Meromorphic Starlike Functions with Alternating and Missing Coefficients 1

Transcription

1 General Mathematics Vol. 14, No. 4 (2006), Meromorphic Starlike Functions with Alternating and Missing Coefficients 1 M. Darus, S. B. Joshi and N. D. Sangle In memoriam of Associate Professor Ph. D. Luciana Lupaş Abstract In this paper we introduced a new class Ω p (n, α) of meromorphic starlike univalent functions with alternating coefficients in the punctured unit disk U = {z : 0 < z < 1}. We obtain among other results are the coefficient inequalities, distortion theorem and class preserving integral operator Mathematics Subject Classification: 30C45 Keywords: meromorphic function, Starlike functions, Distortion theorem 1 Received 24 September, 2006 Accepted for publication (in revised form) 9 November,

2 114 M. Darus, S. B. Joshi and N. D. Sangle 1 Introduction Let A p denote the class of functions of the form: (1) f(z) = a z + a p+k z p+k, (a 0, p N = {1, 2, 3,...}) which are regular in the punctured unit disk U = {z : 0 < z < 1}. Define (2) D 0 f(z) = f(z) (3) Df(z) = D 1 f(z) = a z + (p + k + 2)a p+k z p+k = (z2 f(z)) z (4) D 2 f(z) = D(D 1 f(z)), and for n = 1, 2, 3,... (5) D n f(z) = D(D n f(z)) = a z + (p + k + 2)a p+k z p+k, = (z2 D n f(z)) z Let B n (α), denote the class consisting functions in A p satisfying (6) Re { D n+1 f(z) D n f(z) } 2 < α, (z U, 0 α < 1,n N o = N {0}). Let Λ p be the subclass of A p which consisting of functions of the form (7) f(z) = a z + () p+k a p+k z p+k, (a > 0;a p+k 0, p N).

3 Meromorphic Starlike Functions Further let (8) Ω p (n,α) = B n (α) Λ p. We note that, in [1] Uralegaddi and Somanatha define a class B n (α), which consists functions of the form f(z) = a z + a k z k (a 0) which are analytic in U and obtained inclusion relation, the class preserving integral. In the same year, Uralegaddi and Ganigi [2] considered meromorphic starlike functions with alternating coefficients. Further, recently in [4], Aouf and Darwish also considered meromorphic starlike univalent functions with alternating coefficients and obtained coefficient inequalities, distortion theorem and integral operators. The class of meromorphic functions have been studied by various authors and among are Darwish [3], Aouf and Hossen [5], and Mogra et.al [6], few to mention. In the present paper, we consider functions of the forms (7) and obtain basic properties, which include for example the coefficient inequalities, distortion theorem, closure theorem and integral operators. Finally, the class preserving integral operator of the form (9) F c+1 (z) = (c + 1) 1 0 u c+1 f(uz)du (0 u < 1, 0 < c < ) is considered. Techniques used are similar to those of Aouf and Hossen [5].

4 116 M. Darus, S. B. Joshi and N. D. Sangle 2 Coefficient Inequality Theorem 2.1. Let the function f(z) be defined by (1). If (10) (p + k + 2) n (p + k + α) a p+k (1 α) a, then f(z) B n (α). Proof. It suffices to show that D n+1 f(z) 1 (11) D n f(z) D n+1 f(z) < 1, z < 1, (3 2α) D n f(z) we have D n+1 f(z) 1 D n f(z) D n+1 f(z) (3 2α) D n f(z) = (p + k + 2) n (p + k + 1)a p+k z p+k = (2 2α)a (p + k + 2) n (p + k + 2α 1)a p+k z p+k (p + k + 2) n (p + k + 1) a p+k (2 2α)a. (p + k + 2) n (p + k + 2α 1) a p+k The last expression is bounded by 1 if (p+k+2) n (p+k+1) a p+k (2 2α)a (p+k+2) n (p+k+2α) a p+k, which reduces to (12) (p + k + 2) n (p + k + α) a p+k (1 α) a. But (12) is true by hypothesis. Hence the result follows.

5 Meromorphic Starlike Functions Theorem 2.2. Let the function f(z) be defined by (7). Then f(z) Ω p (n,α) if and only if (13) (p + k + 2) n (p + k + α) a p+k (1 α)a, Proof. In view of Theorem 2.1, it suffices to prove the only if part. Let us assume that f(z) defined by (7) is in Ω p (n,α). Then { } { } D n+1 f(z) D n+1 f(z) 2D n f(z) Re 2 = Re, D n f(z) D n f(z) (14) a (p + k + 2) n (p + k)a p+k z p+k = Re a < α, z < 1. () p+k (p + k + 2) n a p+k z p+k Choose value of z on the real axis so that Dn+1 f(z) D n f(z) 2 is real. Upon clearing the denominator in (14) and letting z 1 through real values, we obtain ( a (p + k + 2) n (p + k)a p+k a + (p + k + 2) n a p+k )α. Thus (p + k + 2) n (p + k + α)a p+k (1 α)a. Hence the result follows. Corollary 1 Let the function f(z) be defined by (7) be in Ω p (n,α). Then (15) a p+k The result is sharp for the function (1 α)a (p + k + 2) n (p + k + α), (16) f(z) = a z + (1 α)a ()p+k (p + k + 2) n (p + k + α) zp+k, (k 1).

