General Mathematics Vol. 15, No. 4 (2007), 131-143 Carlson - Shaffer operator and their applications to certain subclass of uniformly convex functions1 G.Murugusundaramoorthy, T. Rosy and K.Muthunagai Abstract Making use of Carlson - Shaffer operator,we define a new subclass of uniformly convex functions with negative coefficients and obtain the coefficient bounds, extreme points and radius of starlikeness for functions belonging to the generalized class T S(λ, α, β). Furthermore, partial sums fk (z) of functions f (z) in the class S(λ, α, β) are considered and sharp lower bounds for the ratios of real part of f (z) to fk (z) and f ′ (z) to fk′ (z) are determined. 2000 Mathematical Subject Classification: 30C45. Key words: Univalent, convex, starlike, uniformly convex. 1 Introduction Let A denote the class of functions of the form (1.1) f (z) = z + ∞ X an z n n=2 1 Received 16 August, 2007 Accepted for publication (in revised form) 22 December, 2007 131 132 G.Murugusundaramoorthy, T. Rosy and K.Muthunagai which are analytic and univalent in the open disc U = {z : z ∈ C |z| < 1}. Also denote by T the subclass of A consisting of functions of the form (1.2) f (z) = z − ∞ X an z n , (an ≥ 0) n=2 Following Gooodman [3, 4], Rønning [5, 6] introduced and studied the following subclasses (i) A function f ∈ A is said to be in the class Sp (α, β) of uniformly β−starlike functions if it satisfies the condition ¯ ′ ¯ ¾ ½ ′ ¯ zf (z) ¯ zf (z) − α > β ¯¯ − 1¯¯ , z ∈ U, (1.3) Re f (z) f (z) −1 < α ≤ 1 and β ≥ 0. (ii) A function f ∈ A is said to be in the class U CV (α, β) of uniformly β−convex functions if it satisfies the condition ¯ ′′ ¯ ½ ¾ ¯ zf (z) ¯ zf ′′ (z) ¯ , z ∈ U, (1.4) Re 1 + ′ − α > β ¯¯ ′ f (z) f (z) ¯ and −1 < α ≤ 1 and β ≥ 0. Indeed it follows from (1.3) and (1.4) that (1.5) f ∈ U CV (α, β) is equivalent with zf ′ ∈ Sp (α, β). For functions f ∈ A given by (1.1) and g(z) ∈ A given by g(z) = z+ ∞ P n=2 we define the Hadamard product (or Convolution ) of f and g by (1.6) (f ∗ g)(z) = z + ∞ X an bn z n , z ∈ U. n=2 Let φ(a, c; z) be the incomplete beta function defined by (1.7) φ(a, c; z) = z + ∞ X (a)n−1 n=2 (c)n−1 z n , c 6= 0, −1, −2, . . . bn z n , Carlson - Shaffer operator... 133 where (x)n is the Pochhammer symbol defined interms of the Gamma functions, by ( 1 n=0 Γ(x + n) = (1.8) (x)n = Γ(x) x(x + 1)(x + 2) . . . (x + n − 1), n ∈ N Further, for f ∈ A (1.9) L(a, c)f (z) = φ(a, c; z) ∗ f (z) = z + ∞ X (a)n−1 n=2 (c)n−1 an z n , where L(a, c) is called Carlson - Shaffer operator [2] and the operator ∗ stands for the hadamard product (or convolution product) of two power series as given by (1.6). We notice that L(a, a)f (z) = f (z), L(2, 1)f (z) = zf ′ (z). For −1 ≤ α < 1 ,0 ≤ λ ≤ 1 and β ≥ 0, we let S(λ, α, β) be the subclass of A consisting of functions of the form (1.1) and satisfying the analytic criterion ½ ¾ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′ Re −α (1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′ ¯ ¯ ¯ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′ ¯ ¯, z ∈ U > β ¯¯ − 1 (1.10) ¯ (1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′ where L(a, c)f (z) is given by (1.9). We also let T S(λ, α, β) = S(λ, α, β)∩T. By