Estimate on the Second Hankel Functional for a Subclass of Close-to-Convex Functions
with Respect to Symmetric Points
Chuah Puoi Choo and Aini Janteng
School of Science and Technology, Universiti Malaysia Sabah Jalan UMS, 88400 Kota Kinabalu, Sabah, Malaysia
[email protected], aini−[email protected] Abstract
Let S be the class of functions which are analytic, normalised and univalent in the open unit disc D = {z:|z|<1}. In [4], Janteng in- troduced a subclass of close-to-convex functions with respect to (w.r.t) symmetric points denoted by Ks(α), 0≤α <1. In this paper, we give the upper bound for the second Hankel determinant for this particular class of functions.
Mathematics Subject Classification: Primary 30C45
Keywords: close-to-convex w.r.t symmetric points, Hankel determinant, upper bound
1 Introduction
Let S be the class of functionsf which are analytic and univalent in the open unit disc D={z:|z|<1}given by
f(z) =z+ ∞ n=2
anzn (1)
where an is a complex number. In 1976, Noonan and Thomas in [7] defined the qthHankel determinant of f for q≥1 and n≥1 by
Hq(n) =
an an+1 . . . an+q−1 an+1 an+2 . . . an+q
... ... ... ... an+q−1 an+q . . . an+2q−2
This determinant has been considered by several authors. For example, Noor in [8] studied the rate of growth of Hq(n) as n → ∞ for functions f given by (1) with bounded boundary. Later, Ehrenborg in [1] considered the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence and some its properties were discussed thoroughly by Layman in [6].
Observe that the well-known Fekete and Szeg¨o functional isH2(1). Fekete and Szeg¨o in [2] studied the estimates |a3 −μa22| for μ real for f ∈ S. For our discussion in this paper, the Hankel determinant for the case q = 2 and n= 2 are being considered i.e. H2(2).
In [5], Janteng at. el. determined the functional |a2a4−μa23| for the functions f ∈ R. In this paper, we seek upper bound for the functional |a2a4 −μa23| where the μ is real for functions f ∈ Ks(α). The class Ks(α) is defined as follows:
Definition 1.1 ([4]) Let f be given by (1). Then f ∈ Ks(α),0 ≤ α < 1, if there exists a g ∈Cs such that for z ∈D,
Re
2αzf(z)
(g(z)−g(−z)) + 2f(z) (g(z)−g(−z))
>0.
Note: The definition above is also equivalent to the following:
f ∈Ks(α), if there exists ah=zg ∈Ss∗ such that Re
2αz2f(z)
h(z)−h(−z) + 2zf(z) h(z)−h(−z)
>0. (2)
2 Preliminary Results
LetP be the family of all functionspanalytic inD for whichRe p(z)>0 and p(z) = 1 +c1z+c2z2+. . . (3) for z ∈D.
Lemma 2.1 ([9]) If p∈P then |ck| ≤2 for each k.
Lemma 2.2 ([3]) The power series for p(z) given by (3) converges in D to a function in P if and only if the Toeplitz determinants
Dn=
2 c1 c2 . . . cn c−1 2 c1 . . . cn−1
... ... ... ... ... cn c−n+1 c−n+2 . . . 2
, n = 1,2,3, . . . (4)
and c−k = ¯ck, are all negative. They are strictly positive except for p(z) = m
k=1ρkp0(eitkz), ρk>0, tk real andtk=tj fork =j; in this case Dn >0 for n < m−1 and Dn= 0 for n≥m.
This necessary and sufficient condition is due to Carath´eodory and Toeplitz and can be found in [3].
