SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β ERHANDENIZ,MURATC¸AG˘LAR,ANDHALITORHAN Abstract. In the present investigation the authors obtain upper bounds for the second Han- 5 kel determinant H2(2) of theclasses bi-starlikeandbi-convex functions oforder β, represented 1 by Sσ∗(β)and Kσ(β), respectively. In particular, the estimates for the second Hankel determi- 0 nantH2(2)ofbi-starlikeandbi-convexfunctionswhichareimportantsubclassesofbi-univalent 2 functionsarepointedout. r p A 1. Introduction and definitions 7 Let denote the family of functions f analytic in the open unit disk = z C: z <1 of ] A U { ∈ | | } V the form C ∞ (1.1) f(z)=z+ a zn. . n h n=2 X t a Let denote the class of all functions in which are univalent in . The Koebe one-quarter S A U m theorem (see [7]) ensures that the image of under every f contain a disk of radius 1(cid:30)4. So, U ∈S [ every f has an inverse function f−1 satisfying f−1(f(z))=z (z ) and ∈S ∈U 3 f(f−1(w))=w (w <r0(f); r0(f) 1(cid:30)4) | | ≥ v where f 1(w)=w a w2+(2a2 a )w3 (5a3 5a a +a )w4+.... 2 − − 2 2− 3 − 2− 2 3 4 A function f is saidto be bi-univalentin if both f(z)and f 1(z)are univalentin . Let 8 − ∈A U U 6 σ denote the class of bi-univalent functions in given by (1.1). U 1 Twoofthemostfamoussubclassesofunivalentfunctionsaretheclass (β)ofstarlikefunctions ∗ 0 of order β and the class (β)of convex functions of order β. By definitiSon, we have . K 1 zf (z) ′ 0 ∗(β)= f : >β; z ; 0 β <1 S ∈S ℜ f(z) ∈U ≤ 5 (cid:26) (cid:18) (cid:19) (cid:27) 1 and : zf (z) v (β)= f : 1+ ′′ >β; z ; 0 β <1 . Xi K (cid:26) ∈S ℜ(cid:18) f′(z) (cid:19) ∈U ≤ (cid:27) The classes consisting of starlike and convex functions are usually denoted by = (0) and ∗ ∗ r S S = (0), respectively. a K K For 0 β < 1, a function f σ is in the class (β) of bi-starlike functions of order β, or ≤ ∈ Sσ∗ (β)ofbi-convex functions of order β ifbothf anditsinversemapf 1 are,respectively,starlike σ − K or convex of order β. These classes were introduced by Brannan and Taha [2] in 1985. Especially the classes (0) = and (0) = are bi-starlike and bi-convex functions, respectively. In Sσ∗ Sσ∗ Kσ Kσ 1967, Lewin [17] showed that for every functions f σ of the form (1.1), the second coefficient of ∈ f satisfy the inequality a < 1.51. In 1967, Brannan and Clunie [1] conjectured that a √2 2 2 | | | | ≤ for f σ. Later, Netanyahu [18] proved that max a = 4/3. In 1985, Kedzierawski [13] f σ 2 ∈ ∈ | | proved Brannan and Clunie’s conjecture for f . In 1985, Tan [25] obtained the bound for a ∈ Sσ∗ 2 namely a < 1.485 which is the best known estimate for functions in the class σ. Brannan and 2 | | Taha[2]obtainedestimatesontheinitialcoefficients a and a forfunctionsintheclasses (β) | 2| | 3| Sσ∗ and (β). Recently, Deniz [6] and Kumar et al. [15] both extended and improved the results of σ K BrannanandTaha[2]bygeneralizingtheirclassesusingsubordination. Theproblemofestimating coefficients a , n 2 is still open. However,a lot of results for a , a and a were provedfor n 2 3 4 | | ≥ | | | | | | 2000 Mathematics Subject Classification. Primary30C45,30C50;Secondary30C80. Key words and phrases. Bi-univalentfunctions, bi-starlikefunctions of order β,bi-convex functions of order β, secondHankeldeterminant. Corresponding author. [email protected] (ErhanDeniz). 