On a New Subclass of Salagean Type Analytic Functions

Transcription

1 Int. Journal of Math. Analysis, Vol. 3, 2009, no. 14, On a New Subclass of Salagean Type Analytic Functions K. K. Dixit a, V. Kumar b, A. L. Pathak c and P. Dixit b a Department of Mathematics, Janta College, Bakewar Etawah (U.P.) India b Department of Mathematics, Christ Church College Kanpur (U.P.) India c Department of Mathematics, Brahmanand College Kanpur (U.P.) India alpathak@rediffmail.com Abstract Using the Salagean derivative, we introduce and study a n new subclass T,(n, p, γ) of analytic functions with negative coefficients. We obtain coefficient conditions, extreme points, distortion bounds, integral operators, Hadamard product and convex combination for the above class of analytic functions. Mathematics Subject Classification: 30C45, 30C50 Keywords: Analytic, Salagean operator, Uniformly convex, Hadamard product 1 Introduction Let A(j) denote the class of functions of the form f(z) =z + a k z k, (j N = {1, 2, 3,...}) (1.1.1) which are analytic in the unit disc U = {z : z < 1}. For function f(z) in A(j), we define D 0 f(z) =f(z),

2 690 K. K. Dixit, V. Kumar, A. L. Pathak and P. Dixit D 1 f(z) =zf (z). and D n f(z) =D(D n 1 f(z)), (z N). The differential operator D n was introduced by Salagean [9]. With the help of the differential operator D n, we say that a function f(z) belonging to A(j) is in the class S j (n, p, γ) if and only if { } Re (1 + pe iα ) Dn+1 f(z) pe iα γ,0 γ<1,α R, D n f(z) n N 0 = N (0), for some p 0 and for all z U. The operator D n was studied by Sekine [11] and Aouf and Salagean [1]. Let T (j) denote the subclass of A(j) consisting of functions of the form f(z) =z Further, we define the class T j (n, p, γ) by a k z k, (a k 0; j N). (1.1.2) T j (n, p, γ) =S j (n, p, γ) T (j). It is worthmentioning that this class T j (n, p, γ) was defined and studied by I. Magdas [5] for j =1,γ= 0 and further by Dixit and Pathak [2] for γ =0. By giving specific values to j, n, k and γ, we obtain the following important classes studied by various researchers in earlier works. (i) For j =1,n= 1 and γ = 0, we obtain the class of functions f(z) satisfying the condition Re 1+ zf (z) p zf (z) f (z) f (z) z U, studied by Murugusundaramoorthy [6] and K.G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahmanyam and H. Silverman [13]. (ii) For j = 1,n = 0 and γ = 0, we obtain the class of functions f(z) satisfying the condition p zf (z) f(z) 1 zf Re (z),z U. f(z) (iii) For j =1,n=1, p = 1 and γ = 0, we obtain the class of functions f(z) satisfying the condition Re 1+ zf (z) f (z) zf (z) f(z),z U.

3 Subclass of Salagean type analytic functions 691 (iv) For j =1,n=0,p= 1 and γ = 0, we obtain the class of functions f(z) satisfying the condition zf (z) f(z) 1 zf Re (z),z U, f(z) studied by Ronning [8]. (v) For j =1, n =1, p = 0 and γ = 0, we obtain the class of functions f(z) satisfying the condition Re 1+ zf (z) 0. f (z) This is the class of convex functions with negative coefficients studied by Silverman [12]. (vi) For j =1,n=0,p= 0 and γ = 0, we obtain the class of functions f(z) satisfying the condition zf (z) Re 0, f(z) investigated by Robertson [7] and Silverman [12]. In view of these remarks, we see that a study of the class T j (n, p, γ) leads to unified results on properties of various subclasses of univalent functions. 2 Main Results In our first theorem, we introduce a sufficient coefficient bound for analytic functions in T j (n, p, γ). Theorem 2.1: Let the functions f(z) be given by (1.1). Then f(z) T j (n, p, γ) if and only if for some p 0 and 0 γ<1. k n [k(p +1) p γ] a k (), (2.2.1) Proof. Assume that f(z) T j (n, p, γ), then by definition, D n+1 f(z) Re γ p D n+1 f(z) D n f(z) 1 D n f(z),z U,

