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SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β ˘ ERHAN DENIZ, MURAT C ¸ AGLAR, AND HALIT ORHAN
Abstract. In this paper we obtain upper bounds for the second Hankel determinant H2 (2) of the classes bi-starlike and bi-convex functions of order β, which we denote by Sσ∗ (β) and Kσ (β), respectively. In particular, the estimates for the second Hankel determinat H2 (2) of bi-starlike and bi-convex functions which are important subclasses of bi-univalent functions are pointed out.
1. Introduction and definitions Let A denote the family of functions f of the form (1.1)
f (z) = z +
∞ X
an z n
n=2
which are analytic in the open unit disk U = {z : |z| < 1} and let S denote the class of all functions in A which are univalent in U. The Koebe one-quarter theorem (see [7]) ensures that the image of U under every f ∈ S contain a disk of radius 1 4. So, every f ∈ S has an inverse function f −1 satisfying f −1 (f (z)) = z (z ∈ U) and f (f −1 (w)) = w
(|w| < r0 (f ); r0 (f ) ≥ 1 4)
where f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 − (5a32 − 5a2 a3 + a4 )w4 + .... A function f ∈ A is said to be bi-univalent in U if both f (z) and f −1 (z) are univalent in U. Let σ denote the class of bi-univalent functions in U given by (1.1). Two of the most famous subclasses of univalent functions are the class S ∗ (β) of starlike functions of order β and the class K(β) of convex functions of order β. By definition, we have 0 zf (z) S ∗ (β) = f ∈ S : < > β; z ∈ U; 0 ≤ β < 1 f (z) and
zf 00 (z) K(β) = f ∈ S : < 1 + 0 > β; z ∈ U; 0 ≤ β < 1 . f (z) The classes consisting of starlike and convex functions are usually denoted by S ∗ = S ∗ (0) and K = K(0), respectively. For 0 ≤ β < 1, a function f ∈ σ is in the class Sσ∗ (β) of bi-starlike functions of order β, or Kσ (β) of bi-convex functions of order β if both f and its inverse map f −1 are, respectively, starlike or convex of order β. These classes were introduced by Brannan and Taha [2] in 1985. Especially the classes Sσ∗ (0) = Sσ∗ and Kσ (0) = Kσ are bi-starlike and bi-convex functions, respectively. In 1967, Lewin [17] showed that for every functions f ∈ σ of the form (1.1), the second coefficient√of f satisfy the inequality |a2 | < 1.51. In 1967, Brannan and Clunie [1] conjectured that |a2 | ≤ 2 for f ∈ σ. Later, Netanyahu [18] proved that maxf ∈σ |a2 | = 4/3. In 1985, Kedzierawski [13] proved Brannan and Clunie’s conjecture for f ∈ Sσ∗ . In 1985, Tan [25] obtained the bound for a2 namely |a2 | < 1.485 which is the best known estimate for functions in the class σ. Brannan and Taha [2] obtained estimates on the initial coefficients |a2 | and |a3 | for functions in the classes Sσ∗ (β) and Kσ (β). Recently, Deniz [6] extented and Kumar et al. [15] both extended and improved the results of Brannan and Taha [2] by generalizing their classes using subordination. The problem of estimating coefficients |an |, n ≥ 2 is still open. However, a lot of results for |a2 |, |a3 | and |a4 | were
2000 Mathematics Subject Classification. Primary 30C45, 30C50; Secondary 30C80. Key words and phrases. bi-univalent functions, bi-starlike functions of order β, bi-convex functions of order β, second Hankel determinant. 1
˘ ERHAN DENIZ, MURAT C ¸ AGLAR, AND HALIT ORHAN
2
proved for some subclasses of σ (see [3], [5], [9], [11], [21], [23], [24], [26], [27]). Unfortunatelly, they are not sharp. One of the important tools in the theory of univalent functions is Hankel Determinants which are useful, for example, in showing that a function of bounded characteristic in U, i.e., a function which is a ratio of two bounded analytic functions, with its Laurent series around the origin having integral coefficients, is rational [4]. The Hankel determinants [19] Hq (n) (n = 1, 2, ..., q = 1, 2, ...) of the function f are defined by an an+1 ... an+q−1 an+1 an+2 ... an+q Hq (n) = (a1 = 1). .. .. .. . . . an+q−1
an+q
...
