Armenian Journal of Mathematics Volume 4, Number 2, 2012, 98–105
COEFFICIENT INEQUALITY FOR CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS D. Vamshee Krishna* and T. Ramreddy** * Department of Mathematics, School of Humanities and Sciences, K L University, Green Fields, Vaddeswaram-522502, Guntur Dt., Andhra Pradesh, India.
[email protected]
** Department of Mathematics, Kakatiya University, Vidyaranyapuri-506009, Warangal Dt., Andhra Pradesh, India.
[email protected]
Abstract The objective of this paper is to an obtain an upper bound to the second Hankel determinant |a2 a4 − a23 | for the function f , belonging to a certain subclass of analytic functions, using Toeplitz determinants. Key Words: Analytic function, upper bound, second Hankel determinant, positive real function, Toeplitz determinants. Mathematics Subject Classification 2000: 30C45; 30C50.
Introduction Let A denote the class of functions f of the form f (z) = z +
∞ X n=2
98
an z n
(1)
in the open unit disc E = {z : |z| < 1}. Let S be the subclass of A consisting of univalent functions. In 1976, Noonan and Thomas [13] defined the q th Hankel determinant of f for q ≥ 1 and n ≥ 1 as
Hq (n) =
an an+1 .. . an+q−1
an+1 · · · an+2 · · · .. .. . . an+q · · ·
an+q−1 an+q . .. .
(2)
an+2q−2
This determinant has been considered by several authors. For example, Noor [14] determined the rate of growth of Hq (n) as n → ∞ for the functions in S with bounded boundary. Ehrenborg [4] studied the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence and some of its properties were discussed by Layman in [9]. One can easily observe that the Fekete-Szeg¨ o functional is H2 (1). Fekete-Szeg¨ o then further 2 generalized the estimate |a3 − µa2 | with µ real and f ∈ S. Ali [2] found sharp bounds on the first four coefficients and sharp estimate for the Fekete-Szeg¨ o functional |γ3 − tγ22 |, where P n t is real, for the inverse function of f defined as f −1 (w) = w + ∞ n=2 γn w to the class of f (α). For our discussion in strongly starlike functions of order α(0 < α ≤ 1) denoted by ST this paper, we consider the Hankel determinant in the case of q = 2 and n = 2, known as the second Hankel determinant a2 a3 = |a2 a4 − a23 |. a3 a4
(3)
Janteng, Halim and Darus [8] have considered the functional |a2 a4 −a23 | and found a sharp bound for the function f in the subclass RT of S, consisting of functions whose derivative has a positive real part studied by Mac Gregor [10]. In their work, they have shown that if f ∈ RT then |a2 a4 − a23 | ≤ 49 . These authors [7] also obtained the second Hankel determinant and sharp bounds for the familiar subclasses of S, namely, starlike and convex functions denoted by ST and CV and shown that |a2 a4 − a23 | ≤ 1 and |a2 a4 − a23 | ≤ 81 respectively. Mishra and Gochhayat [11] have obtained the sharp bound to the non- linear functional |a2 a4 − a23 | for the class ofn analytic ofunctions denoted by Rλ (α, ρ)(0 ≤ ρ ≤ 1, 0 ≤ λ < 1, |α| < π2 ), λ
defined as Re eiα Ωz fz (z)
> ρ cos α, using the fractional differential operator denoted by
Ωλz , defined by Owanand Srivastava [15]. oThese authors have shown that, if f ∈ Rλ (α, ρ) 2 2 2 2 then |a2 a4 − a23 | ≤ (1−ρ) (2−λ)9(3−λ) cos α . Similarly, the same coefficient inequality was calculated for certain subclasses of analytic functions by many authors ([1], [3], [12]). Motivated by the above mentioned results obtained by different authors in this direction, in this paper, we consider a certain subclass of analytic functions and obtain an upper bound to the functional |a2 a4 − a23 | for the function f belonging to this class , defined as follows. Definition1.1.A function f (z) ∈ A is said to be in the class Q(α, β, γ) with α , β > 0 and 99
0 ≤ γ < α + β ≤ 1, if it satisfies the condition that f (z) 0 Re α + βf (z) ≥ γ, z
∀z ∈ E
(4)
This class was considered and studied by Zhi- Gang Wang, Chun-yi gao and Shao-Mou yuan [17]. We first state some preliminary Lemmas required for proving our result.
