Applied Mathematical Sciences, Vol. 2, 2008, no. 35, 1739-1748 Subclass of Univalent Functions with Negative Coefficients and Starlike with Respect to Symmetric and Conjugate Points S. M. Khairnar and Meena More Department of Mathematics Maharashtra Academy of Engineering Alandi - 412 105, Pune (M. S.), India smkhairnar2007@gmail.com meenamore@maepune.com Abstract Let S be the class of functions analytic and univalent in the open unit disc given by f(z) =z + a k z k and a k C. In this paper we have studied three subclasses Ss M(α, β, δ),s c M(α, β, δ) and S scm(α, β, δ) consisting of analytic functions with negative coefficients and starlike with respect to symmetric points, starlike with respect to conjugate points and starlike with respect to symmetric conjugate points, respectively. Here we discuss coefficient inequality, growth, distortion, extreme points, convex combination and convolution properties of the three classes. Mathematics Subject Classification: 30C45, 30C50 Keywords: Analytic functions, univalent functions, starlike with respect to symmetric points, distortion bounds, convolution theorem 1. Introduction Let S be the class of functions analytic and univalent in the open unit disc given by f(z) =z + a k z k (1)
1740 S. M. Khairnar and M. More and a k C.M be the subclass of S consisting of functions f of the form a k z k (2) where a k 0,a k IR. In [5], Sakaguchi introduced the class of analytic functions which are univalent and starlike { with respect } to symmetric points. This class is denoted by Ss zf and satisfies Re > 0 for z D. This definition has given rise to many generalized and extended classes of functions. The f(z) f( z) subclasses Ss M(α, β, δ),s c M(α, β, δ) and S scm(α, β, δ) consisting of analytic functions with negative coefficients were introduced by Halim and et. al. in [1], and are respectively starlike with respect to symmetric points, starlike with respect to conjugate points, and starlike with respect to symmetric conjugate points. Here α, β satisfy the conditions 0 α<1, 0 <β<1, 0 δ<1 and 0 < 2(1 β) < 1. This paper extends the results in [2] to other properties namely, distortion, convex combination and convolution. Let S be the 1+αβ subclass ( ) of S consisting of functions starlike in D. Notice that f S iff Re z f > 0 for z D. f { Consider S} s, the subclass of S consisting of functions given by (1) satisfying Re zf (z) > 0,z D. These functions are called starlike with respect f(z) f( z) to symmetric points and were introduced by Sakaguchi in [5]. The same class is also considered by Robertson [8], Stankiewicz [4], Wu [14], Owa et. al. [12], and Aini Janteng, M. Darus [2]. El-Ashwah and Thomas in [10], have introduced two other subclasses namely Sc and S sc. In [13], Sudharsan et. al. and Aini Janteng, M. Darus in [2] have discussed the subclass Ss (α, β, δ) of functions f analytic and univalent in Id given by (1) and satisfying the condition zf (z) (1 + δ) f(z) f( z) <β αzf (z) +(1 δ) f(z) f( z) for some 0 α 1, 0 <β 1 and z ID. However, in this paper we consider the subclass M defined by (2). Definition 1.1 : A function f Ss M(α, β, δ) is said to be starlike with respect to symmetric points if it satisfies zf (z) (1 + δ) f(z) f( z) <β αzf (z) +(1 δ) f(z) f( z) for z D. Definition 1.2 : A function f S c M(α, β, δ) is said to be starlike with respect
Subclass of univalent functions 1741 to conjugate points if it satisfies zf (z) (1 + δ) f(z)+f(z) <β αzf (z) +(1 δ) f(z)+f(z) for z D. Definition 1.3. A function f Ssc M(α, β, δ) is said to be starlike with respect to symmetric conjugate points if it satisfies zf (z) (1 + δ) f(z) f( z) <β αzf (z) +(1 δ) for z ID. f(z) f( z) Notice that the above conditions imposed on α, β and δ in the introduction are necessary to ensure that these classes form a subclass of S. First we state the preliminary results similar to those obtained by Halim et. al. in [1], required for proving our main results. 2. Preliminaries Theorem 2.1: f S s M(α, β, δ) if and only if a k 1. (3) Corollary 2.1 : If f Ss M(α, β, δ) then a k k(1 + βα)+[1 β + δ(1 + β)](( 1) k 1), k 2. Theorem 2.2 : f Sc M(α, β, δ) if and only if Corollary 2.2 : If f S c M(α, β, δ) then a k k(1 + βα)+2[β 1 δ(1 + β)] a k 1. (4) k(1 + βα)+2[β 1 δ(1 + β)], k 2. Theorem 2.3: f Ssc M(α, β, δ) if and only if a k 1. (5)
1742 S. M. Khairnar and M. More Corollary 2.3 : If f Ssc M(α, β, δ) then a k k(1 + βα)+[1 β + δ(1 + β)](( 1) k 1), k 2. 3. Growth and Distortion Theorems Theorem 3.1 : Let the function f be defined by (2) and belong to the class Ss M(α, β, δ). Then for {z :0< z = r<1}, r r 2 f(z) r + Proof. Let f(z) = z f(z) z + since (3) implies Similarly f(z) z a k z k a k z k r + r 2 a k z k r 2. 2 a k r + r a k. z z 2 a k = r r 2 Hence the result. The result is sharp for (6) 2 a k r r. (7) f(z) =z z 2 at z = ±r. Next we state similar results for functions belonging to Sc M(α, β, δ) and SscM(α, β, δ). Method of proof is same as in Theorem 3.1. Theorem 3.2 : Let the function f be defined by (2) and belong to the class Sc M(α, β, δ). Then for {z :0< z = r<1}, r 2[β(α +1) δ(1 + β)] r2 f(z) r + 2[β(α +1) δ(1 + β)] r2.