an+2q−2
This determinant was discussed by several authors with q = 2. For example, we can know that the functional H2 (1) = a3 − a22 is known as the Fekete-Szeg¨o functional and they consider the further generalized functional a3 − µa22 where µ is some real number (see, [8]). In 1969, Keogh and Merkes [14] proved the Fekete-Szeg¨o problem for the classes S ∗ and K. Someone can see the Fekete-Szeg¨ o problem for the classes S ∗ (β) and K(β) at special cases in the paper of Orhan et.al. [20]. On the other hand, recently Zaprawa [28], [29] have studied on Fekete-Szeg¨o problem for some classes of bi-univalent functions. In special cases, he gave Fekete-Szeg¨o problem for the classes Sσ∗ (β) and Kσ (β). In 2014, Zaprawa [28] proved the following resuts for µ ∈ R, 3 1 1 − β; 2 ∗ 2 ≤µ≤ 2 f ∈ Sσ (β) ⇒ a3 − µa2 ≤ 3 2(1 − β) |µ − 1| ; µ ≥ 2 and µ ≤ 21 and f ∈ Kσ (β) ⇒ a3 − µa22 ≤
1−β 3 ;
2 3
≤ µ ≤ 43 (1 − β) |µ − 1| ; µ ≥ and µ ≤ 4 3
2 3
.
The second Hankel determinant H2 (2) is given by H2 (2) = a2 a4 − a23 . The bounds for the second Hankel determinant H2 (2) obtained for the classes S ∗ and K in [12]. Recently, Lee et al. [16] established the sharp bound to |H2 (2)| by generalizing their classes using subordination. In their paper, one can find the sharp bound to |H2 (2)| for the functions in the classes S ∗ (β) and K(β). In this paper, we seek upper bound for the functional H2 (2) = a2 a4 − a23 for functions f belongs to the classes Sσ∗ (β) and Kσ (β). Let P be the class of functions with positive real part consisting of all analytic functions P : U → C satisfying p(0) = 1 and 0. We need the following results about the functions belonging to the class P: Lemma 1.1. [22] If the function p ∈ P is given by the series p(z) = 1 + c1 z + c2 z 2 + ...
(1.2)
then the sharp estimate |ck | ≤ 2 (k = 1, 2, ...) holds. Lemma 1.2. [10] If the function p ∈ P is given by the series (1.2), then (1.3)
2c2
= c21 + x(4 − c21 )
(1.4)
4c3
2 = c31 + 2(4 − c21 )c1 x − c1 (4 − c21 )x2 + 2(4 − c21 ) 1 − |x| z,
for some x, z with |x| ≤ 1 and |z| ≤ 1. 2. Main results Our first main result for the class
Sσ∗ (β)
is the following:
Theorem 2.1. Let f (z) given by (1.1) be in the class Sσ∗ (β), 0 ≤ β < 1. Then i h i h 1 4 1 − β 2 1 + 4 (1 − β)2 , β ∈ 0, 1 − √ 3 2 2 a2 a4 − a23 ≤ (2.1) (β−1)2 3 1 √ 1 − , 1 . , β ∈ 2 2 (1−2(1−β) ) 2 2
SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β
3
Proof. Let f ∈ Sσ∗ (β) and g = f −1 . Then (2.2)
zf 0 (z) wg 0 (w) = β + (1 − β)p(z) and = β + (1 − β)q(w) f (z) g(w)
where p(z) = 1 + c1 z + c2 z 2 + ... and q(w) = 1 + d1 w + d2 w2 + ... in P. Now, equating the coefficients in (2.2), we have (2.3)
a2
=
(1 − β)c1 ,
(2.4)
2a3 −
=
(1 − β)c2 ,
(2.5)
3a4 − 3a3 a2 +
a22 a32
=
(1 − β)c3
and −a2
=
(1 − β)d1 ,
− 2a3
=
(1 − β)d2 ,
−10a32 + 12a3 a2 − 3a4
=
(1 − β)d3 .
(2.6) 3a22
(2.7) (2.8)
From (2.3) and (2.6), we arrive at c1 = −d1
(2.9) and
a2 = (1 − β)c1 .