1 2 ERHANDENIZ,MURATC¸AG˘LAR,ANDHALITORHAN some subclasses of σ (see [3], [5], [9], [11], [21], [23], [24], [26], [27]). Unfortunatelly, none of them are not sharp. One of the important tools in the theory of univalent functions is Hankel Determinants which are utility, for example, in showing that a function of bounded characteristic in , i.e., a function U whichisaratiooftwoboundedanalyticfunctions,withitsLaurentseriesaroundtheoriginhaving integral coefficients, is rational [4]. The Hankel determinants [19] H (n)(n = 1,2,..., q = 1,2,...) q of the function f are defined by a a ... a n n+1 n+q 1 − a a ... a n+1 n+2 n+q Hq(n)= .. .. ..  (a1 =1). . . .    a a ... a   n+q−1 n+q n+2q−2    This determinant was discussed by several authors with q =2. For example, we can know that the functional H (1) = a a2 is known as the Fekete-Szego¨ functional and they consider the 2 3 − 2 further generalized functional a µa2 where µ is some real number (see, [8]). In 1969, Keogh 3 − 2 and Merkes [14] proved the Fekete-Szego¨ problem for the classes and . Someone can see the ∗ S K Fekete-Szego¨ problem for the classes (β) and (β) at special cases in the paper of Orhan et.al. ∗ S K [20]. On the other hand, very recently Zaprawa [28], [29] have studied on Fekete-Szego¨ problem for some classes of bi-univalent functions. In special cases, he gave Fekete-Szego¨ problem for the classes (β) and (β). In 2014, Zaprawa[28] proved the following resuts for µ R, Sσ∗ Kσ ∈ 1 β; 1 µ 3 f ∈Sσ∗(β)⇒ a3−µa22 ≤ 2(1 β−) µ 1 ; µ 23≤and≤µ2 1 (cid:26) − | − | ≥ 2 ≤ 2 (cid:12) (cid:12) and (cid:12) (cid:12) f ∈Kσ(β)⇒ a3−µa22 ≤ (1 β1)−3βµ; 1 ; µ 234≤anµd≤µ34 2 . (cid:26) − | − | ≥ 3 ≤ 3 (cid:12) (cid:12) The second Hankel determ(cid:12)inant H2(cid:12)(2) is given by H2(2) = a2a4 − a23. The bounds for the second Hankel determinant H (2) obtained for the classes and in [12]. Recently, Lee et al. 2 ∗ S K [16] established the sharp bound to H (2) by generalizing their classes using subordination. In 2 | | their paper, one can find the sharp bound to H (2) for the functions in the classes (β) and 2 ∗ | | S (β). K Inthispaper,weseekupperboundforthefunctionalH (2)=a a a2 forfunctionsf belonging 2 2 4− 3 to the classes (β)and (β). Sσ∗ Kσ Let be the class of functions with positive real part consisting of all analytic functions : P P C satisfying p(0)=1 and p(z)>0. U → ℜ To establish our main results, we shall require the following lemmas. Lemma 1.1. [22] If the function p is given by the series ∈P (1.2) p(z)=1+c z+c z2+... 