1
Preliminary Results
Let P denote the class of functions p analytic in E for which Re{p(z)} > 0, " # ∞ X p(z) = (1 + c1 z + c2 z 2 + c3 z 3 + ...) = 1 + cn z n , ∀z ∈ E.
(5)
n=1
Lemma 1 ([16]) If p ∈ P, then |ck | ≤ 2, for each k ≥ 1. Lemma 2 ([5]) The power series for p given in (5) converges in the unit disc E to a function in P if and only if the Toeplitz determinants 2 Dn =
c−1 .. .
c1 2 .. .
··· ··· .. .
c2 c1 .. .
c−n c−n+1 c−n+2 · · ·
cn cn−1 .. .
, n = 1, 2, 3....
2
P and c−k = ck , are all non-negative. They are strictly positive except for p(z) = m k=1 ρk p0 (exp(itk )z), . ρk > 0, tk real and tk 6= tj , for k 6= j; in this case Dn > 0 for n < (m − 1) and Dn = 0 for n ≥ m. This necessary and sufficient condition is due to Caratheodory and Toeplitz, can be found in [5]. We may assume without restriction that c1 > 0. On using Lemma 2.2, for n = 2 and n = 3 respectively, we get 2 c1 c2 D2 = c1 2 c1 = [8 + 2Re{c21 c2 } − 2 | c2 |2 − 4c21 ] ≥ 0, c2 c1 2 which is equivalent to 2c2 = {c21 + x(4 − c21 )},
for some x, |x| ≤ 1.
2 c1 c2 c3 c1 2 c1 c2 D3 = . c2 c1 2 c1 c3 c2 c1 2 100
(6)
Then D3 ≥ 0 is equivalent to |(4c3 − 4c1 c2 + c31 )(4 − c21 ) + c1 (2c2 − c21 )2 ≤ 2(4 − c21 )2 − 2|(2c2 − c21 )|2 .
(7)
From the relations (6) and (7), after simplifying, we get 4c3 = {c31 + 2c1 (4 − c21 )x − c1 (4 − c21 )x2 + 2(4 − c21 )(1 − |x|2 )z} for some real value of
2
z, with
|z| ≤ 1. (8)
Main Result
Theorem 1 If f (z) = z + then
P∞
n=2
an z n ∈ Q(α, β, γ), (α, β > 0
|a2 a4 −
a23 |
and
0 ≤ γ < α + β ≤ 1)
4(α + β − γ)2 ≤ . (α + 3β)2
P n Proof. Since f (z) = z + ∞ n=2 an z ∈ Q(α, β, γ), from the definition 1.1, there exists an analytic function p ∈ P in the unit disc E with p(0) = 1 and Re{p(z)} > 0 such that f (z) αf (z) + βzf 0 (z) − γz 0 α + βf (z) − γ = p(z) ⇒ = p(z) (9) z (α + β − γ)z Replacing f (z), f 0 (z) and p(z) with their equivalent series expressions in the relation (9), we have " ( ) ( ) # "( )# ∞ ∞ ∞ X X X α z+ an z n + βz 1 + nan z n−1 − γz = (α + β − γ)z 1+ cn z n n=2
n=2
n=1
Upon simplification, we obtain
(α + 2β)a2 + (α + 3β)a3 z + (α + 4β)a4 z 2 + ... = (α + β − γ) c1 + c2 z + c3 z 2 + ... . (10)
Equating the coefficients of like powers of z 0 , z 1 and z 2 respectively on both sides of (10), we get (α + β − γ) (α + β − γ) (α + β − γ) c 1 ; a3 = c 2 ; a4 = c3 ] (11) [a2 = (α + 2β) (α + 3β) (α + 4β) Considering the second Hankel functional |a2 a4 − a23 | for the function f ∈ Q(α, β, γ) and substituting the values of a2 , a3 and a4 from the relation (11), we have 2 (α + β − γ) (α + β − γ) (α + β − γ) 2 2 c1 × c3 − c |a2 a4 − a3 | = (α + 2β) (α + 4β) (α + 3β)2 2 Upon simplification, we obtain |a2 a4 − a23 | =