(2.10)
Now, from (2.4), (2.7) and (2.10), we get that (2.11)
2
a3 = (1 − β) c21 +
(1 − β) (c2 − d2 ) . 4
Also, from (2.5) and (2.8), we find that 2 5 1 3 2 (2.12) a4 = (1 − β) c31 + (1 − β) c1 (c2 − d2 ) + (1 − β) (c3 − d3 ) . 3 8 6 Thus, we can easily establish that a2 a4 − a23 = − 1 (1 − β)4 c41 + 1 (1 − β)3 c21 (c2 − d2 ) 3 8 1 1 2 2 2 (2.13) + (1 − β) c1 (c3 − d3 ) − (1 − β) (c2 − d2 ) . 6 16 According to Lemma 1.2 and (2.9), we write (2.14)
2c2 = c21 + x(4 − c21 ) 2d2 = d21 + x(4 − d21 )
=⇒ c2 − d2 = 0
and c31 c1 − c1 (4 − c21 )x − (4 − c21 )x2 . 2 2 Using (2.14) and (2.15) in (2.13), we have a2 a4 − a23 = − 1 (1 − β)4 c41 + 1 (1 − β)2 c41 3 12
(2.15)
(2.16)
c3 − d3 =
1 1 2 2 2 2 2 2 2 − (1 − β) c1 (4 − c1 )x − (1 − β) c1 (4 − c1 )x . 6 12
Since p ∈ P, so |c1 | ≤ 2. Letting c1 = c, we may assume without restriction that c ∈ [0, 2]. Thus, applying the triangle inequality on (2.16), with µ = |x| ≤ 1, we obtain a2 a4 − a23 ≤ 1 (1 − β)4 c4 + 1 (1 − β)2 c4 3 12 1 1 2 2 2 (2.17) + (1 − β) c (4 − c2 )µ + (1 − β) c2 (4 − c2 )µ2 = F (µ). 6 12 Differentiating F (µ), we get 1 1 2 2 F 0 (µ) = (1 − β) c2 (4 − c2 ) + (1 − β) c2 (4 − c2 )µ. 6 6
˘ ERHAN DENIZ, MURAT C ¸ AGLAR, AND HALIT ORHAN
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Using elementary calculus, one can show that F 0 (µ) > 0 for µ > 0. It implies that F is an increasing function and thus, the upper bound for F (µ) corresponds to µ = 1, in which case 1 1 1 4 2 2 (2.18) F (µ) ≤ (1 − β) c4 + (1 − β) c4 + (1 − β) c2 (4 − c2 ) = G(c). 3 12 4 Assume that G(c) has a maximum value in an interior of c ∈ [0, 2], by elementary calculation we find nh i o 2 2 2 (2.19) G0 (c) = (1 − β) 2 (1 − β) − 1 c3 + 3c . 3 q 3 Then G0 (c) = 0 implies the real critical point c01 = 0 or c02 = 1−2(1−β) 2. After some calculations following cases: h we concluded i 1 Case 1: When β ∈ 0, 1 − 2√2 , we observe that c02 ≥ 2, that is, c02 is out of the interval (0, 2). Therefore the maximum value of G(c) occurs at c01 = 0 or c = c02 which contradicts our assumption of having the maximum value at the interior point of c ∈ [0, 2]. Since G is an increasing function in the interval [0, 2], maximum point of G must be on the boundary of c ∈ [0, 2], that is, c = 2. Thus, we have i h 4 2 1 − β 2 1 + 4 (1 − β) . max G(c) = G(2) = 0≤c≤2 3 1 Case 2: When β ∈ 1 − 2√2 , 1 , we observe that c02 < 2, that is, c02 is interior of the interval [0, 2]. Since G00 (c02 ) < 0, the maximum value of G(c) occurs at c = c02 . Thus, we have ! s 2 3 (β − 1) 3 . max G(c) = G(c02 ) = G = 2 0≤c≤2 2 1 − 2 (1 − β)2 1 − 2 (1 − β) This completes the proof of the Theorem 2.1.
For α = 0, Theorem 2.1 readily yields the following coefficient estimates for bi-starlike functions. Corollary 2.2. Let f (z) given by (1.1) be in the class Sσ∗ . Then a2 a4 − a23 ≤ 20 . 3 Our second main result for the class Kσ (β) is following: Theorem 2.3. Let f (z) given by (1.1) be in the class Kσ (β), 0 ≤ β < 1. Then i h i h 1 1 − β 2 1 + (1 − β)2 , β ∈ 0, 1 − √1 6 2 a2 a4 − a23 ≤ (2.20) (1−β)2 3 1 √ , β ∈ 1 − , 1 . 8 (2−(1−β)2 ) 2 Proof. Let f ∈ Kσ (β) and g = f −1 . Then (2.21)
1+
wg 00 (w) zf 00 (z) = β + (1 − β)p(z) and 1 + = β + (1 − β)q(w) f 0 (z) g 0 (w)
where p(z) = 1 + c1 z + c2 z 2 + ... and q(w) = 1 + d1 w + d2 w2 + ... in P. Now, equating the coefficients in (2.21), we have (2.22)
2a2
=
(1 − β)c1 ,
(2.23)
6a3 −
=
(1 − β)c2 ,
(2.24)
12a4 − 18a3 a2 +
4a22 8a32
=
(1 − β)c3
and −2a2
=
(1 − β)d1 ,
− 6a3
=
(1 − β)d2 ,
−32a32 + 42a3 a2 − 12a4
=
(1 − β)d3 .