1 2 then the sharp estimate c 2 (k =1,2,...) holds. k | |≤ Lemma 1.2. [10] If the function p is given by the series (1.2), then ∈P (1.3) 2c = c2+x(4 c2) 2 1 − 1 (1.4) 4c = c3+2(4 c2)c x c (4 c2)x2+2(4 c2) 1 x2 z, 3 1 − 1 1 − 1 − 1 − 1 −| | (cid:16) (cid:17) for some x, z with x 1 and z 1. | |≤ | |≤ 2. Main results Our first main result for the class (β) as follows: Sσ∗ Theorem 2.1. Let f(z) given by (1.1) be in the class (β), 0 β <1. Then Sσ∗ ≤ 4(1 β)2 4β2 8β+5 , β 0,29 √137 (2.1) a a a2 3 − − ∈ −32 (cid:12) 2 4− 3(cid:12)≤ (1−β)2 (cid:0)1163ββ22−−1246ββ+−57 ,(cid:1) β ∈ h29−3√2137,1i. (cid:12) (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)  SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β 3 Proof. Let f (β)and g =f 1. Then ∈Sσ∗ − zf (z) wg (w) ′ ′ (2.2) =β+(1 β)p(z)and =β+(1 β)q(w) f(z) − g(w) − where p(z)=1+c z+c z2+... and q(w)=1+d w+d w2+... in . 1 2 1 2 P Comparing coefficients in (2.2), we have (2.3) a = (1 β)c , 2 1 − (2.4) 2a a2 = (1 β)c , 3− 2 − 2 (2.5) 3a 3a a +a3 = (1 β)c 4− 3 2 2 − 3 and (2.6) a = (1 β)d , 2 1 − − (2.7) 3a2 2a = (1 β)d , 2− 3 − 2 (2.8) 10a3+12a a 3a = (1 β)d . − 2 3 2− 4 − 3 From (2.3) and (2.6), we arrive at (2.9) c = d 1 1 − and (2.10) a =(1 β)c . 2 1 − Now, from (2.4), (2.7) and (2.10), we get that (1 β) (2.11) a =(1 β)2c2+ − (c d ). 3 − 1 4 2− 2 Also, from (2.5) and (2.8), we find that 2 5 1 (2.12) a = (1 β)3c3+ (1 β)2c (c d )+ (1 β)(c d ). 4 3 − 1 8 − 1 2− 2 6 − 3− 3 Thus, we can easily establish that 1 1 a a a2 = (1 β)4c4+ (1 β)3c2(c d ) 2 4− 3 −3 − 1 8 − 1 2− 2 (cid:12) (cid:12) (cid:12) (cid:12)1 1 (2.13) (cid:12) (cid:12) +(cid:12)(cid:12) (1 β)2c1(c3 d3) (1 β)2(c2 d2)2 . 6 − − − 16 − − (cid:12) (cid:12) (cid:12) According to Lemma 1.2 and (2.9), we write (cid:12) 2c =c2+x(4 c2) 4 c2 (2.14) 2d22=d121+x(4−−d121) (cid:27)=⇒c2−d2 = −2 1(x−y) and 4c = c3+2(4 c2)c x c (4 c2)x2+2(4 c2) 1 x2 z, 3 1 − 1 1 − 1 − 1 − 1 −| | (cid:16) (cid:17) 4d = d3+2(4 d2)d y d (4 d2)y2+2(4 d2) 1 y 2 w, 3 1 − 1 1 − 1 − 1 − 1 −| | (cid:16) (cid:17) (2.15) c3 c 4 c2 c 4 c2 4 c2 c d = 1 + 1 − 1 (x+y) 1 − 1 (x2+y2)+ − 1 1 x2 z 1 y 2 w . 3 3 − 2 2 − 2 2 −| | − −| | (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:17) 4 ERHANDENIZ,MURATC¸AG˘LAR,ANDHALITORHAN for some x,y, z,w with x 1, y 1, z 1 and w 1.Using (2.14) and (2.15) in (2.13), and | |≤ | |≤ | |≤ | |≤ applying the triangle inequality we have 1 1 a a a2 = (1 β)4c4+ (1 β)3c2(4 c2)(x y) 2 4− 3 −3 − 1 16 − 1 − 1 − (cid:12) (cid:12)(cid:12)+1(1 β)(cid:12)(cid:12)2c1(cid:12)(cid:12)(cid:12) c31 + (4−c21)c1(x+y) (4−c21)c1(x2+y2)+ (4−c21) (1 x2)z (1 y 2)w 6 − 2 2 − 4 2 −| | − −| | (cid:20) (cid:16) (cid:17)(cid:21) 1 (1 β)2(4 c2)2(x y)2 −64 − − 1 − (cid:12) 1 1 (cid:12) 1 (1 β)4c4+ (1 β)2c4+(cid:12) (1 β)2c (4 c2) ≤ 3 − 1 12 − 1 (cid:12) 6 − 1 − 1 1 1 + (1 β)3c2(4 c2)+ (1 β)2c2(4 c2) (x + y ) 16 − 1 − 1 12 − 1 − 1 | | | | (cid:20) (cid:21) 1 1 1 + (1 β)2c2(4 c2) (1 β)2c (4 c2) (x2+ y 2)+ (1 β)2(4 c2)2(x + y )2. 