4 692 K. K. Dixit, V. Kumar, A. L. Pathak and P. Dixit 1 k n+1 a k z k 1 1 k n+1 a k z k 1 Re γ p 1 1 k n a k z k 1 1 k n a k z k 1 p k n+1 a k z k 1 1 k n a k z k 1 k n a k z k 1. (2.2.2) Choosing values of z on the real axis so that the left side of (2.2) is real and letting z 1,we get ( ) 1 k n+1 a k γ + γ k n a k p (k n+1 k n ) a k k n [(p +1)k p γ] (). Conversely, suppose that (2.1) is true for z U. it is sufficient to prove that p D n+1 f(z) 1 D n+1 D n f(z) Re f(z) 1 (). D n f(z) We have p D n+1 f(z) 1 D n+1 D n f(z) Re f(z) 1 (p +1) D n+1 f(z) D n f(z) 1 D n f(z) k n (k 1) a k z k+1 This completes the proof (p +1) (p +1) 1 1 k n a k z k+1 k n (k 1) a k, by (2.1). k n a k

5 Subclass of Salagean type analytic functions 693 Remark: For j = 1 and γ = 0, Theorem 2.1 was proved by magdas [5] and further by Dixit and Pathak [2] for γ =0. Corollary 2.2: Let the function f(z) defined by (1.1) is in the class T j (n, p, γ). Then 0 a k The result is sharp for the functions f(z) =z k n [(p +1)k p γ]. k n [(p +1)k p γ] zk. (2.2.3) We prove the following Theorems 2.3, 2.4, 2.5 and 2.6 by using techniques adapted by Dixit and Pathak [2]. Theorem 2.3: Let 0 p 1 p 2, j N and n N 0. Then T j (n, p 1,γ) T j (n, p 2,γ). Theorem 2.4: For p 0, j N, n N, T j (n +1,p,γ) T j (n, p, γ). Theorem 2.5: The class T j (n, p, γ) is closed under conves linear combinations. Theorem 2.6: Let the function f(z) defined by (1.1) be in the class T,(n, p, γ). Then for z = r<1, and D i f(z) r D i f(z) r + (j +1) n i [(p + 1)(j +1) p γ] rj+1 (2.2.4) (j +1) n i [(p + 1)(j +1) p γ] rj+1 (2.2.5) for z U and 0 i n. The equalities in (2.4) and (2.5) are attained for the function f(z) given by D i f(z) = z or equivalently, f(z) =z ()z j+1, (z = ±r) (2.2.6) (j +1) n i [(p + 1)(j +1) p γ] ()z j+1, (z = ±r). (j +1) n [(p + 1)(j +1) p γ]

6 694 K. K. Dixit, V. Kumar, A. L. Pathak and P. Dixit Corollary 2.7: Let the function f(z) defined by (1.1) be in the class T j (n, p, γ). Then for z = r<1, f(z) r ()r j+1 (j +1) n [(p + 1)(j +1) p γ] (2.2.7) and f(z) r + ()r j+1, (z U). (2.2.8) (j +1) n [(p + 1)(j +1) p γ] The equalities in (2.7) and (2.8) are attained for the function given by f(z) =z ()r j+1 (j +1) n [(p + 1)(j +1) p γ]. Proof. Taking i = 0 in Theorem 2.6, we immediately obtain (2.7) and (2.8). Corollary 2.8: Let the function f(z) defined by (1.1) be in the class T j (n, p, γ). Then for z = r<1, f (z) 1 ()r j (j +1) n 1 [(p + 1)(j +1) p γ] (2.2.9) and f (z) 1+ ()r j (j +1) n 1 [(p + 1)(j +1) p γ]. (2.2.10) by The equalities in (2.9) and (2.10) are attained for the function f(z) given f(z) =z ()z j+1 (j +1) n [(p + 1)(j +1) p γ]. Proof. Setting i = 1 in Theorem 2.6 and making use of the definition (1.1), we arrive at Corollary 2.8. Remark: Corollaries 2.7 and 2.8 for j = 1 and γ = 0, were proved by Magdas [5] and further by Dixit and Pathak [2] for γ =0. Theorem 2.9: Let f j (z) =z and f k (z) =z k n [(p +1)k p γ] zk,