Subclass of univalent functions 1743 the result is sharp for 2[β(α +1) δ(1 + β)] z2 at z = ±r. Theorem 3.3 : Let the function f be defined by (2) and belong to the class Ssc M(α, β, δ). Then for {z :0< z = r<1}, r r 2 f(z) r + The result is sharp for r 2. z 2 at z = ±r. Next we state the distortion theorems. Theorem 3.4 : Let f Ss M(α, β, δ). Then for {z :0< z = r<1} 1 r f (z) 1+ (1 + βα) The result is sharp for z 2. r. (1 + βα) Theorem 3.5 : Let f be the function defined by (2) and belonging to the class Sc M(α, β, δ). Then for {z :0< z = r<1} 1 β(α +1) δ(1 + β) r f (z) 1+ β(α +1) δ(1 + β) r. The result is sharp for 2[β(α +1) δ(1 + β)] z2 at z = ±r. Theorem 3.6 : Let f be the function defined by (2) and belonging to the class S scm(α, β, δ). Then for {z :0< z = r<1} 1 r f (z) 1+ (1 + βα) The result is sharp for r. (1 + βα) z 2 at z = ±r.
1744 S. M. Khairnar and M. More 4. Closure Theorems All the three subclasses discussed here are closed under convex linear combinations. We prove for the class Ss M(α, β, δ). It can easily be proved similarly for Sc M(α, β, δ) and S scm(α, β, δ). Theorem 4.1 : Consider f j (z) = z a k,j z k Ss M(α, β, δ) for j = 1, 2,,l then g(z) = l c j f j (z) Ss M(α, β, δ) where l c j =1. Proof. Let g(z) = l c j (z = z = z z k ) a k,j z k l c j a k,j e k z k where e k = Now g(z) Ss M(α, β, δ) since e k l l c j a k,j. c j a k,j β(α + 2(1 δ) (1 + 2δ) l c j = 1 since 5. Extreme Points k(1 + βα)+[1 β + δ(1 + β)]( 1) k 1) a k,j 1. β(α+2(1 δ)) (1+2δ) Theorem 5.1 : Let f 1 (z) =z, f k (z) =z k(1+βα)+[1 β+δ(1+β)](( 1) k 1) zk for k 2. Then f Ss M(α, β, δ) if and only if it can be expressed in the form f(z) = λ k f k (z) where λ k 0 and λ k =1. Proof. Let f(z) = λ k f k (z) = z λ k k(1 + βα)+[1 β + δ(1 + β)](( 1) k 1) zk. (8)
Subclass of univalent functions 1745 Now f(z) Ss M(α, β, δ) since { } k(1 + βα)+[1 β + δ(1 + β)](( 1) k 1) λ k = λ k =1 λ 1 1. Conversely, suppose that f S sm(α, β, δ). Then by Corollary 2.1 set a k k(1 + βα)+[1 β + δ(1 + β)](( 1) k 1), k 2 λ k = k(1 + βα)+[1 β + δ(1 + β)](( 1)k 1) a k, k 2 (9) and λ 1 =1 λ k then f(z) = λ k f k (z). Similarly extreme points for functions belonging to S sm(α, β, δ) and S scm(α, β, δ) are found. Methods of proving Theorem 5.2 and Theorem 5.3 are similar to that of Theorem 5.1. Theorem 5.2 : Let f 1 (z) =z, f k (z) =z k(1 + βα)+2[β 1 δ(1 + β)] zk, k 2. Then f Sc M(α, β, δ) if and only if it can be expressed in the form f(z) = λ k f k (z) where λ k 0 and λ k =1. Theorem 5.3 : Let f 1 (z) =z, f k (z) =z k(1 + βα)+[1 β + δ(1 + β)](( 1) k 1) zk, k 2. Then f Ssc M(α, β, δ) if and only if it can be expressed in the form f(z) = λ k f k (z) where λ k 0 and λ k =1. 6. Convolution Theorems The three subclasses S sm(α, β, δ),s c M(α, β, δ) and S scm(α, β, δ) are closed under convolution. We prove for the class S sm(α, β, δ).