24 − 1 − 1 − 12 − 1 − 1 | | | | 64 − − 1 | | | | (cid:20) (cid:21) Sincep ,so c 2.Lettingc =c,wemayassumewithoutrestrictionthatc [0,2].Thus, 1 1 ∈P | |≤ ∈ for λ= x 1 and µ= y 1 we obtain | |≤ | |≤ a a a2 T +T (λ+µ)+T (λ2+µ2)+T (λ+µ)2 =F(λ,µ) 2 4− 3 ≤ 1 2 3 4 where (cid:12) (cid:12) (cid:12) (cid:12) (1 β)2 T = T (c)= − 1+4(1 β)2 c4 2c3+8c 0, 1 1 12 − − ≥ 1 h(cid:16) (cid:17) i T = T (c)= (1 β)2c2(4 c2)(7 3β) 0, 2 2 48 − − − ≥ 1 T = T (c)= (1 β)2c(4 c2)(c 2) 0, 3 3 24 − − − ≤ 1 T = T (c)= (1 β)2(4 c2)2 0. 4 4 64 − − 1 ≥ Now we need to maximize F(λ,µ) in the closed square S = (λ,µ):0 λ 1,0 µ 1 . { ≤ ≤ ≤ ≤ } Since T <0 and T +2T >0 for c [0,2), we conclude that 3 3 4 ∈ F F (F )2 <0. λλ µµ λµ · − Thus the function F cannot have a local maximum in the interior of the square S. Now, we investigate the maximum of F on the boundary of the square S. For λ=0 and 0 µ 1 (similarly µ=0 and 0 λ 1), we obtain ≤ ≤ ≤ ≤ F(0,µ)=G(µ)=(T +T )µ2+T µ+T . 3 4 2 1 i. The case T +T 0 : In this case for 0 < µ < 1 and any fixed c with 0 c < 2, it is 3 4 ≥ ≤ clear that G(µ)=2(T +T )µ+T >0, that is, G(µ) is an increasing function. Hence, for fixed ′ 3 4 2 c [0,2), the maximum of G(µ) occurs at µ=1, and ∈ maxG(µ)=G(1)=T +T +T +T . 1 2 3 4 ii. The case T +T < 0 : Since T +2(T +T ) 0 for 0 < µ < 1 and any fixed c with 3 4 2 3 4 ≥ 0 c <2, it is clear that T +2(T +T )< 2(T +T )µ+T < T and so G(µ)> 0. Hence for 2 3 4 3 4 2 2 ′ ≤ fixed c [0,2), the maximum of G(µ) occurs at µ=1. ∈ Also for c=2 we obtain 4 (2.16) F(λ,µ)= (1 β)2(4β2 8β+5). 3 − − Taking into account the value (2.16), and the cases i and ii, for 0 µ 1 and any fixed c with ≤ ≤ 0 c 2, ≤ ≤ maxG(µ)=G(1)=T +T +T +T . 1 2 3 4 For λ=1 and 0 µ 1 (similarly µ=1 and 0 λ 1), we obtain ≤ ≤ ≤ ≤ F(1,µ)=H(µ)=(T +T )µ2+(T +2T )µ+T +T +T +T . 3 4 2 4 1 2 3 4 Similarly to the above cases of T +T , we get that 3 4 maxH(µ)=H(1)=T +2T +2T +4T . 1 2 3 4 SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β 5 Since G(1) H(1) for c [0,2], maxF(λ,µ)=F(1,1) on the boundary of the square S. Thus ≤ ∈ the maximum of F occurs at λ=1 and µ=1 in the closed square S. Let K :[0,2] R → (2.17) K(c)=maxF(λ,µ)=F(1,1)=T +2T +2T +4T . 1 2 3 4 Substituting the values of T ,T ,T and T in the function K defined by (2.17), yield 1 2 3 4 (1 β)2 K(c)= − 16β2 26β+5 c4+24(2 β)c2+48 . 48 − − Assume that K(c) has a maximum(cid:2)(cid:0)value in an int(cid:1)erior of c [0,2], by el(cid:3)ementary calculation ∈ we find (1 β)2 (2.18) K′(c)= − 16β2 26β+5 c3+12(2 β)c . 12 − − As a result of some calculations we ca(cid:2)(cid:0)n do the followin(cid:1)g examine: (cid:3) Case 1: Let 16β2 26β+5 0, that is, β 0,13 √89 . Therefore K (c) > 0 for c (0,2). − ≥ ∈ −16 ′ ∈ Since K is an increasing function in the intervalh(0,2), maiximum point of K must be on the boundary of c [0,2], that is, c=2. Thus, we have ∈ 4 maxK(c)=K(2)= (1 β)2 4β2 8β+5 . 