(2.25) 8a22
(2.26) (2.27)
From (2.22) and (2.25), we arrive at (2.28)
c1 = −d1
SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β
5
and 1 (1 − β)c1 . 2 Now, from (2.23), (2.26) and (2.29), we get that 1 1 2 (2.30) a3 = (1 − β) c21 + (1 − β) (c2 − d2 ) . 4 12 Also, from (2.24) and (2.27), we find that 5 5 1 3 2 (2.31) a4 = (1 − β) c31 + (1 − β) c1 (c2 − d2 ) + (1 − β) (c3 − d3 ) . 48 48 24 Thus, we can easily establish that a2 a4 − a23 = − 1 (1 − β)4 c41 + 1 (1 − β)3 c21 (c2 − d2 ) 96 96 1 1 2 2 2 (2.32) + (1 − β) c1 (c3 − d3 ) − (1 − β) (c2 − d2 ) . 48 144 (2.29)
a2 =
Using (2.14) and (2.15) in (2.32), we have a2 a4 − a23 = − 1 (1 − β)4 c41 + 1 (1 − β)2 c41 96 96 1 1 2 2 (2.33) − (1 − β) c21 (4 − c21 )x − (1 − β) c21 (4 − c21 )x2 . 48 96 Since p ∈ P, so |c1 | ≤ 2. Letting c1 = c, we may assume without restriction that c ∈ [0, 2]. Thus, applying the triangle inequality on (2.16), with µ = |x| ≤ 1, we obtain a2 a4 − a23 n o 1 1 2 2 2 ≤ (2.34) (1 − β) 1 + (1 − β) c4 + (1 − β) c2 (4 − c2 ) 2µ + µ2 = F (µ). 96 96 Differentiating F (µ), we get 1 1 2 2 F 0 (µ) = (1 − β) c2 (4 − c2 ) + (1 − β) c2 (4 − c2 )µ. 48 48 Using elementary calculus, one can show that F 0 (µ) > 0 for µ > 0. It implies that F is an increasing function and thus, the upper bound for F (µ) corresponds to µ = 1, in which case n o 1 1 2 2 2 (2.35) F (µ) ≤ (1 − β) 1 + (1 − β) c4 + (1 − β) c2 (4 − c2 ) = G(c). 96 32 Then, nh i o 1 2 2 (2.36) G0 (c) = (1 − β) (1 − β) − 2 c3 + 6c . 24 q 6 0 Setting G (c) = 0, since 0 ≤ c ≤ 2, we have the real critical poins c01 = 0 or c02 = 2−(1−β) 2. Proceeding similarlyhas in the iearlier proof, it follows that: Case 1: When β ∈ 0, 1 − √12 , the maximum value of G(c) corresponds to c = 2. i h 1 2 max G(c) = G(2) = 1 − β 2 1 + (1 − β) . 0≤c≤2 6 q 6 Case 2: When β ∈ 1 − √12 , 1 , the maximum value of G(c) corresponds to c = c02 = 2−(1−β) 2. ! s 2 6 3 (1 − β) . max G(c) = G(c02 ) = G = 2 0≤c≤2 8 2 − (1 − β)2 2 − (1 − β) This completes the proof of the Theorem 2.3.
For α = 0, Theorem 2.3 readily yields the following coefficient estimates for bi-convex functions. Corollary 2.4. Let f (z) given by (1.1) be in the class Kσ . Then a2 a4 − a23 ≤ 1 . 6
6
˘ ERHAN DENIZ, MURAT C ¸ AGLAR, AND HALIT ORHAN
Remark 2.5. We have noticed an interesting equality given by (2.14) in this paper. When we use this equality, especially for determined of upper bound of |a3 | , we see that bound is better than earlier exists. References [1] A. Brannan and J. G. Clunie, Aspects of contemporary complex analysis Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham, July 120, 1979, Academic Press New York, London, 1980. [2] D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions, in: S.M. Mazhar, A. Hamoui, N.S. Faour (Eds.), Math. Anal. and Appl., Kuwait; February 18–21, 1985, in: KFAS Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp. 53–60. see also Studia Univ. Babe¸s-Bolyai Math. 31 (2) (1986), 70–77. [3] S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions, C. R. Acad. Sci. Paris, Ser. I, 352 (6) (2014), pp. 479–484. [4] D. G. Cantor, Power series with integral coefficients, Bull. Amer. Math. Soc. 69 (1963), 362–366. [5] M. C ¸ a˘ glar, H. Orhan and N. Ya˘ gmur, Coefficient bounds for new subclasses of bi-univalent functions, Filomat 27:7 (2013), 1165-1171. [6] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal. 2(1) (2013), 49–60. [7] P. L. Duren, Univalent functions,Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York, 1983. [8] M. Fekete and G. Szeg¨ o, Eine Bemerkung uber ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85-89. [9] B.A. Frasin, M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011) 1569-1573. [10] U. Grenander and G. Szeg¨ o, Toeplitz forms and their applications, California Monographs in Mathematical Sciences Univ. California Press, Berkeley, 1958. [11] S. G. Hamidi, J. M. Jahangiri, Faber polynomial coefficient estimates for analytic bi-close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I 352(2014) 17–20. [12] A, Janteng, S. A. Halim and M. Darus, Hankel Determinant for starlike and convex functions. Int. J. Math. Anal. 1(13) (2007), 619-625. [13] A. W. Kedzierawski, Some remarks on bi-univalent functions, Ann. Univ.Mariae Curie-Sklodowska Sect. A 39 (1985), 77–81 (1988). [14] F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12. [15] S. S. Kumar, V. Kumar and V. Ravichandran, Estimates for the initial coefficients of bi-univalent functions, Tamsui Oxford J. Inform. Math. Sci. 29(4) (2013), 487-504. [16] S. K. Lee, V. Ravichandran and S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Ineq. Appl. 2013, 2013:281. [17] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68. [18] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, Arch. Rational Mech. Anal. 32 (1969), 100–112. [19] J.W. Noonan and D.K. Thomas, On the second Hankel determinant of areally mean p-valent functions,Trans. Amer. Math. Soc., 223(2) (1976), 337-346. [20] H. Orhan, E. Deniz and D. Raducanu, The Fekete Szeg¨ o problem for subclasses of analytic functions defined by a differential operator related to conic domains, Comput. Math. Appl. 59 (2010), 283-295. [21] H. Orhan, N. Magesh, and V. K. Balaji, Initial coefficient bounds for a general class of bi-univalent functions, arxiv:1303.2504v2. [22] Ch. Pommerenke, Univalent Functions, Vandenhoeck and Rupercht, G¨ ottingen, 1975. [23] H. M. Srivastava, A.K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010) 1188–1192. [24] H. M. Srivastava, S. Bulut, M. C ¸ a˘ glar, N. Ya˘ gmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat 27:5 (2013), 831-842. [25] D. L. Tan, Coefficient estimates for bi-univalent functions, Chinese Ann. Math. Ser. A 5 (5) (1984), 559–568. [26] Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and biunivalent functions, Appl. Math. Lett. 25 (2012) 990–994. [27] Q.