suitably specializing the values of λ, (a) and (c), the class S(λ, α, β) can be reduces to the class studied earlier by Rønning [5, 6]. Also choosing α = 0 and β = 1 the class coincides with the classes studied in [10] and [11] respectively. The main object of this paper is to study the coefficient bounds, extreme points and radius of starlikeness for functions belonging to the generalized class T S(λ, α, β). Furthermore, partial sums fk (z) of functions f (z) in the class S(λ, α, β) are considered and sharp lower bounds for the ratios of real part of f (z) to fk (z) and f ′ (z) to fk′ (z) are determined. 134 2 G.Murugusundaramoorthy, T. Rosy and K.Muthunagai Basic Properties In this section we obtain a necessary and sufficient condition for functions f (z) in the classes S(λ, α, β) and T S(λ, α, β). Theorem 2.1. A function f (z) of the form (1.1) is in S(λ, α, β) if (2.1) ∞ X (a)n−1 |an | ≤ 1 − α, (1 + λ(n − 1))[n(1 + β) − (α + β)] (c) n−1 n=2 −1 ≤ α < 1, 0 ≤ λ ≤ 1, β ≥ 0. Proof. It sufficies to show that ¯ ¯ ¯ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′ ¯ β ¯¯ − 1¯¯ ′ (1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z)) ¾ ½ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′ −1 −Re (1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′ ≤1−α We have ¯ ¯ ¯ ¯ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′ ¯ − 1 β ¯¯ ¯ (1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′ −Re ½ ¾ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′ −1 (1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′ ¯ ¯ ¯ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′ ¯ ¯ ¯ ≤ (1 + β) ¯ − 1 ¯ (1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′ (1 + β) ≤ ∞ P n−1 |a | (n − 1)[1 + λ(n − 1)] (a) (c)n−1 n n=2 ∞ P 1− n=2 . n−1 |a | [1 + λ(n − 1)] (a) (c)n−1 n Carlson - Shaffer operator... 135 This last expression is bounded above by (1 − α) if ∞ X [1 + λ(n − 1)][n(1 + β) − (α + β)] n=2 (a)n−1 |an | ≤ 1 − α, (c)n−1 and hence the proof is complete. Theorem 2.2. A necessary and sufficient condition for f (z) of the form (1.2) to be in the class T S(λ, α, β), −1 ≤ α < 1, 0 ≤ λ ≤ 1, β ≥ 0 is that (2.2) ∞ X (1 + λ(n − 1))[n(1 + β) − (α + β)] n=2 (a)n−1 an ≤ 1 − α, (c)n−1 Proof. In view of Theorem 1, we need only to prove the necessity. If f ∈ T S(λ, α, β) and z is real then 1− ∞ P n−1 n[1 + λ(n − 1)] (a) a z n−1 (c)n−1 n n=2 ∞ P 1− −α≥ [1 + λ(n − n=2 n−1 1)] (a) a z n−1 (c)n−1 n ¯ ¯ ∞ ¯ ¯P (a)n−1 ¯ (n − 1)[1 + λ(n − 1)] (c) |an | ¯ n−1 ¯ ¯ n=2 ¯ ≥ β ¯¯ ∞ ¯ P (a) n−1 ¯ ¯ 1− |a | [1 + λ(n − 1)] n ¯ ¯ (c)n−1 n=2 Letting z → 1 along the real axis, we obtain the desired inequality ∞ X (1 + λ(n − 1))[n(1 + β) − (α + β)] n=2 (a)n−1 an ≤ 1 − α. (c)n−1 Theorem 2.3. Let f (z) defined by (1.2) and g(z) defined by g(z) = z − ∞ P bn z n be in the class T S(λ, α, β). Then the function h(z) defined by n=2 h(z) = (1 − µ)f (z) + µg(z) = z − ∞ X qn z n , n=2 where qn = (1 − µ)an + µbn , 0 ≤ µ < 1 is also in the class T S(λ, α, β). 