0 c 2 3 − − ≤ ≤ (cid:0) (cid:1) Case 2: Let 16β2 26β + 5 < 0, that is, β 13 √89,1 . Then K (c) = 0 implies the − ∈ −16 ′ real critical point c01 = 0 or c02 = 16−β122(22−6ββ+)5. Whe(cid:16)n β ∈ 13−(cid:17)1√689,29−3√2137 , we observe that − c02 ≥ 2, that is, c02 is out of the inqterval (0,2). Therefore th(cid:16)e maximum valueiof K(c) occurs at c =0 or c=c which contradicts our assumption of having the maximum value at the interior 01 02 point of c [0,2]. Since K is an increasing function in the interval (0,2), maximum point of K ∈ must be on the boundary of c [0,2], that is, c=2. Thus, we have ∈ 4 maxK(c)=K(2)= (1 β)2 4β2 8β+5 . 0 c 2 3 − − ≤ ≤ (cid:0) (cid:1) When β 29 √137,1 we observe that c < 2, that is, c is interior of the interval [0,2]. ∈ −32 02 02 Since K′′(c02)(cid:16)<0, the ma(cid:17)ximum value of K(c) occurs at c=c02. Thus, we have 12(2 β) 13β2 14β 7 maxK(c)=K(c )=K − − =(1 β)2 − − . 0≤c≤2 02 s16β2−26β+5! − (cid:18)16β2−26β+5(cid:19) This completes the proof of the Theorem 2.1. (cid:3) Forβ =0,Theorem2.1readilyyieldsthefollowingcoefficientestimatesforbi-starlikefunctions. Corollary 2.2. Let f(z) given by (1.1) be in the class . Then Sσ∗ 20 a a a2 . 2 4− 3 ≤ 3 Our second main result for the class (cid:12) (β) is fo(cid:12)llowing: (cid:12)σ (cid:12) K Theorem 2.3. Let f(z) given by (1.1) be in the class (β), 0 β <1. Then σ K ≤ (1 β)2 5β2+8β 32 (2.19) a a a2 − − 2 4− 3 ≤ 24 3β2 3β 4 (cid:18) − − (cid:19) (cid:12) (cid:12) Proof. Let f (β)and g (cid:12)=f 1. Th(cid:12)en σ − ∈K zf (z) wg (w) ′′ ′′ (2.20) 1+ =β+(1 β)p(z)and 1+ =β+(1 β)q(w) f (z) − g (w) − ′ ′ where p(z)=1+c z+c z2+... and q(w)=1+d w+d w2+... in . 1 2 1 2 P Now, equating the coefficients in (2.20), we have (2.21) 2a = (1 β)c , 2 1 − (2.22) 6a 4a2 = (1 β)c , 3− 2 − 2 (2.23) 12a 18a a +8a3 = (1 β)c 4− 3 2 2 − 3 6 ERHANDENIZ,MURATC¸AG˘LAR,ANDHALITORHAN and (2.24) 2a = (1 β)d , 2 1 − − (2.25) 8a2 6a = (1 β)d , 2− 3 − 2 (2.26) 32a3+42a a 12a = (1 β)d . − 2 3 2− 4 − 3 From (2.21) and (2.24), we arrive at (2.27) c = d 1 1 − and 1 (2.28) a = (1 β)c . 2 1 2 − Now, from (2.22), (2.25) and (2.28), we get that 1 1 (2.29) a = (1 β)2c2+ (1 β)(c d ). 3 4 − 1 12 − 2− 2 Also, from (2.23) and (2.26), we find that 5 5 1 (2.30) a = (1 β)3c3+ (1 β)2c (c d )+ (1 β)(c d ). 4 48 − 1 48 − 1 2− 2 24 − 3− 3 Thus, we can easily establish that 1 1 a a a2 = (1 β)4c4+ (1 β)3c2(c d ) 2 4− 3 −96 − 1 96 − 1 2− 2 (cid:12) (cid:12) (cid:12) (cid:12)1 1 (2.31) (cid:12) (cid:12) +(cid:12)(cid:12) (1 β)2c1(c3 d3) (1 β)2(c2 d2)2 . 48 − − − 144 − − (cid:12) (cid:12) Using (2.14) and (2.15) in (2.31), we have (cid:12) (cid:12) 1 1 a a a2 = (1 β)4c4+ (1 β)3c2(4 c2)(x y) 2 4− 3 −96 − 1 192 − 1 − 1 − (cid:12) (cid:12)(cid:12)+ 1 (1 β(cid:12)(cid:12))2c(cid:12)(cid:12)(cid:12)1 c31 + (4−c21)c1(x+y) (4−c21)c1(x2+y2)+ (4−c21) (1 x2)z (1 y 2)w 48 − 2 2 − 4 2 −| | − −| | (cid:20) (cid:16) (cid:17)(cid:21) 1 (1 β)2(4 c2)2(x y)2 −288 − − 1 − (cid:12) 1 1 (cid:12) 1 (1 β)4c4+ (1 β)2c4+(cid:12) (1 β)2c (4 c2) ≤ 96 − 1 96 − 1 (cid:12) 48 − 1 − 1 1 1 + (1 β)3c2(4 c2)+ (1 β)2c2(4 c2) (x + y ) 192 − 1 − 1 96 − 1 − 1 | | | | (cid:20) (cid:21) 1 1 1 + (1 β)2c2(4 c2) (1 β)2c (4 c2) (x2+ y 2)+ (1 β)2(4 c2)2(x + y )2. 