1) + 2(d1 + ¿2)c1(4 — c1)|y| + + {(d1 + d2)c1 + 2d1d — 4d2} (4 — c2)|y|2|. (15)
Using the values of d1, d2 given in (14), we can write
¿1 + ¿2 = 1 and {(¿1 + ¿2 )c2 + 2d1 C1 — 4^2} = = c1 + 2(p + 2)2c1 + 4(p + 1)(p + 3).
(16)
Substituting the values from (16) and value of d1 from (14) to the right-hand side of (15), we have
4
¿1C1C3 + ¿2 c2
< |c4 + 2(p + 2)2c1 (4 — c2) + 2c1 (4 — c1)|y| +
+ {c1 + 2(p + 2)2C1 + 4(p + 1)(p + 3)}(4 — c1 )|y|2|. (17)
Consider {c2 + 2(p + 2)2c1 + 4(p + 1)(p + 3)} =
{01 + (p + 2)2}2 — (Vp4 + 8p3 + 20p2 + 16p + 4)2
C1 + j(p + 2)2 + ( Vp4 + 8p3 + 20p2 + 16p + 4
x
x
C1 + {(p + 2)2 — (vV + 8p3 + 20p2 + 16p + 4)
4
Noting that (cl + a)(cl + b) > (cl - a)(cl - b), where a, b > 0, and cl e [0, 2] in the above expression, we obtain
{c2 + 2(p + 2)2ci + 4(p + 1)(p + 3)} >
> {c2 - 2(p + 2)2ci + 4(p +1)(p + 3)} . (18)
From expressions (17) and (18), we get
d1 c1c3 + d2c2 < |c1 + 2(p + 2)2c1 (4 - c2) + 2c1(4 - cf)|y| +
+ {c2 - 2(p + 2)2c1 + 4(p + 1)(p + 3)} (4 - c2)|y|2|. (19)
Choosing c1 = c G [0, 2], replacing |y| by ß on the right-hand side of (19), we obtain
4
d1 c1c3 + d2 c
< [e4 + 2(p + 2)2c(4 - e2) + 2e2(4 - e2)i+
+ {e2 - 2(p + 2)2e + 4(p + 1)(p + 3)} (4 - c2)i2] =
= F(e, i) , 0 < i = |y| < 1 and 0 < e < 2. (20)
Next, we maximize function F (e, i) on the closed region [0, 2] x [0,1]. Differentiating F (e, i) given in the right-hand side of (20) partially with respect to i, we get
d F
= 2 [e2 + {e2 - 2(p + 2)2e + 4(p + 1)(p + 3)} ¡i] (4 - e2). (21)
For 0 < i < 1, for fixed e with 0 < e < 2 and p G N, from (21), we observe that ^f^ > 0. Therefore, F(e, i) becomes an increasing function of i and hence it cannot have a maximum value at any point in the interior of the closed region [0, 2] x [0,1]. The maximum value of F (e, i) occurs on the boundary i.e., when i = 1. Therefore, for fixed e G [0, 2], we have
max F(e,i)= F(e, 1) = G(e). (22)
0<^<1
Replacing i by 1 in F (e, i), it simplifies to
G(e) = -2e4 - 4p(p + 4)e2 + 16(p + 1)(p + 3), (23)
G'(e) = -8e3 - 8p(p + 4)e. (24)
From (24), we observe that G'(c) < 0 for every c G [0, 2] with p G N. Consequently, G(c) becomes a decreasing function of c, whose maximum value occurs at c = 0 only. From (23), the maximum value of G(c) at c = 0 is obtained to be
Gmax = G(0) = 16(p + 1)(p + 3). (25)
Simplifying expressions (20) and (25), we get
dicica + d2c2 < 4(p +1)(p + 3). (26)
From relations (13) and (26), along with the value of t in (14), upon simplification, we obtain
|aP+iap+3 - aP+2|<
2p(1 - a) p + 2
(27)
By setting c1 = c = 0 and selecting y = 1 in the expressions (7) and (9), we find that c2 = 2 and c3 = 0, respectively. Substituting the values c2 = 2, c3 = 0, and d2 = — (p + 1)(p + 3) in (26), we observe that equality is attained, which shows that our result is sharp. For the values c2 = 2 and c3 = 0, from (6) we derive the extremal function given by
1 + z2
g(z) = 1 + 2z2 + 2z4 + ... = 1+^.
1 — z2
So that from (10), we have
f' (z) — pazp-1 , 2 4 1 + z2
J K 1 1 = 1 + 2z2 + 2z4 + ... =
p(1 — a)zp-1 ' ' 1 — z
This completes the proof of our Theorem.
Remark 1. If p = 1 and a = 0 in (27) then |a2a4 — a^ | < |; this coincides with the result of Janteng et al. [7].
a
¿3! < 9
same as that of Vamshee Krishna and RamReddy [14].
Remark 2. If p = 1 in (27) then |a2a4 — a3| < 4(19a) , this result is
2p
Remark 3. If a = 0 in (27) then |ap+1 ap+3 — ap+2| < p+P^ , this result
coincides with the result obtained by Vamshee Krishna and RamReddy [15].
2
Acknowledgment. The authors express sincere thanks to the esteemed
Referee(s) for their careful readings, valuable suggestions and comments,
which helped to improve the paper.
References
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[4] Noor K. I. Hankel determinant problem for the class of functions with bounded boundary rotation. Rev. Roumaine Math. Pures Appl., 1983, vol. 28, no. 8, pp. 731-739.
[5] Layman J. W. The Hankel transform and some of its properties J. Integer Seq., 2001, vol. 4, no. 1, pp. 1-11.
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[8] MacGregor T. H. Functions whose derivative have a positive real part. Trans. Amer. Math. Soc., 1962, vol. 104, no. 3, pp. 532-537.
[9] Duren P. L. Univalent functions. vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, USA, 1983.
[10] Pommerenke Ch. Univalent functions. Gottingen: Vandenhoeck and Ruprecht, 1975.
[11] Simon B. Orthogonal polynomials on the unit circle, part 1. Classical theory. vol. 54, American mathematical society colloquium publications. Providence (RI): American Mathematical Society; 2005.
[12] Grenander U., Szego G. Toeplitz forms and their applications. 2nd ed. New York (NY): Chelsea Publishing Co.; 1984.
[13] Libera R. J., Zlotkiewicz E. J. Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 1983, vol. 87, pp. 251257.
[14] Vamshee Krishna D., RamReddy T. Coefficient inequality for a function whose derivative has a positive real part of order alpha. Mathematica Bohemica, 2015, vol. 140, no. 1, pp. 43-52.
[15] Vamshee Krishna D., RamReddy T. Coefficient inequality for certain p-valent analytic functions. Rocky Mountain J. Math., 2014, vol. 44, no. 6, pp. 1941-1959.
Received January 10, 2016.
In revised form, July 03, 2016.
Accepted July 03, 2016.
GIT, GITAM University Visakhapatnam 530 045, A. P., India E-mail: vamsheekrishna1972@gmail.com
Kakatiya University Warangal 506 009, T. S., India E-mail: reddytr2@gmail.com