-H. Xu, H.-G. Xiao and H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218 (2012), 11461-11465. [28] P. Zaprawa, On the Fekete-Szeg¨ o problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), 169–178. [29] P. Zaprawa, Estimates of initial coefficients for bi-univalent functions, Abstr. Appl. Anal., 2014, Article ID 357480, 6 pages. Department of Mathematics, Faculty of Science and Letters, Kafkas University, Kars, Turkey. E-mail address: [email protected], [email protected] Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey. E-mail address: [email protected]
Abstract. In this paper we obtain upper bounds for the second Hankel determinant H2 (2) of the classes bi-starlike and bi-convex functions of order β, which we denote by Sσ∗ (β) and Kσ (β), respectively. In particular, the estimates for the second Hankel determinat H2 (2) of bi-starlike and bi-convex functions which are important subclasses of bi-univalent functions are pointed out.
1. Introduction and definitions Let A denote the family of functions f of the form (1.1)
f (z) = z +
∞ X
an z n
n=2
which are analytic in the open unit disk U = {z : |z| < 1} and let S denote the class of all functions in A which are univalent in U. The Koebe one-quarter theorem (see [7]) ensures that the image of U under every f ∈ S contain a disk of radius 1 4. So, every f ∈ S has an inverse function f −1 satisfying f −1 (f (z)) = z (z ∈ U) and f (f −1 (w)) = w
(|w| < r0 (f ); r0 (f ) ≥ 1 4)
where f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 − (5a32 − 5a2 a3 + a4 )w4 + .... A function f ∈ A is said to be bi-univalent in U if both f (z) and f −1 (z) are univalent in U. Let σ denote the class of bi-univalent functions in U given by (1.1). Two of the most famous subclasses of univalent functions are the class S ∗ (β) of starlike functions of order β and the class K(β) of convex functions of order β. By definition, we have 0 zf (z) S ∗ (β) = f ∈ S : < > β; z ∈ U; 0 ≤ β < 1 f (z) and
zf 00 (z) K(β) = f ∈ S : < 1 + 0 > β; z ∈ U; 0 ≤ β < 1 . f (z) The classes consisting of starlike and convex functions are usually denoted by S ∗ = S ∗ (0) and K = K(0), respectively. For 0 ≤ β < 1, a function f ∈ σ is in the class Sσ∗ (β) of bi-starlike functions of order β, or Kσ (β) of bi-convex functions of order β if both f and its inverse map f −1 are, respectively, starlike or convex of order β. These classes were introduced by Brannan and Taha [2] in 1985. Especially the classes Sσ∗ (0) = Sσ∗ and Kσ (0) = Kσ are bi-starlike and bi-convex functions, respectively. In 1967, Lewin [17] showed that for every functions f ∈ σ of the form (1.1), the second coefficient√of f satisfy the inequality |a2 | < 1.51. In 1967, Brannan and Clunie [1] conjectured that |a2 | ≤ 2 for f ∈ σ. Later, Netanyahu [18] proved that maxf ∈σ |a2 | = 4/3. In 1985, Kedzierawski [13] proved Brannan and Clunie’s conjecture for f ∈ Sσ∗ . In 1985, Tan [25] obtained the bound for a2 namely |a2 | < 1.485 which is the best known estimate for functions in the class σ. Brannan and Taha [2] obtained estimates on the initial coefficients |a2 | and |a3 | for functions in the classes Sσ∗ (β) and Kσ (β). Recently, Deniz [6] extented and Kumar et al. [15] both extended and improved the results of Brannan and Taha [2] by generalizing their classes using subordination. The problem of estimating coefficients |an |, n ≥ 2 is still open. However, a lot of results for |a2 |, |a3 | and |a4 | were
2000 Mathematics Subject Classification. Primary 30C45, 30C50; Secondary 30C80. Key words and phrases. bi-univalent functions, bi-starlike functions of order β, bi-convex functions of order β, second Hankel determinant. 1
˘ ERHAN DENIZ, MURAT C ¸ AGLAR, AND HALIT ORHAN
2
proved for some subclasses of σ (see [3], [5], [9], [11], [21], [23], [24], [26], [27]). Unfortunatelly, they are not sharp. One of the important tools in the theory of univalent functions is Hankel Determinants which are useful, for example, in showing that a function of bounded characteristic in U, i.e., a function which is a ratio of two bounded analytic functions, with its Laurent series around the origin having integral coefficients, is rational [4]. The Hankel determinants [19] Hq (n) (n = 1, 2, ..., q = 1, 2, ...) of the function f are defined by an an+1 ... an+q−1 an+1 an+2 ... an+q Hq (n) = (a1 = 1). .. .. .. . . . an+q−1
an+q
...