192 − 1 − 1 − 96 − 1 − 1 | | | | 576 − − 1 | | | | (cid:20) (cid:21) Since p , so c 2. Taking c = c, we may assume without restriction that c [0,2]. Thus, 1 1 ∈ P | | ≤ ∈ for λ= x 1 and µ= y 1 we obtain | |≤ | |≤ a a a2 M +M (λ+µ)+M (λ2+µ2)+M (λ+µ)2 =Ψ(λ,µ) 2 4− 3 ≤ 1 2 3 4 where (cid:12) (cid:12) (cid:12) (cid:12) (1 β)2 M = M (c)= − 1+(1 β)2 c4 2c3+8c 0, 1 1 96 − − ≥ 1 h(cid:16) (cid:17) i M = M (c)= (1 β)2c2(4 c2)(3 β) 0, 2 2 192 − − − ≥ 1 M = M (c)= (1 β)2c(4 c2)(c 2) 0, 3 3 192 − − − ≤ 1 M = M (c)= (1 β)2(4 c2)2 0. 4 4 576 − − 1 ≥ ThereforeweneedtomaximizeΨ(λ,µ)intheclosedsquareS= (λ,µ):0 λ 1,0 µ 1 . { ≤ ≤ ≤ ≤ } To show that the maximum of Ψ we can follow the maximum of F in the Theorem 2.1. Thus the maximum of Ψ occurs at λ=1 and µ=1 in the closed square S. Let Φ:[0,2] R defined by → (2.32) Φ(c)=maxΨ(λ,µ)=Ψ(1,1)=M +2M +2M +4M . 1 2 3 4 SECOND HANKEL DETERMINANT FOR BI-STARLIKE AND BI-CONVEX FUNCTIONS OF ORDER β 7 Substituting the values of M ,M ,M and M in the function Φ given by (2.32), yield 1 2 3 4 (1 β)2 Φ(c)= − 3β2 3β 4 c4+4(8 3β)c2+32 . 288 − − − Assume that Φ(c) has a maximum(cid:2)(cid:0)value in an in(cid:1)terior of c [0,2], by (cid:3)elementary calculation ∈ we find (1 β)2 Φ′(c)= − 3β2 3β 4 c3+2(8 3β)c . 72 − − − Setting Φ(c)=0, since 0<c<2, and(cid:2)(cid:0)3β2 3β 4<(cid:1) 0 and 8 3β >(cid:3)0 for every β [0,1) we ′ − − − ∈ have the real criticalpoin c03 = 3β2(23β3−β8)4. Since c03 ≤2 for every β ∈[0,1) and so Φ′′(c03)<0, − − the maximum value of Φ(c) corrqesponds to c=c03, that is, 2(3β 8) (1 β)2 5β2+8β 32 maxΦ(c)=Φ(c )=Φ − = − − . 0<c<2 03 s3β2−3β−4! 24 (cid:18) 3β2−3β−4 (cid:19) On the other hand, (1 β)2 (1 β)2 Φ(0)= − and Φ(2)= − β2 2β+2 . 9 6 − Consequently, since Φ(0)<Φ(2) Φ(c ) we obtain max Φ(cid:0)(c)=Φ(c )(cid:1). ≤ 03 0 c< 2 03 This completes the proof of the Theorem 2.3. ≤ ≤ (cid:3) Forβ =0,Theorem2.3readilyyieldsthefollowingcoefficientestimatesforbi-convexfunctions. Corollary 2.4. Let f(z) given by (1.1) be in the class . Then σ K 1 a a a2 . 2 4− 3 ≤ 3 (cid:12) (cid:12) (cid:12) Referen(cid:12)ces [1] A.BrannanandJ.G.Clunie,AspectsofcontemporarycomplexanalysisProceedingsoftheNATOAdvanced StudyInstituteheldattheUniversityofDurham,Durham,July120,1979,AcademicPressNewYork,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, PergamonPress,ElsevierScienceLimited,Oxford,1988,pp.53–60.seealsoStudiaUniv.Babe¸s-BolyaiMath. 