(α + β − γ)2 × |(α + 3β)2 c1 c3 − (α + 2β)(α + 4β)c22 | (12) (α + 2β)(α + 3β)2 (α + 4β) 101
The expression (12) is equivalent to |a2 a4 − a23 | =
(α + β − γ)2 d1 c1 c3 + d2 c22 . × (α + 2β)(α + 3β)2 (α + 4β)
(13)
Where {d1 = (α + 3β)2 ; d2 = −(α + 2β)(α + 4β)}
(14)
Substituting the values of c2 and c3 from (6) and (8) respectively from Lemma 2 in the right hand side of (13), we have d1 c1 c3 + d2 c22 = |d1 c1 × 1 {c31 + 2c1 (4 − c21 )x − c1 (4 − c21 )x2 + 2(4 − c21 )(1 − |x|2 )z}+ 4 1 d2 × {c21 + x(4 − c21 )}2 |. 4 Using the facts that |z| < 1 and |xa + yb| ≤ |x||a| + |y||b|, where x, y, a and b are real numbers, after simplifying, we get 4 d1 c1 c3 + d2 c22 ≤ |(d1 + d2 )c41 + 2d1 c1 (4 − c21 ) + 2(d1 + d2 )c21 (4 − c21 )|x|− (d1 + d2 )c21 + 2d1 c1 − 4d2 (4 − c21 )|x|2 |. (15) Using the values of d1 , d2 , d3
and d4 from the relation (14), upon simplification, we obtain
{(d1 + d2 ) = β 2 ; d1 = (α + 3β)2 }; (d1 + d2 )c21 + 2d1 c1 − 4d2 = β 2 c21 + 2(α + 3β)2 c1 + 4(α + 2β)(α + 4β) .
(16) (17)
Consider
β 2 c21 + 2(α + 3β)2 c1 + 4(α + 2β)(α + 4β) 4(α + 2β)(α + 4β) 2(α + 3β)2 2 2 c1 + . = β c1 + β2 β2 # " 4 2 2 (α + 3β) 4(α + 2β)(α + 4β) (α + 3β) − . = β2 + c1 + β2 β4 β2 = β2
"
(α + 3β)2 c1 + β2
s
2 −
# α4 + 49β 4 + 50α2 β 2 + 84αβ 3 + 12α3 β . β4
After simplifying, the above expression reduces to
β 2 c21 + 2(α + 3β)2 c1 + 4(α + 2β)(α + 4β) s " ( )# 2 4 + 49β 4 + 50α2 β 2 + 84αβ 3 + 12α3 β (α + 3β) α = β 2 c1 + + × β2 β4 s " ( )# (α + 3β)2 α4 + 49β 4 + 50α2 β 2 + 84αβ 3 + 12α3 β c1 + − . (18) β2 β4 102
Since c1 ∈ [0, 2], using the result (c1 + a)(c1 + b) ≥ (c1 − a)(c1 − b), where a, b ≥ 0 in the right hand side of (18), upon simplification, we obtain
β 2 c21 + 2(α + 3β)2 c1 + 4(α + 2β)(α + 4β) ≥ β 2 c21 − 2(α + 3β)2 c1 + 4(α + 2β)(α + 4β) (19)
From the relations (17) and (19), we get − (d1 + d2 )c21 + 2d1 c1 − 4d2 ≤ − β 2 c21 − 2(α + 3β)2 c1 + 4(α + 2β)(α + 4β) . (20) Substituting the calculated values from (16) and (20) in the right hand side of (15), we obtain 4|d1 c1 c3 + d2 c22 | ≤ |β 2 c41 + 2(α + 3β)2 c1 (4 − c21 ) + 2β 2 c21 (4 − c21 )|x| − β 2 c21 − 2(α + 3β)2 c1 + 4(α + 2β)(α + 4β) (4 − c21 )|x|2 |. Choosing c1 = c ∈ [0, 2], applying Triangle inequality and replacing | x | by µ in the right hand side of the above inequality, it reduces to 4|d1 c1 c3 + d2 c22 | ≤ [β 2 c4 + 2(α + 3β)2 c(4 − c2 ) + 2β 2 c2 (4 − c2 )µ + β 2 c2 − 2(α + 3β)2 c + 4(α + 2β)(α + 4β) (4 − c2 )µ2 ]. = F (c, µ),