1746 S. M. Khairnar and M. More Theorem 6.1 : Let f,g SsM(α, β, δ) where f(z) =z a k z k and g(z) =z b k z k then f g SsM(α, β, γ) for γ k +(1+δ)(( 1) k 1)[N] 2 +(1+2δ)[D] 2 (α + 2(1 δ))[d] 2 (αk +(δ 1)(( 1) k 1)[N] 2 where N = and D =. Proof : We have f Ss M(α, β, δ) if and only if a k 1. (10) Similarly g Ss M(α, β, δ) if and only if b k 1. (11) To find a smallest number γ such that k(1 + γα) =[1 γ + δ(1 + γ)](( 1) k 1) a k b k 1 (12) γ(α + 2(1 δ)) (1 + 2δ) By Cauchy Schwarz inequality (10) and (11) imply ak b k 1 (13) (12) will hold for k(1 + γα)+[1 γ + δ(1 + γ)](( 1) k 1) a k b k γ(α + 2(1 δ)) (1 + 2δ) k(1 + βα)+[1 β + δ(1 + β)](( 1)k 1) ak b k. That is if ak b k [γ(α + 2(1 δ)) (1 + 2δ)][k(1 + βα)+[1 β + δ(1 + β)](( 1)k 1)] [][k(1 + γα)+[1 γ + δ(1 + γ)](( 1) k 1)] (14) (13) implies ak b k. (15)
Subclass of univalent functions 1747 Thus it is enough to show that k(1 + βα)+[1 β + δ(1 + β)](( 1)k 1) [γ(α + 2(1 δ)) (1 + 2δ)][k(1 + βα)+[1 β + δ(1 + β)](( 1)k 1)] [][k(1 + γα)+[1 γ + δ(1 + γ)](( 1) k 1)] which simplifies to γ [k +(1+δ)( 1) k 1)][N] 2 +(1+2δ)[D] 2 (α + 2(1 δ))[d] 2 (αk +(δ 1)(( 1) k 1)[N] 2 where N = and (16) D = k(1 + βα)+[1 β + δ(1 + β)](( 1) k 1) (17) Theorem 6.2 : Let f,g Sc M(α, β, δ) then f g S c M(α, γ, δ) where γ (k 2(1 + δ))[n] 2 +[D] 2 (α +2(δ 1))[D] 2 (kα + 2(1 δ))[n] 2 where N = and D = k(1 + βα)+2[β(1 δ) (1 + δ)]. Convolution theorem for subclass Ssc M(α, β, δ) is similar to Theorem 6.1. REFERENCES [1] S. A. Halim, A. Janteng and M. Darus, Coefficient properties for classes with negative coefficients and starlike with respect to other points, Proceeding of 13-th Mathematical Sciences National Symposium, UUM, 2 (2005), 658-663. [2] Aini Janteng, M. Darus, Classes with negative coefficients and starlike with respect to other points, International Mathematical Forum, 2, (2007), No. 46, 2261-2268. [3] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51(1) (1975), 109-116. [4] J. Stankiewicz, Some remarks on functions starlike with respect to symmetric points. Ann. Univ. Marie Curie Sklodowska, 19(7) (1965), 53-59. [5] E. K. Sakaguchi, On certain univalent mapping, J. Math. Soc. Japan, 11 (1959), 72-75.
1748 S. M. Khairnar and M. More [6] K. Al Shaqsi and M. Darus, On subclass of harmonic starlike functions with respect to K-symmetric points, International Mathematical Forum, 2 (2007), No. 57, 2799-2805. [7] K. Al Shaqsi and M. Darus, On subclass of close-to-convex functions, Int. Jou. Contemp. Math. Sci., 2 (2007), No. 15, 745-757. [8] M.I.S. Robertson, Applications of the subordination principle to univalent functions, Pacific J. of Math., 11 (1961), 315-324. [9] V. Ravichandran, Starlike and convex functions with respect to conjugate points, Acta Math. Acad. Paedagog, Nyíregyházi, 20 (2004), 31-37. [10] R. M. El. Ashwah and D. K. Thomas, Some subclasses of close-to-convex functions, J. Ramanujan Math. Soc., 2 (1987), 86-100. [11] P. L. Duren, Univalent Functions, Grundlehren Math. Wiss., Vol. 259, Springer, New York, 1983. [12] S. Owa, Z. Wu and F. Ren, A note on certain subclass of Sakaguchi functions, Bull. de la Royale de liege, 57 (3) (1988), 143-150. [13] T. V. Sudharsan, P. Balasubrahmanayan and K. G. Subramanian, On functions starlike with respect to symmetric and conjugate points, Taiwanese Journal of Mathematics, 2(1) (1998), 57-68. [14] Z. Wu, On classes of Sakaguchi functions and Hadamard products, Sci. Sinica Ser. A, 30 (1987), 128-135. Received: December 25, 2007