7 Subclass of Salagean type analytic functions 695 (k j +1; n N 0 ), for p 0. Then f(z) is in the class T j (n, p, γ) if and only if it can be expressed in the form f(z) = μ k f k (z), (2.2.11) k=j where μ k 0, (k j) and μ k =1. k=j Proof. Assume that f(z) = μ k f k (z) =z k=j k n [(p +1)k p γ] μ kz k. Then it follows that k n () [(p +1)k p γ] k n [(p +1)k p γ] μ k =(1 γ) μ k =(1 γ)(1 μ j ). So, by Theorem 2.1, f(z) T j (n, p, γ). Conversely, assume that the function f(z) defined by (1.1) belongs to the class T j (n, p, γ). Then and Setting a k k n [(p +1)k p γ], (k j +1; n N 0). μ k = kn [(p +1)k p γ], (k j +1; n N 0 ), μ j =1 we can see that f(z) can be expressed in the form (2.11). This completes the proof of Theorem 2.9. We employ the techniques by Dixit and Pathak [2] in the proof of Theorem 2.10, 2.11, 2.12, 2.13, 2.14 and μ k,

8 696 K. K. Dixit, V. Kumar, A. L. Pathak and P. Dixit Theorem 2.10: Let the function f(z) defined by (1.1) be in the class T j (n, p, γ). Then f(z) is close-to-convex of order ρ(0 ρ<1) in z <r 1, where r 1 = r 1 (n, p, γ, ρ) = inf k [ ] (1 ρ)k n 1 1/(k 1) {(p +1)k p γ}, (k j +1). The result is sharp with the extremal function f(z) given by (2.3). (2.2.12) Theorem 2.11: Let the function f(z) defined by (1.1) be in the class T j (n, p, γ). Then f(z) is starlike of order ρ(0 ρ<1) in z <r 2, where r 2 = r 2 (n, p, γ, ρ) = inf k [ ] 1/(k 1) 1 ρ {(p +1)k p γ} k, (k j +1). k ρ (2.2.13) The result is sharp with the extremal function f(z) given by (2.3). Theorem 2.12: Let the function f(z) defined by (1.1) be in the class T j (n, p, γ). Then f(z) is convex of order ρ(0 ρ<1) in z <r 3, where r 3 = r 3 (n, p, γ, ρ) = inf k [ ] 1/(k 1) 1 ρ {(p +1)k p γ} kn 1, (k j +1). k ρ (2.2.14) The result is sharp with the extremal function f(z) given by (2.3). Theorem 2.13: Let the function f(z) defined by (1.1) be in the class T j (n, p, γ) and let c be a real number such that c> 1. Then the function f(z) defined by F (z) = c +1 z z also belongs to the class T j (n, p, γ). 0 t c 1 f(t)dt, (c > 1) (2.2.15) Theorem 2.14: Let the function F (z) =z a k z k, (a k 0, j N), be in the class T j (n, p, γ) and let c be a real number such that c> 1. Then the function f(z) given by (2.15) is univalent in z <R, where [ ] (c +1)k R n 1 1(k 1) {(p +1)k p γ} inf, (k j +1). (2.2.16) k (c + k)()

9 Subclass of Salagean type analytic functions 697 The result is sharp. Let the functions f ν (z)(ν =1, 2) be given by f ν (z) =z a k z k, (a k j 0; ν =1, 2). (2.2.17) The modified Hadamard product of f 1 (z) and f 2 (z) is defined by f 1 f 2 (z) =z a k,1 a k,2 z k. (2.2.18) Theorem 2.15: Let f 1 (z) T j (n, p, γ) and f 2 (z) T j (n, p, γ). f 2 (z) T j (n, ξ, γ), where ξ = ξ(j, n, p, q, γ) Then f 1 = (j +1)n [(p + 1)(j +1) p γ][(q + 1)(j +1) q γ] (j +1 γ)(). j() (2.2.19) and The result is best possible for the functions f 1 (z) =z f 2 (z) =z (j +1) n [(p + 1)(j +1) p γ] zj+1 (2.2.20) (j +1) n [(q + 1)(j +1) q γ] zj+1 (2.2.21) Corollary 2.16: Let each of the functions f ν (z)(ν =1, 2) defined by (2.4) be in the class T j (n, p, γ). Then f 1 f 2 (z) T j (n, ξ, γ), where ξ = (j +1)n [(p + 1)(j +1) p γ] 2 (j +1 γ)(). (2.2.22) The result is sharp. Proof. Setting q = p in Theorem 2.15, we arrive at Corollary The result is sharp for the functions f ν (z) =z (j +1) n [(p + 1)(j +1) p γ] zj+1, (ν =1, 2).