for 0 ≤ µ = |x| ≤ 1. (21)
Where F (c, µ) = [β 2 c4 + 2(α + 3β)2 c(4 − c2 ) + 2β 2 c2 (4 − c2 )µ + β 2 c2 − 2(α + 3β)2 c + 4(α + 2β)(α + 4β) (4 − c2 )µ2 ]. (22) We next maximize the function F (c, µ) on the closed square [0, 2] × [0, 1]. Differentiating F (c, µ) in (22) partially with respect to µ, we get ∂F = [2β 2 c2 (4 − c2 ) + 2 β 2 c2 − 2(α + 3β)2 c + 4(α + 2β)(α + 4β) (4 − c2 )µ]. (23) ∂µ For 0 < µ < 1 , for fixed c with 0 < c < 2 and α, β > 0, from (23), we observe that ∂F > 0. ∂µ Consequently, F (c, µ) is an increasing function of µ and hence it cannot have a maximum value at any point in the interior of the closed square [0, 2] × [0, 1]. Further, for a fixed c ∈ [0, 2], we have max F (c, µ) = F (c, 1) = G(c)(say). (24) 0≤µ≤1
From the relations (22) and (24), upon simplification, we obtain G(c) = F (c, 1) = −2β 2 c4 − 4β(α2 + 5β 2 + 6αβ)c2 + 16(α + 2β)(α + 4β) . 103
(25)
G0 (c) = −8β 2 c3 − 8β(α2 + 5β 2 + 6αβ)c . G00 (c) = −24β 2 c2 − 8β(α2 + 5β 2 + 6αβ) .
(26) (27)
From the expression (26), we observe that G0 (c) ≤ 0, for all values of c ∈ [0, 2] and for fixed values of α, β > 0, where (0 < α + β ≤ 1).Therefore, G(c) is a monotonically decreasing function of c in 0 ≤ c ≤ 2, attains its maximum value at c = 0. From the expression (25), we have G-maximum value at c = 0 is given by max G(c) = G(0) = 16(α + 2β)(α + 4β).
0≤c≤2
(28)
Considering, only the maximum value of G(c) at c = 0, from the relations (21) and (28), after simplifying, we get |d1 c1 c3 + d2 c22 | ≤ 4(α + 2β)(α + 4β). From the expressions (13) and (29), upon simplification, we obtain 4(α + β − γ)2 2 |a2 a4 − a3 | ≤ . (α + 3β)2
(29)
(30)
This completes the proof of our Theorem. Remark. For the choice of α = (1 − σ). β = σ and γ = 0, we get(α, h β, γ) i= ((1 − σ), σ, 0), 4 2 for which, from (30), upon simplification, we obtain |a2 a4 − a3 | ≤ (1+2σ) 2 , for 0 ≤ σ ≤ 1. This result is a special case to that of Murugusundaramoorthy and Magesh [12]. Acknowledgments The authors would like to thank the esteemed referee for his/her valuable suggestions and comments in the preparation of this paper.
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