International Journal of Algebra, Vol. 6, 2012, no. 3, 117-122 Quasi-Convex Functions with Respect to Symmetric Conjugate Points 1 Chew Lai Wah, 2 Aini Janteng and 3 Suzeini Abdul Halim 1,2 School of Science and Technology Universiti Malaysia Sabah 88400 Kota Kinabalu, Sabah, Malaysia 3 Institute of Mathematical Sciences, Universiti Malaya 50603 Kuala Lumpur, Malaysia 1 wawa03087@hotmail.com, 2 aini jg@ums.edu.my, 3 suzeini@um.edu.my Abstract Let C sc (A, B) denote the class of functions f which are analytic in an open unit disc D = {z : z < 1} and satisfying the condition 1+Az (f(z) f( z)) 1+Bz, 1 B < A 1, z D. In this paper, we consider the class Ksc(A, B) consisting of analytic functions f and satisfying (z)) 1+Az (g(z) g( z)) 1+Bz, g C sc(a, B), 1 B<A 1, z D. (zf The aims of paper are to determine coefficient estimates and distortion bounds for the class Ksc(A, B). Mathematics Subject Classification: Primary 30C45 Keywords: convex with respect to symmetric conjugate points, quasiconvex with respect to symmetric conjugate points, coefficient estimates. 1 Introduction Let U be the class of functions which are analytic in the open unit disc D = {z : z < 1} given by w(z) = b k z k and satisfying the conditions w(0) = 0, w(z) < 1, z D.
118 Chew Lai Wah, Aini Janteng and Suzeini Abdul Halim Let S denote the class of functions f which are analytic and univalent in D of the form f(z) =z + a n z n,z D. (1) n=2 Also, let Ssc be the subclass of S consisting of functions given by (1) satisfying { zf } (z) Re > 0, z D. f(z) f( z) These functions are called starlike with respect to symmetric conjugate points and were introduced by El-Ashwah and Thomas in 1987. The class can be extended to other class in D, namely convex functions with respect to symmetric conjugate points. Let C sc denote the class of convex functions with respect to symmetric conjugate points and satisfying the condition { (zf (z)) } Re > 0, z D. (f(z) f( z)) In 2010, Loo and Janteng introduced a subclass of Ssc which were denoted by Ssc (A, B). Let denote Ssc (A, B) the class of functions of the form (1) and satisfying the condition 2zf (z) f(z) f( z) 1+Az 1+Bz, 1 B<A 1, z D. Lim and Janteng in 2011 considered the class of functions of the form (1) denoted by C sc (A, B) and satisfying the condition 1+Az (f(z) f( z)) 1+Bz, 1 B<A 1, z D. In this paper, let consider K sc(a, B) be the class of functions of the form (1) and satisfying the condition (g(z) g( z)) 1+Az 1+Bz,g C sc(a, B), 1 B<A 1, z D. Obviously Ksc (A, B) is a subclass of the class quasi-convex with respect to symmetric conjugate points, Ksc = K sc (1, 1). By definition of subordination, it follows that f Ksc (A, B) if and only if = 1+Aw(z) (g(z) g( z)) 1+Bw(z) = P (z), w U (2)
Quasi-convex functions 119 where P (z) =1+ n=1 p n z n. (3) We study the class Ksc (A, B) and obtain coefficient estimates and distortion bounds. 2 Preliminary Results We need the following preliminary results, required for proving our theorems. Lemma 2.1 ([3]) If P (z) is given by (3) then p n (). (4) Theorem 2.1 ([4]) Let f C sc (A, B), then for n 1, b 2n n 1 () ( +2j), (2n)n!2 n and b 2n+1 n 1 () ( +2j). (2n +1)n!2 n 3 Main Results First, we give the coefficient inequality for the class K sc(a, B). Theorem 3.1 Let f K sc(a, B), then for n 1, Proof. a 2n Since g C sc (A, B), it follows that n 1 () ( +2j). (5) (2n)n!2 n 2(zg (z)) =(g(z) g( z)) K(z) for z D, with Re K(z) > 0 where K(z) =1+c 1 z + c 2 z 2 + c 3 z 3 +... Upon equating coefficients, we obtain 2 2 b 2 = c 1, 3(2)b 3 = c 2, (6) 4 2 b 4 = c 3 +3b 3 c 1, 5(4)b 5 = c 4 +3b 3 c 2. (7)