31(2)(1986), 70–77. [3] S.Bulut,Faberpolynomialcoefficientestimatesforacomprehensivesubclassofanalyticbi-univalentfunctions, C.R.Acad.Sci.Paris,Ser.I,352(6)(2014), pp.479–484. [4] D.G.Cantor,Powerserieswithintegralcoefficients,Bull.Amer.Math.Soc.69(1963), 362–366. [5] M.C¸a˘glar,H.OrhanandN.Ya˘gmur,Coefficientboundsfornewsubclassesofbi-univalentfunctions,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.FeketeandG.Szego¨,EineBemerkunguberungeradeschlichteFunktionen,J.LondonMath.Soc.8(1933), 85-89. [9] B.A.Frasin,M.K.Aouf,Newsubclassesofbi-univalentfunctions,Appl.Math.Lett.24(2011)1569-1573. [10] U. Grenander and G. Szego¨, Toeplitz forms and their applications, California Monographs in Mathematical Sciences Univ.CaliforniaPress,Berkeley,1958. [11] S.G.Hamidi,J.M.Jahangiri,Faberpolynomialcoefficientestimatesforanalyticbi-close-to-convexfunctions, C.R.Acad.Sci.Paris,Ser.I352(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,Someremarksonbi-univalentfunctions,Ann.Univ.MariaeCurie-Sk lodowskaSect. A39 (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, TamsuiOxfordJ.Inform.Math.Sci.29(4)(2013), 487-504. [16] S.K.Lee,V.RavichandranandS.Supramaniam,BoundsforthesecondHankeldeterminantofcertainunivalent functions,J.Ineq.Appl.2013, 2013:281. [17] M.Lewin,Onacoefficient problemforbi-univalentfunctions,Proc.Amer.Math.Soc.18(1967), 63–68. 8 ERHANDENIZ,MURATC¸AG˘LAR,ANDHALITORHAN [18] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalentfunctionin|z|<1,Arch.RationalMech.Anal.32(1969), 100–112. [19] J.W.NoonanandD.K.Thomas,OnthesecondHankeldeterminantofareallymeanp-valentfunctions,Trans. Amer.Math.Soc.,223(2)(1976), 337-346. [20] H. Orhan, E. Deniz and D. Raducanu, The Fekete Szego¨ problem for subclasses of analytic functions defined byadifferentialoperatorrelatedtoconicdomains,Comput.Math.Appl.59(2010), 283-295. [21] H.Orhan,N.Magesh,andV.K.Balaji,Initialcoefficient boundsforageneral classofbi-univalentfunctions, Filomat,(2014) inpress. [22] Ch.Pommerenke,UnivalentFunctions,Vandenhoeck andRupercht,G¨ottingen, 1975. [23] H.M.Srivastava,A.K.Mishra,P.Gochhayat, Certainsubclassesofanalyticandbi-univalentfunctions,Appl. Math.Lett.23(2010) 1188–1192. [24] H.M.Srivastava, S.Bulut,M.C¸a˘glar,N.Ya˘gmur,Coefficientestimatesforageneralsubclassofanalyticand bi-univalentfunctions,Filomat27:5(2013), 831-842. [25] D.L.Tan,Coefficientestimatesforbi-univalentfunctions,ChineseAnn.Math.Ser.A5(5)(1984), 559–568. [26] Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi- univalentfunctions,Appl.Math.Lett. 25(2012)990–994. [27] Q.-H.Xu, H.-G.Xiaoand H. M. Srivastava, A certain general subclass of analytic and bi-univalentfunctions andassociatedcoefficientestimateproblems,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 Stevin21(2014), 169–178. [29] P. Zaprawa, Estimates of initial coefficients for bi-univalent functions, Abstr. Appl. Anal., 2014, Article ID 357480, 6pages. DepartmentofMathematics,Facultyof Science andLetters, KafkasUniversity,Kars, Turkey. E-mail address: [email protected] (Erhan Deniz), [email protected] (Murat C¸a˘glar) DepartmentofMathematics,Facultyof Science,AtaturkUniversity, Erzurum,25240,Turkey. E-mail address: [email protected] (Halit Orhan)