6 118 M. Darus, S. B. Joshi and N. D. Sangle 3 Distortion Theorem Theorem 3.1. Let the function f(z) be defined by (7) be in Ω p (n,α). Then for 0 < z < 1, we have (17) a r (1 α)a (p + 3) n (p + α + 1) r f(z) a r + (1 α)a (p + 3) n (p + α + 1) r, where equality holds for the function (18) f(z) = a z + (1 α)a (p + 3) n (p + α + 1) zp, (z = ir,r), and (19) a r 2 (1 α)a (p + 3) n (p + α + 1) f (z) a r 2 + (1 α)a (p + 3) n (p + α + 1), where equality holds for the function f(z) given by (18) at z = ir, r. Proof. In view of Theorem 2.2, we have (20) a p+k (1 α)a (p + k + 2) n (p + k + α). Thus, for 0 < z < 1, (21) and (22) f(z) a r + r a p+k a r + (1 α)a (p + 3) n (p + α + 1) r f(z) a r r a p+k a r (1 α)a (p + 3) n (p + α + 1) r.

7 Meromorphic Starlike Functions Thus (17) follows. Since (p+3) n (p+α+1) (p+k) a p+k (p+k+2) n (p+k+α) a p+k (1 α) a where (p+k+2)n (p+k+α) p+k follows that is an increasing function of k, from Theorem 2.2, it (23) (p + k) a r + (1 α)a (p + 3) n (p + α + 1). Hence (24) f (z) a r + (p + k)a 2 p+k r p+k a r + (p + k)a p+k a r + (1 α)a (p + 3) n (p + α + 1) and (25) f (z) a r (p + k)a 2 p+k r p+k a r (p + k)a p+k a r (1 α)a (p + 3) n (p + α + 1) Thus (19) follows. It can be easily seen that the function f(z) defined by (18) is extremal for the theorem.

8 120 M. Darus, S. B. Joshi and N. D. Sangle 4 Closure Theorem Let the function f j (z) be defined for j {1, 2, 3,...,m}, by (26) f j (z) = a,j z for z U. + () p+k a p+k,j z p+k, (a,j > 0;a p+k,j 0) Now, we shall prove the following result for the closure of function in the class Ω p (n,α). Theorem 4.1. Let the functions f j (z) be defined by (26) be in the class Ω p (n,α) for every j {1, 2, 3,...,m}. Then the function F(z) defined by (27) F(z) = b z + () p+k b p+k z p+k, (b > 0;b p+k 0, p N) is a member of the class Ω p (n,α), where (28) b = 1 m a,j and b p+k = 1 m j=1 a p+k,j (k = 1, 2,...). j=1 Proof. Since f j (z) Ω p (n,α), it follows from Theorem 2.2, that (29) (p + k + 2) n (p + k + α) a p+k,j (1 α) a,j, for every j {1, 2, 3,...,m}. Hence, = (p + k + 2) n (p + k + α)b p+k = ( ) (p + k + 2) n 1 (p + k + α) a p+k,j = m j=1

9 Meromorphic Starlike Functions = 1 m ( ) (p + k + 2) n (p + k + α)a p+k,j j=1 ( ) 1 (1 α) a,j = (1 α)b, m which (in view of Theorem 2.2) implies that F(z) Ω p (n,α). j=1 Theorem 4.2. The class Ω p (n,α) is closed under convex linear combination. Proof. Let the f j (z)(j = 1, 2) defined by (26) be in the class Ω p (n,α), it is sufficient to prove that the function (30) H(z) = λf 1 (z) + (1 λ)f 2 (z) (0 λ 1) is also in the class Ω p (n,α). Since, for (0 λ 1), (31) H(z) = λa,1 + (1 λ)a,2 z {λa p+k,1 + (1 λ)a p+k,2 } z p+k, we observe that (p + k + 2) n (p + k + α) {λa p+k,1 + (1 λ)a p+k,2 } = = λ (p+k+2) n (p+k+α)λa p+k,1 +(1 λ) (p+k+2) n (p+k+α)λa p+k,2 (1 α) {λa,1 + (1 λ)a,2 } with the aid of Theorem 2.2. Hence H(z) Ω p (n,α). This completes the proof of Theorem 4.2.