10 698 K. K. Dixit, V. Kumar, A. L. Pathak and P. Dixit Theorem 2.17: Let the function f ν (z)(ν =1, 2) defined by (2.17) be in the class T j (n, p, γ), then the function h(z) =z belong to the class T j (n, η, γ), where (a 2 k,1 + a2 k,2 )zk, (2.2.23) η = η(j, n, p, γ) = (j +1)n {(p + 1)(j +1) p γ} 2 2{(j +1) γ}(). 2j() (2.2.24) The result is sharp for the functions f ν (z), (ν =1, 2) given by f ν (z) =z (j +1) n [(p + 1)(j +1) p γ] 2 zj+1 Proof. By virtue of Theorem 2.1, we have [k n {(p +1)k p γ}] 2 a 2 k,1 and that is [ k n {(p +1)k p γ}] 2 () 2 (2.2.25) [k n {(p +1)k p γ}] 2 a 2 k,2 [ k n {(p +1)k p γ}a k,2 ] 2 () 2. (2.2.26) It follows from (2.25) and (2.26) that 1 2() 2 [kn {(p +1)k p γ}] 2 (a 2 k,1 + a2 k,2 ) 1. (2.2.27) Therefore, we need to find the largest η such that k n [(η +1)k η γ] 1 = γ η(k 1) + k γ 1 2() 2 [kn {(p +1)k p γ}] 2, k n [(p +1)k p γ]2 2()

11 Subclass of Salagean type analytic functions 699 η kn [(p +1)k p γ] 2 2(k γ)(). (2.2.28) 2()(k 1) Since right hand side of (2.28) is an increasing function of k, we readily have η (j +1)n [(p + 1)(j +1) p γ] 2 {(j +1) γ}() 2j() (2.2.29) and Theorem 2.17 follows at once. References [1] Aouf M.K. and Salagean G.S., Prestarlike functions which negative coefficients, Rev. Roum. Math. Pures. Appl., XLIV (1999), No. 4, [2] Dixit K.K. and Pathak A.L., A new subclass of analytic functions defined by Salagean operator, Bull. Cal. Math. Soc., accepted (2008). [3] Goodman A.W., On uniformly starlike functions, J. Math. and Appl., 155 (1991), [4] Goodman A.W., On uniformly convex functions, Ann. Polon. Math., 56, (1991). [5] Madas I., On α-type uniformly convex functions, Studia Univ. Babes- Bolyai, Mathematica, Vol. XLIV (1991), No. 1, [6] Murugusundaramoorthy G., Studies on classes of analytic functions with negative coefficients, Ph.D. Thesis, University of Madras, [7] Robertson M.S., On the theory of univalent functions, Annals of Math., 37 (1936), [8] Ronning F., Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), No. 1, [9] Salagean G.S., Subclass of univalent functions, Lecture Notes in Mathematics, Springer-Verlag, 1013 (1983), [10] Schild A. and Silverman H., Convolutions of univalent functions with negative coefficients, Ann. Univ. Mariae Curie-Sklodowaska Sect. A, 29 (1975), [11] Sekine T., Generalization of certain subclasses of analytic functions, Inter. J. Math. Math. Sci., Vol. 10 (1987) No. 4,

12 700 K. K. Dixit, V. Kumar, A. L. Pathak and P. Dixit [12] Silverman H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), [13] Subramanian K.G., Murugusundarmoorthy G., Balasubrahmanyam P. and silverman H., Subclasses of uniformly convex and uniformly starlike functions Math. Japonica, (1996), No. 3, Received: October, 2008