3 Main Result
Theorem 3.1 Let f ∈Ks(α), and 0≤α <1. Then,
|a2a4−μa23| ≤
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((2α+1)2−μ(α+1)(3α+1))3
((2α+1)2−2μ(α+1)(3α+1))2(α+1)(2α+1)2(3α+1)
− μ((2α+1)2−μ(α+1)(3α+1))2
((2α+1)2−2μ(α+1)(3α+1))2(2α+1)2 − (2α+1)μ 2, if μ≤0;
(2α+1)2−μ(α+1)(3α+1)
(α+1)(2α+1)2(3α+1) , if 0≤μ≤ 2(α+1)(3α+1)(2α+1)2 ;
(2α+1)μ 2 if 2(α+1)(3α+1)(2α+1)2 ≤μ≤ (α+1)(3α+1)(2α+1)2 ;
− ((2α+1)2−μ(α+1)(3α+1))3
((2α+1)2−2μ(α+1)(3α+1))2(α+1)(2α+1)2(3α+1)
+ μ((2α+1)2−μ(α+1)(3α+1))2
((2α+1)2−2μ(α+1)(3α+1))2(2α+1)2 +(2α+1)μ 2, if μ≥ (α+1)(3α+1)(2α+1)2 . Proof.
Since h∈Ss∗, it follows from (2) that ∃p∈P such that
2zh(z) = (h(z)−h(−z))p(z) (5) for some z ∈D. Equating coefficients in (5) yields
b2 = c1
2, b3 = c2
2, b4 = c3
4 + c1c2
8 (6)
It also follows from (2) that
2αz2f(z) + 2zf(z) = (h(z)−h(−z))p(z) (7) Equating coefficients in (7) yields
2(α+ 1)a2 =c1, 3(2α+ 1)a3 =c2+b3, 4(3α+ 1)a4 =c3+b3c1 (8) From (6) and (8),
|a2a4−μa23|=
c1c3
8(α+ 1)(3α+ 1) + c21c2
16(α+ 1)(3α+ 1) − μc22 4(2α+ 1)2
(9)
We make use of Lemma 2.2 to obtain the proper bound on (9). We begin by rewriting (4) for the cases n = 2 and n= 3,
D2 =
2 c1 c2 c1 2 c1
¯c2 2 c1
= 8 + 2Rec21 −2|c2|2−4c21 ≥0, which is equivalent to
2c2 =c21+x(4−c21) (10) for some x,|x| ≤1.
Further, D3 ≥0 is equivalent to
|(4c3−4c1c2+c31)(4−c21) +c1(2c2−c21)2| ≤2(4−c21)2−2|2c2−c21|2 (11) and from (11) and (10), we have
4c3 =c31+ 2(4−c21)c1x−c1(4−c21)x2 + 2(4−c21)(1− |x|2)z, (12) for some value of z,|z| ≤1.
Suppose c1 =cand c∈[0,2]. Using (10) and (12), we obtain c1c3
8(α+ 1)(3α+ 1) + c21c2
16(α+ 1)(3α+ 1)− μc22 4(2α+ 1)2
=
((2α+ 1)2−μ(α+ 1)(3α+ 1))c4
16(α+ 1)(2α+ 1)2(3α+ 1) + (3(2α+ 1)2 −4μ(α+ 1)(3α+ 1))c2(4−c2)x 32(α+ 1)(2α+ 1)2(3α+ 1)
−((2α+ 1)2c2+ 2μ(α+ 1)(3α+ 1)(4−c2))(4−c2)x2
32(α+ 1)(2α+ 1)2(3α+ 1) + c(4−c2)(1− |x|2)z 16(α+ 1)(3α+ 1)
≤ |(2α+ 1)2−μ(α+ 1)(3α+ 1)|c4
16(α+ 1)(2α+ 1)2(3α+ 1) + c(4−c2) 16(α+ 1)(3α+ 1)
+|3(2α+ 1)2−4μ(α+ 1)(3α+ 1)|c2(4−c2)ρ 32(α+ 1)(2α+ 1)2(3α+ 1)
+(|(2α+ 1)2−2μ(α+ 1)(3α+ 1)|c2−2(2α+ 1)2c+ 8|μ|(α+ 1)(3α+ 1))(4−c2)ρ2 32(α+ 1)(2α+ 1)2(3α+ 1)
≡F(ρ)
(13)