an+2q−2
This determinant was discussed by several authors with q = 2. For example, we can know that the functional H2 (1) = a3 − a22 is known as the Fekete-Szeg¨o functional and they consider the further generalized functional a3 − µa22 where µ is some real number (see, [8]). In 1969, Keogh and Merkes [14] proved the Fekete-Szeg¨o problem for the classes S ∗ and K. Someone can see the Fekete-Szeg¨ o problem for the classes S ∗ (β) and K(β) at special cases in the paper of Orhan et.al. [20]. On the other hand, recently Zaprawa [28], [29] have studied on Fekete-Szeg¨o problem for some classes of bi-univalent functions. In special cases, he gave Fekete-Szeg¨o problem for the classes Sσ∗ (β) and Kσ (β). In 2014, Zaprawa [28] proved the following resuts for µ ∈ R, 3 1 1 − β; 2 ∗ 2 ≤µ≤ 2 f ∈ Sσ (β) ⇒ a3 − µa2 ≤ 3 2(1 − β) |µ − 1| ; µ ≥ 2 and µ ≤ 21 and f ∈ Kσ (β) ⇒ a3 − µa22 ≤
1−β 3 ;
2 3
≤ µ ≤ 43 (1 − β) |µ − 1| ; µ ≥ and µ ≤ 4 3
2 3
.
The second Hankel determinant H2 (2) is given by H2 (2) = a2 a4 − a23 . The bounds for the second Hankel determinant H2 (2) obtained for the classes S ∗ and K in [12]. Recently, Lee et al. [16] established the sharp bound to |H2 (2)| by generalizing their classes using subordination. In their paper, one can find the sharp bound to |H2 (2)| for the functions in the classes S ∗ (β) and K(β). In this paper, we seek upper bound for the functional H2 (2) = a2 a4 − a23 for functions f belongs to the classes Sσ∗ (β) and Kσ (β). Let P be the class of functions with positive real part consisting of all analytic functions P : U → C satisfying p(0) = 1 and 0. We need the following results about the functions belonging to the class P: Lemma 1.1. [22] If the function p ∈ P is given by the series p(z) = 1 + c1 z + c2 z 2 + ...
(1.2)
then the sharp estimate |ck | ≤ 2 (k = 1, 2, ...) holds. Lemma 1.2. [10] If the function p ∈ P is given by the series (1.2), then (1.3)
2c2
= c21 + x(4 − c21 )
(1.4)
4c3
2 = c31 + 2(4 − c21 )c1 x − c1 (4 − c21 )x2 + 2(4 − c21 ) 1 − |x| z,
for some x, z with |x| ≤ 1 and |z| ≤ 1. 2. Main results Our first main result for the class
Sσ∗ (β)
is the following:
Theorem 2.1. Let f (z) given by (1.1) be in the class Sσ∗ (β), 0 ≤ β < 1. Then i h i h 1 4 1 − β 2 1 + 4 (1 − β)2 , β ∈ 0, 1 − √ 3 2 2 a2 a4 − a23 ≤ (2.1) (β−1)2 3 1 √ 1 − , 1 . , β ∈ 2 2 (1−2(1−β) ) 2 2
SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β
3
Proof. Let f ∈ Sσ∗ (β) and g = f −1 . Then (2.2)
zf 0 (z) wg 0 (w) = β + (1 − β)p(z) and = β + (1 − β)q(w) f (z) g(w)
where p(z) = 1 + c1 z + c2 z 2 + ... and q(w) = 1 + d1 w + d2 w2 + ... in P. Now, equating the coefficients in (2.2), we have (2.3)
a2
=
(1 − β)c1 ,
(2.4)
2a3 −
=
(1 − β)c2 ,
(2.5)
3a4 − 3a3 a2 +
a22 a32
=
(1 − β)c3
and −a2
=
(1 − β)d1 ,
− 2a3
=
(1 − β)d2 ,
−10a32 + 12a3 a2 − 3a4
=
(1 − β)d3 .
