Properties of Starlike Functions with Respect to Conjugate Points

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 5, 1, no. 38, Properties of Starlike Functions with Respect to Conjugate Points Suci Aida Fitri Mad Dahhar and Aini Janteng School of Science and Technology, Universiti Malaysia Sabah Locked Bag No. 73, Kota Kinabalu, Sabah, Malaysia suciaida aini Abstract Let 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 zf (z) 1+Az 1+Bz f(z)+f( z), 1 B<A 1, z D. The aim of paper are to determine distortion bounds and preserving property for the class Sc (A, B). Mathematics Subject Classification: Primary 3C45 Keywords: starlike with respect to conjugate points, distortion bounds, preserving property. 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 k=1 w() =, w(z) < 1, z D. 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= Also, let Ss be the subclass of S consisting of functions given by (1) satisfying { zf } (z) Re >, z D. f(z) f( z)

2 1856 Suci Aida Fitri Mad Dahhar and Aini Janteng These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi in El-Ashwah and Thomas in [1], introduced two other classes namely the class Sc consisting of functions starlike with respect to conjugate points and Ssc consisting of functions starlike with respect to symmetric conjugate points. Further, let f,g U. Then we say that f is subordinate to g, and we write f g, if there exists a function w Usuch that f(z) =g(w(z)) for all z D. Specially, if g is univalent in D, then f g if and only if f() = g() and f(d) g(d). In terms of subordination, Mad Dahhar and Janteng in 9 introduced a subclass of S c denoted by S c (A, B). Let S c (A, B) denote the class of functions of the form (1) and satisfying the condition zf (z) f(z)+f( z) 1+Az, 1 B<A 1, z D. 1+Bz By definition of subordination it follows that f Sc (A, B) if and only if where zf (z) f(z)+f( z) = 1+Aw(z) 1+Bw(z) P (z) =1+ n=1 = P (z), w U () p n z n. (3) We study the class S c (A, B) and obtain distortion bounds and preserving property. Preliminary Result We need the following preliminary lemma, required for proving our result. Lemma.1 ([]) Let N(z) be analytic and D(z) starlike in D and N() = D() =. Then ( N (z) D (z) 1) ( ) A B N (z) < 1 implies D (z) ( N(z) D(z) 1) ( ) A B N(z) < 1, z D. D(z)

3 Properties of starlike functions Main Result We give the distortion bounds and preserving property for the class S c (A, B). Theorem 3.1 Let f Sc (A, B), then for z = r, <r<1, 1 Ar (1 Br)(1 + r) f (z) 1+Ar (1 + Br)(1 r) (4) and r 1 At r (1 Bt)(1 + t) dt f(z) 1 At dt. (5) (1 + Bt)(1 t) The bounds are sharp. Proof. Put h(z) = f(z)+f( z). Then from (), we obtain zf (z) = h(z) Since h is starlike, it follows that (see [4]) 1+Aw(z) 1+Bw(z). (6) r (1 + r) h(z) r (1 r). (7) 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. (8) Applying results (7) and (8) in (6) we obtain (4). Next, set z = r, and upon elementary integration of (4) will give the results in (5). The extremal functions corresponding to the left and right sides of (4) and (5) are, respectively f(z) = (1 At) (1 Bt)(1 + t) dt and f(z) = (1 + At) (1 + Bt)(1 t) dt. Theorem 3. If f Sc (A, B) then F S c (A, B), where F (z) = z f(t) dt.

4 1858 Suci Aida Fitri Mad Dahhar and Aini Janteng Proof. With the given F above, consider zf (z) F (z)+f ( z) = 1 zf (z) [ f(t)dt f(t)dt + f( t)dt ]. Suppose, we let N(z) and D(z) be the numerator and denominator functions respectively. It can be shown that D(z) = 1 [ ] f(t)dt + f( t)dt is starlike. Furthermore, N (z) D (z) = zf (z) f(z)+f( t) with f Sc (A, B). Thus N (z) D (z) = 1+Aw(z) 1+Bw(z), w U. This implies that ( N (z) D (z) 1) ( ) A B N (z) < 1. D (z) Hence, by Lemma.1, we have ( N(z) D(z) 1) ( ) A B N(z) < 1, z D. D(z) or equivalently, Thus F S c (A, B). N(z) D(z) = 1+Aw 1(z) 1+Bw 1 (z) w 1 U. References [1] El-Ashwah, R.M. and Thomas, D.K. : Some subclasses of close-to-convex functions, J. Ramanujan Math. Soc., (1987): [] Goel, R.M. and Mehrok, B.C. : A subclass of starlike functions with respect to symmetric points, Tamkang J. Math., 13(1)(198): 11-4.

5 Properties of starlike functions 1859 [3] Mad Dahhar, S.A.F. and Janteng, A. : A subclass of starlike functions with respect to conjugate points, International Mathematical Forum, 4(8)(9): [4] Ravichandran, V. : Starlike and convex functions with respect to conjugate points, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, (4): [5] Sakaguchi, K. : On a certain univalent mapping, J. Math. Soc. Japan, 11(1959): Received: April, 9