with ρ=|x| ≤1 and α >0. This gives rise to
F(ρ) =
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⎩
(3(2α+1)2−4μ(α+1)(3α+1))2c2(4−c2) 32(α+1)(2α+1)2(3α+1)
+{[(2α+1)2−2μ(α+1)(3α+1)]c2−2(2α+1)2c−8μ(α+1)(3α+1)}(4−c2)ρ
16(α+1)(2α+1)2(3α+1) ifμ≤0;
[3(2α+1)2−4μ(α+1)(3α+1)]c2(4−c2) 32(α+1)(2α+1)2(3α+1)
+{[(2α+1)2−2μ(α+1)(3α+1)]c2−2(2α+1)2c+8μ(α+1)(3α+1)}(4−c2)ρ
16(α+1)(2α+1)2(3α+1) if 0≤μ≤2(α+1)(3α+1)(2α+1)2 ;
[3(2α+1)2−4μ(α+1)(3α+1)]c2(4−c2) 32(α+1)(2α+1)2(3α+1)
+{[2μ(α+1)(3α+1)−(2α+1)2]c2−2(2α+1)2c+8μ(α+1)(3α+1)}(4−c2)ρ
16(α+1)(2α+1)2(3α+1) if 2(α+1)(3α+1)(2α+1)2 ≤μ≤ 4(α+1)(3α+1)3(2α+1)2 ;
[4μ(α+1)(3α+1)−3(2α+1)2]c2(4−c2) 32(α+1)(2α+1)2(3α+1)
+{[2μ(α+1)(3α+1)−(2α+1)2]c2−2(2α+1)2c+8μ(α+1)(3α+1)}(4−c2)ρ
16(α+1)(2α+1)2(3α+1) ifμ≥ 4(α+1)(3α+1)3(2α+1)2 .
and again for all the cases above, F(ρ)> 0 for ρ >0 and consequently F is an increasing function and MaxρF(ρ) =F(1).
Now, let G(c) =F(1)
= |(2α+ 1)2−μ(α+ 1)(3α+ 1)|c4
16(α+ 1)(2α+ 1)2(3α+ 1) + c(4−c2) 16(α+ 1)(3α+ 1) +|3(2α+ 1)2−4μ(α+ 1)(3α+ 1)|c2(4−c2)
32(α+ 1)(2α+ 1)2(3α+ 1)
+{|(2α+ 1)2−2μ(α+ 1)(3α+ 1)|c2−2(2α+ 1)2c+ 8|μ|(α+ 1)(3α+ 1)}(4−c2) 32(α+ 1)(2α+ 1)2(3α+ 1)
(14) (i) First, let us consider the case μ≤0.
Eq. (14) gives
G(c) = −c{[(2α+ 1)2−2μ(α+ 1)(3α+ 1)]c2−4[(2α+ 1)2−μ(α+ 1)(3α+ 1)]} 4(α+ 1)(2α+ 1)2(3α+ 1)
Elementary calculation reveals that G attains its maximum value at c = 4[(2α+1)2−μ(α+1)(3α+1)]
(2α+1)2−2μ(α+1)(3α+1) .
The upper bound for (13) corresponds toρ= 1 andc=
4[(2α+1)2−μ(α+1)(3α+1)]
(2α+1)2−2μ(α+1)(3α+1) , in which case
c1c3
8(α+ 1)(3α+ 1) + c21c2
16(α+ 1)(3α+ 1) − μc22 4(2α+ 1)2
≤ ((2α+ 1)2−μ(α+ 1)(3α+ 1))3
((2α+ 1)−2μ(α+ 1)(3α+ 1))2(α+ 1)(2α+ 1)2(3α+ 1)
− μ((2α+ 1)2−μ(α+ 1)(3α+ 1))2
((2α+ 1)2−2μ(α+ 1)(3α+ 1))2(2α+ 1)2− μ (2α+ 1)2
(15)
(ii) Secondly, we consider the case 0≤μ≤ 2(α+1)(3α+1)(2α+1)2 . This gives
G(c) = −c[(2α+ 1)2−2μ(α+ 1)(3α+ 1)](c2−4) 4(α+ 1)(2α+ 1)2(3α+ 1)
where G attains its maximum value at c= 2. Hence, we obtain c1c3
8(α+ 1)(3α+ 1) + c21c2
16(α+ 1)(3α+ 1)− μc22 4(2α+ 1)2
≤ (2α+ 1)2−μ(α+ 1)(3α+ 1) (α+ 1)(2α+ 1)2(3α+ 1)
(iii) To prove the third result, we consider two cases.