(2.6) 3a22
(2.7) (2.8)
From (2.3) and (2.6), we arrive at c1 = −d1
(2.9) and
a2 = (1 − β)c1 .
(2.10)
Now, from (2.4), (2.7) and (2.10), we get that (2.11)
2
a3 = (1 − β) c21 +
(1 − β) (c2 − d2 ) . 4
Also, from (2.5) and (2.8), we find that 2 5 1 3 2 (2.12) a4 = (1 − β) c31 + (1 − β) c1 (c2 − d2 ) + (1 − β) (c3 − d3 ) . 3 8 6 Thus, we can easily establish that a2 a4 − a23 = − 1 (1 − β)4 c41 + 1 (1 − β)3 c21 (c2 − d2 ) 3 8 1 1 2 2 2 (2.13) + (1 − β) c1 (c3 − d3 ) − (1 − β) (c2 − d2 ) . 6 16 According to Lemma 1.2 and (2.9), we write (2.14)
2c2 = c21 + x(4 − c21 ) 2d2 = d21 + x(4 − d21 )
=⇒ c2 − d2 = 0
and c31 c1 − c1 (4 − c21 )x − (4 − c21 )x2 . 2 2 Using (2.14) and (2.15) in (2.13), we have a2 a4 − a23 = − 1 (1 − β)4 c41 + 1 (1 − β)2 c41 3 12
(2.15)
(2.16)
c3 − d3 =
1 1 2 2 2 2 2 2 2 − (1 − β) c1 (4 − c1 )x − (1 − β) c1 (4 − c1 )x . 6 12
Since p ∈ P, so |c1 | ≤ 2. Letting c1 = c, we may assume without restriction that c ∈ [0, 2]. Thus, applying the triangle inequality on (2.16), with µ = |x| ≤ 1, we obtain a2 a4 − a23 ≤ 1 (1 − β)4 c4 + 1 (1 − β)2 c4 3 12 1 1 2 2 2 (2.17) + (1 − β) c (4 − c2 )µ + (1 − β) c2 (4 − c2 )µ2 = F (µ). 6 12 Differentiating F (µ), we get 1 1 2 2 F 0 (µ) = (1 − β) c2 (4 − c2 ) + (1 − β) c2 (4 − c2 )µ. 6 6
˘ ERHAN DENIZ, MURAT C ¸ AGLAR, AND HALIT ORHAN
4
Using elementary calculus, one can show that F 0 (µ) > 0 for µ > 0. It implies that F is an increasing function and thus, the upper bound for F (µ) corresponds to µ = 1, in which case 1 1 1 4 2 2 (2.18) F (µ) ≤ (1 − β) c4 + (1 − β) c4 + (1 − β) c2 (4 − c2 ) = G(c). 3 12 4 Assume that G(c) has a maximum value in an interior of c ∈ [0, 2], by elementary calculation we find nh i o 2 2 2 (2.19) G0 (c) = (1 − β) 2 (1 − β) − 1 c3 + 3c . 3 q 3 Then G0 (c) = 0 implies the real critical point c01 = 0 or c02 = 1−2(1−β) 2. After some calculations following cases: h we concluded i 1 Case 1: When β ∈ 0, 1 − 2√2 , we observe that c02 ≥ 2, that is, c02 is out of the interval (0, 2). Therefore the maximum value of G(c) occurs at c01 = 0 or c = c02 which contradicts our assumption of having the maximum value at the interior point of c ∈ [0, 2]. Since G is an increasing function in the interval [0, 2], maximum point of G must be on the boundary of c ∈ [0, 2], that is, c = 2. Thus, we have i h 4 2 1 − β 2 1 + 4 (1 − β) . max G(c) = G(2) = 0≤c≤2 3 1 Case 2: When β ∈ 1 − 2√2 , 1 , we observe that c02 < 2, that is, c02 is interior of the interval [0, 2]. Since G00 (c02 ) < 0, the maximum value of G(c) occurs at c = c02 . Thus, we have ! s 2 3 (β − 1) 3 . max G(c) = G(c02 ) = G = 2 0≤c≤2 2 1 − 2 (1 − β)2 1 − 2 (1 − β) This completes the proof of the Theorem 2.1.