10 122 M. Darus, S. B. Joshi and N. D. Sangle Theorem 4.3. Let f o (z) = a (32) z f p+k (z) = a z + (1 α)a (33) ()p+k (p + k + 2) n (p + k + α) zp+k, (k 1). Then f(z) Ω p (n,α) if and only if it can be expressed in the form (34) f(z) = where λ p+k f p+k (z), λ p+k (k 0) and λ k = 1. Proof. Let f(z) = λ p+k f p+k (z), where λ p+k (k 0) and λ k = 1. Then ( ) f(z) = λ p+k f p+k (z) = λ o f o (z) + λ p+k f p+k (z) = 1 λ p+k + + k=1 { } a λ p+k z + (1 α)a ()p+k (p + k + 2) n (p + k + α) zp+k = Since (35) = a z + (1 α)a ()p+k (p + k + 2) n (p + k + α) zp+k (p + k + 2) n (1 α)a λ p+k (p + k + α) (p + k + 2) n (p + k + α) = (1 α)a + λ p+k = (1 α)a (1 λ o ) (1 α)a,

11 Meromorphic Starlike Functions by Theorem 2.2, f(z) Ω p (n,α). Conversely, we suppose that f(z) defined by (7) is in the class Ω p (n,α). Then by using (15), we get (36) a p+k Setting (1 α)a, (k 1). (p + k + 2) n (p + k + α) (37) λ p+k = (p + k + 2)n (p + k + α) (1 α)a a p+k, (k 1). and (38) λ o = 1 λ p+k, we have (34). This completes the proof of the Theorem Integral Operator In his section we consider integral transforms of functions in the class Ω p (n,α). Theorem 5.1. Let the function f(z) be defined by (7) be in the class Ω p (n,α), then the integral transforms (39) F c+1 (z) = (c + 1) are in Ω p (n,α), where (40) δ(α,c,p) = 1 The result is sharp for the function 0 u c+1 f(uz)du (0 u < 1, 0 < c < ), (p + k + 3)(p + α + 1) (p + 1)(1 α)(c + 1). (p + k + 3)(p + α + 1) + (c + 1)(1 α) (41) f(z) = a z + (1 α)a ()p+k (p + 3) n (p + α + 1) zp.

12 124 M. Darus, S. B. Joshi and N. D. Sangle Proof. Let (42) F c+1 (z) = (c+1) 1 0 u c+1 f(uz)du = a z (c + 1) p + k + c + 2 a p+kz p+k. In view of Theorem 2.2, it is sufficient to show that ( ) (p + k + 2) n (p + k + δ) c + 1 (43) a p+k 1. (1 δ)a p + k + c + 2 Since f(z) Ω p (n,α), we have (p + k + 2) n (p + k + α) (44) a p+k 1. (1 α)a Thus (42) will be satisfy if or (45) δ (p + k + δ)(c + 1) (1 δ)(p + k + c + 2) p + k + α 1 α for each k, (p + k + c + 2)(p + k + α) (p + k)(1 α)(c + 1). (p + k + c + 2)(p + k + α) + (c + 1)(1 α) For each α, p, and c fixed, let = Then F(k) = (p + k + c + 2)(p + k + α) (p + k)(1 α)(c + 1). (p + k + c + 2)(p + k + α) + (c + 1)(1 α) F(k + 1) F(k) = (c + 1)(1 α)(p + k + 1)(p + k + 2) [(p + k + c + 2)(p + k + α) + (c + 1)(1 α)][(p + k + c + 3)(p + k + α + 1) + (c + 1)(1 α)] > for each k. Hence, F(k) is an increasing function of k. Since F(1) = the result follows. (p + c + 3)(p + α + 1) (p + 1)(1 α)(c + 1), (p + c + 3)(p + α + 1) + (c + 1)(1 α)

13 Meromorphic Starlike Functions References [1] B. A. Uralegaddi and C. Somanatha, New criteria for meromorphic starlike univalent functions, Bull. Austral. Math. Soc. 43, (1991), [2] B. A. Uralegaddi and M. D. Ganigi, Meromorphic starlike functions with alternating coefficients, Rend. Math. 11 (7), Roma(1991), [3] H. E. Darwish, Meromorphic p-valent starlike functions with negative coefficients, Indian J. Pure Appl. Math. 33(7), (2002), [4] M. K. Aouf and H. E. Darwish, Meromorphic starlike univalent functions with alternating coefficients, Utilitas Math. 47, (1995), [5] M. K. Aouf and H. M. Hossen, New criteria for meromorphic p-valent starlike functions, Tsukuba J. Math. 17(2), (1993), 1-6. [6] M. L. Mogra, T. R. Reddy and O. P. Juneja, Meromorphic univalent functions with positive coefficients, Bull. Austral. Math. Soc. 32, (1985), School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi, 43600, Selangor D.E., Malaysia address:

14 126 M. Darus, S. B. Joshi and N. D. Sangle Department of Mathematics Walchand College of Engineering Sangli (M. S), India Department of Mathematics Annasaheb Dange College of Engineering Ashta, Sangli (M. S), India address: navneet