First, consider that 2(α+1)(3α+1)(2α+1) ≤μ≤ 4(α+1)(3α+1)3(2α+1) . From eq.(14), we have
G(c) = c[(2α+ 1)2−2μ(α+ 1)(3α+ 1)]
2(α+ 1)(2α+ 1)2(3α+ 1) In this case, Gattains its maximum value at c= 0.
Next, consider the case 4(α+1)(3α+1)3(2α+1)2 ≤ μ ≤ (α+1)(3α+1)(2α+1)2 . For this case, Eq.(14) gives rise to
G(c) = c{[3(2α+ 1)2 −4μ(α+ 1)(3α+ 1)]c2−4[(2α+ 1)2−μ(α+ 1)(3α+ 1)]} 4(α+ 1)(2α+ 1)2(3α+ 1)
where G attains its maximum value at c= 0.
In both cases, the upper bound is attained as c1c3
8(α+ 1)(3α+ 1) + c21c2
16(α+ 1)(3α+ 1) − μc22 4(2α+ 1)2
≤ μ (2α+ 1)2
(iv) Finally, consider μ≥ (α+1)(3α+1)(2α+1)2 . Here, G attains its maximum value atc=
4[(2α+1)2−μ(α+1)(3α+1)]
(2α+1)2−2μ(α+1)(3α+1) . Hence, c1c3
8(α+ 1)(3α+ 1) + c21c2
16(α+ 1)(3α+ 1) − μc22 4(2α+ 1)2
≤ − ((2α+ 1)2−μ(α+ 1)(3α+ 1))3
((2α+ 1)2−2μ(α+ 1)(3α+ 1))2(α+ 1)(2α+ 1)2(3α+ 1) + μ((2α+ 1)2−μ(α+ 1)(3α+ 1))2
((2α+ 1)2−2μ(α+ 1)(3α+ 1))2(2α+ 1)2+ μ (2α+ 1)2
(16) This completes the proof of theorem.
Acknowledgement
This work was supported by FRG0268-ST-2/2010 Grant, Malaysia. The au- thors express their gratitude to the referee for his valuable comments.
References
[1] Ehrenborg, R.: The Hankel determinant of exponential polynomials, American Mathematical Monthly, 107 (2000): 557-560.
[2] Fekete, M. and Szeg¨o, G.: Eine bemerkung ¨uber ungerade schlichte funk- tionen,J. London. Math. Soc., 8(1933): 85-89.
[3] Grenander, U. and Szeg¨o, G.: Toeplitz forms and their application,Berke- ley and Los Angeles: Univ. of California Press, (1958).
[4] Janteng, A.: Assortment of problems for certain classes of analytic func- tions,PhD Thesis, University of Malaya, Kuala Lumpur, Malaysia, (2006) [5] Janteng, A., Halim, S.A. and Darus, M.: Estimate on the second hankel functional for functions whose derivative has positive real part, Journal of Quality Measurement and Analysis, 4(1)(2008): 189-195.
[6] Layman, J.W.: The Hankel transform and some of its properties, J. of Integer Sequences, 4(2001): 1-11.
[7] Noonan, J.W. and Thomas, D.K.: On the second Hankel determinant of a really meanp-valent functions,Transaction of the American Mathematical Society, 223(1976): 337-346.
[8] Noor, K.I.: Hankel determinant problem for the class of function with bounded boundary rotation, Rev. Roum. Math. Pures et Appl., 28(8)(1983): 731-739.
[9] Pommerenke, Ch.: Univalent functions, Vandenhoeck and Ruprecht, G¨ottingen, (1975)
Received: October, 2012