For α = 0, Theorem 2.1 readily yields the following coefficient estimates for bi-starlike functions. Corollary 2.2. Let f (z) given by (1.1) be in the class Sσ∗ . Then a2 a4 − a23 ≤ 20 . 3 Our second main result for the class Kσ (β) is following: Theorem 2.3. Let f (z) given by (1.1) be in the class Kσ (β), 0 ≤ β < 1. Then i h i h 1 1 − β 2 1 + (1 − β)2 , β ∈ 0, 1 − √1 6 2 a2 a4 − a23 ≤ (2.20) (1−β)2 3 1 √ , β ∈ 1 − , 1 . 8 (2−(1−β)2 ) 2 Proof. Let f ∈ Kσ (β) and g = f −1 . Then (2.21)
1+
wg 00 (w) zf 00 (z) = β + (1 − β)p(z) and 1 + = β + (1 − β)q(w) f 0 (z) g 0 (w)
where p(z) = 1 + c1 z + c2 z 2 + ... and q(w) = 1 + d1 w + d2 w2 + ... in P. Now, equating the coefficients in (2.21), we have (2.22)
2a2
=
(1 − β)c1 ,
(2.23)
6a3 −
=
(1 − β)c2 ,
(2.24)
12a4 − 18a3 a2 +
4a22 8a32
=
(1 − β)c3
and −2a2
=
(1 − β)d1 ,
− 6a3
=
(1 − β)d2 ,
−32a32 + 42a3 a2 − 12a4
=
(1 − β)d3 .
(2.25) 8a22
(2.26) (2.27)
From (2.22) and (2.25), we arrive at (2.28)
c1 = −d1
SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β
5
and 1 (1 − β)c1 . 2 Now, from (2.23), (2.26) and (2.29), we get that 1 1 2 (2.30) a3 = (1 − β) c21 + (1 − β) (c2 − d2 ) . 4 12 Also, from (2.24) and (2.27), we find that 5 5 1 3 2 (2.31) a4 = (1 − β) c31 + (1 − β) c1 (c2 − d2 ) + (1 − β) (c3 − d3 ) . 48 48 24 Thus, we can easily establish that a2 a4 − a23 = − 1 (1 − β)4 c41 + 1 (1 − β)3 c21 (c2 − d2 ) 96 96 1 1 2 2 2 (2.32) + (1 − β) c1 (c3 − d3 ) − (1 − β) (c2 − d2 ) . 48 144 (2.29)
a2 =
Using (2.14) and (2.15) in (2.32), we have a2 a4 − a23 = − 1 (1 − β)4 c41 + 1 (1 − β)2 c41 96 96 1 1 2 2 (2.33) − (1 − β) c21 (4 − c21 )x − (1 − β) c21 (4 − c21 )x2 . 48 96 Since p ∈ P, so |c1 | ≤ 2. Letting c1 = c, we may assume without restriction that c ∈ [0, 2]. Thus, applying the triangle inequality on (2.16), with µ = |x| ≤ 1, we obtain a2 a4 − a23 n o 1 1 2 2 2 ≤ (2.34) (1 − β) 1 + (1 − β) c4 + (1 − β) c2 (4 − c2 ) 2µ + µ2 = F (µ). 96 96 Differentiating F (µ), we get 1 1 2 2 F 0 (µ) = (1 − β) c2 (4 − c2 ) + (1 − β) c2 (4 − c2 )µ. 48 48 Using elementary calculus, one can show that F 0 (µ) > 0 for µ > 0. It implies that F is an increasing function and thus, the upper bound for F (µ) corresponds to µ = 1, in which case n o 1 1 2 2 2 (2.35) F (µ) ≤ (1 − β) 1 + (1 − β) c4 + (1 − β) c2 (4 − c2 ) = G(c). 96 32 Then, nh i o 1 2 2 (2.36) G0 (c) = (1 − β) (1 − β) − 2 c3 + 6c . 24 q 6 0 Setting G (c) = 0, since 0 ≤ c ≤ 2, we have the real critical poins c01 = 0 or c02 = 2−(1−β) 2. Proceeding similarlyhas in the iearlier proof, it follows that: Case 1: When β ∈ 0, 1 − √12 , the maximum value of G(c) corresponds to c = 2. i h 1 2 max G(c) = G(2) = 1 − β 2 1 + (1 − β) . 0≤c≤2 6 q 6 Case 2: When β ∈ 1 − √12 , 1 , the maximum value of G(c) corresponds to c = c02 = 2−(1−β) 2. ! s 2 6 3 (1 − β) . max G(c) = G(c02 ) = G = 2 0≤c≤2 8 2 − (1 − β)2 2 − (1 − β) This completes the proof of the Theorem 2.3.
For α = 0, Theorem 2.3 readily yields the following coefficient estimates for bi-convex functions. Corollary 2.4. Let f (z) given by (1.1) be in the class Kσ . Then a2 a4 − a23 ≤ 1 . 6
6
˘ ERHAN DENIZ, MURAT C ¸ AGLAR, AND HALIT ORHAN
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