120 Chew Lai Wah, Aini Janteng and Suzeini Abdul Halim For (2) and (3), we have 2(1+2 2 a 2 z+3 2 a 3 z 2 +4 2 a 4 z 3 +5 2 a 5 z 4 +...+ a 2n z 2n 1 +(2n+1) 2 a 2n+1 z 2n +...) = 2(1 + 3b 3 z 2 +5b 5 z 4 +7b 7 z 6 +... +(2n 1)a 2n 1 z 2n 2 + a 2n+1 z 2n +...) (1 + p 1 z + p 2 z 2 +... + p 2n z 2n + p 2n+1 z 2n+1 +...) Equating the coefficients of like powers of z, we have 2 2 a 2 = p 1, 3 2 a 3 = p 2 +3b 3 (8) 4 2 a 4 = p 3 +3b 3 p 1, 5 2 a 5 = p 4 +3b 3 p 2 +5b 5 (9) a 2n = p 2n 1 +3b 3 p 2n 3 +5b 5 p 2n 5 +... +(2n 1)b 2n 1 p 1 (10) (2n +1) 2 a 2n+1 = p 2n +3b 3 p 2n 2 +5b 5 p 2n 4 +... +(2n 1)b 2n 1 p 2. Easily using Lemma 2.1 and (8), we get a 2 2(2), a 3 3(2). Again by applying (6), (7) and followed by Lemma 2.1, we get from (9) a 4 ()( +2), a 5 4(4)(2) ()( +2). 5(4)(2) It follows that (5) hold for n=1,2. We now prove (5) using induction. Equation (10) in conjuction with Lemma 2.1 yield a 2n [ ] n 1 1+ (2k +1) b 2k+1. (11) We assume that (5) holds for k=3,4,...,(n-1). Then from (11) and Theorem 2.1, we obtain a 2n n 1 1+ k!2 k In order to complete the proof, it is sufficient to show that (2m) 2 m 1 1+ k!2 k ( +2j) = ( +2j). (12) (2m)m!2 m m 1 (A B+2j), (m =3, 4,..., n). (13)
Quasi-convex functions 121 (13) is valid for m =3. Let us suppose that (13) is true for all m, 3 <m (n 1). Then from (12) = = = n 1 1+ ( ) n 1 2 1+ n (2(n 1)) 2 + ( n 1 n (n 1)!2 n 1 n 2 k!2 k n 2 k!2 k ( +2j) ) 2 n 2 2(n 1)(n 1)!2 n 1 2n 2 (n 1)!2 n 1 = (2n)n!2 n n 1 n 2 ( +2j) ( +2j) ( +2j) Thus, (13) holds for m = n and hence (5) follows. ( +2j) ( +2j)+ ( ) +2(n 1) Theorem 3.2 Let f K sc(a, B), then for z = r, 0 <r<1, 1 r r 0 Proof. 1 t 2 (1 At) (1 + t 2 ) 2 (1 Bt) dt f (z) 1 r r 0 Put h(z) = g(z) g( z) 2. Then from (2), we obtain (zf (z)) = h (z) Since h is odd and starlike, it follows that (see [1]) 2 1+t 2 (1 + At) (1 t 2 ) 2 (1 + Bt) n 2 (n 1)!2 n 1 dt. (14) 1+Aw(z) 1+Bw(z). (15) 1 r 2 (1 + r 2 ) 2 h (z) 1+r2 (1 r 2 ) 2. (16) Furthermore, for w U, it can also be easily established that 1 Ar 1 Br 1+Aw(z) 1+Bw(z) 1+Ar 1+Br. (17) ( +2j)
122 Chew Lai Wah, Aini Janteng and Suzeini Abdul Halim Next, applying results (16) and (17) in (15) we obtain 1 r 2 (1 + r 2 ) 2 1 Ar 1 Br (zf (z)) 1+r2 (1 r 2 ) 2 1+Ar 1+Br. (18) Finally, setting z = r, and integrating (18) gives our result. Acknowledgement The authors Aini Janteng and Suzeini Abdul Halim are partially supported by FRG0268-ST-2/2010 Grant, Malaysia. References [1] Duren, P.L. (1983). Univalent functions. New York: Springer-Verlag. [2] El-Ashwah, R.M. and Thomas, D.K. : Some subclasses of close-to-convex functions, J. Ramanujan Math. Soc., 2(1987): 86-100. Goel, R.M. and Mehrok, B.C. : A subclass of starlike functions with respect to symmetric points, Tamkang J. Math., 13(1)(1982): 11-24. [3] Goel, R.M. and Mehrok, B.C. : A subclass of starlike functions with respect to symmetric points, Tamkang J. Math., 13(1)(1982): 11-24. [4] Lim, C.S.J. and Janteng, A. : Some properties for subclass of convex functions with respect to symmetric conjugate points, Intern. J. of Math. Analysis, (2011) (in press). [5] Loo, C.P. and Janteng, A. : Subclass of starlike functions with respect to symmetric conjugate points, Intern. J. of Algebra, 5(16)(2011): 755-762. Received: August, 2011