136 G.Murugusundaramoorthy, T. Rosy and K.Muthunagai Proof. Let the function (2.3) fj (z) = z − ∞ X an, j z n , an, j ≥ 0, j = 1, 2 , n=2 be in the class T S(λ, α, β). It is sufficient to show that the function g(z) defined by g(z) = µf1 (z) + (1 − µ)f2 (z), 0 ≤ µ ≤ 1, is in the class T S(λ, α, β). Since ∞ X [µan,1 + (1 − µ)an,2 ]z n , g(z) = z − n=2 an easy computation with the aid of Theorem 2.2 gives, ∞ X [1 + λ(n − 1)][n(β + 1) − (α + β)] n=2 + ∞ X (a)n−1 µan,1 (c)n−1 [1 + λ(n − 1)][n(β + 1) − (α + β)] n=2 (a)n−1 (1 − µ)an,2 (c)n−1 ≤ µ(1 − α) + (1 − µ)(1 − α) ≤ 1 − α, which implies that g ∈ T S(λ, α, β). Hence T S(λ, α, β) is convex. Theorem 2.4. (Extreme points) Let f1 (z) = z and (2.4) (1 − α)(c)n−1 fn (z) = z − z n for n = 2, 3, 4, . . . . (1 + nλ − λ)[n(1 + β) − (α + β)](a)n−1 Then f (z) ∈ T S(λ, α, β) if and only if f (z) can be expressed in the form ∞ ∞ P P µn = 1. µn fn (z), where µn ≥ 0 and f (z) = n=1 n=1 The proof of Theorem 2.4, follows on lines similar to the proof of the theorem on extreme points given in Silverman [8]. Next we prove the following closure theorem. Carlson - Shaffer operator... 137 Theorem 2.5. (Closure theorem ) Let the functions fj (z) (j = 1, 2, . . . m) defined by (2.3) be in the classes T S(λ, αj , β) (j = 1, 2, . . . m) respectively. Then the function h(z) defined by à m ! ∞ 1 X X an,j z n h(z) = z − m n=2 j=1 is in the class T S(λ, α, β), where α = min {αj } where −1 ≤ αj < 1. 1≤j≤m Proof. Since fj (z) ∈ T S(λ, αj , β) (j = 1, 2, 3, . . . m) by applying Theorem 2.2, to (2.3) we observe that à ! ∞ m X (a)n−1 1 X (1 + λ(n − 1))[n(1 + β) − (α + β)] an,j (c) m n−1 n=2 j=1 ! Ã∞ m 1 X X (a)n−1 = (1 + λ(n − 1))[n(1 + β) − (α + β)] an,j m j=1 n=2 (c)n−1 m 1 X (1 − αj ) ≤ 1 − α ≤ m j=1 which in view of Theorem 2.2, again implies that h(z) ∈ T S(λ, α, β) and so the proof is complete. Theorem 2.6. Let f ∈ T S(λ, α, β). Then 1. f isnstarlike o of order δ(0 ≤ δ < 1) in the disc |z| < r1 ; that is, zf ′ (z) Re > δ, (|z| < r1 ; 0 ≤ δ < 1), where f (z) r1 = inf n≤2 ½ (a)n−1 (c)n−1 µ 1−δ n−δ ¶ (1 + nλ − λ)[n(1 + β) − (α + β)] 1−α 1 ¾ n−1 . 2. f isnconvex ofoorder δ (0 ≤ δ < 1) in the disc |z| < r2 , that is ′′ (z) Re 1 + zff ′ (z) > δ, (|z| < r2 ; 0 ≤ δ < 1), where r2 = inf n≤2 ½ (a)n−1 (1 − δ)(1 + nλ − λ)[n(1 + β) − (α + β)] (c)n−1 n(n − δ) 1 ¾ n−1 . 138 G.Murugusundaramoorthy, T. Rosy and K.Muthunagai Each of these results are sharp for the extremal function f (z) given by (2.4). Proof. Given f ∈ A, and f is starlike of order δ, we have ¯ ¯ ′ ¯ ¯ zf (z) ¯ < 1 − δ. ¯ − 1 (2.5) ¯ ¯ f (z) For the left hand side of (2.5) we have ¯ ¯ ′ ¯ zf (z) ¯ ¯ ¯≤ − 1 ¯ f (z) ¯ ∞ P (n − 1)an |z|n−1 n=2 1− n=2 1−δ . an |z|n−1 n=2 The last expression is less than 1 − δ if ∞ X n−δ ∞ P an |z|n−1 < 1. Using the fact, that f ∈ T S(λ, α, β) if and only if ∞ X (1 + λ(n − 1))[n(1 + β) − (α + β)] (a)n−1 1−α n=2 (c)n−1 an < 1. We can say (2.5) is true if n − δ n−1 (1 + λ(n − 1))[n(1 + β) − (α + β)] (a)n−1 . |z| < 1−δ 1−α (c)n−1 Or, equivalently, |z|n−1 < (1 − δ)(1 + λ(n − 1))[n(1 + β) − (α + β)] (a)n−1 (n − δ)(1 − α) (c)n−1 which yields the starlikeness of the family. (ii) Using the fact that f is convex if and only if zf ′ is starlike, we can prove (ii), on lines similar to the proof of (i). Carlson - Shaffer operator... 3 139 Partial Sums Following the earlier works by Silverman [8] and Silvia [9] on partial sums of analytic functions. We consider in this section partial sums of functions in the class T S(λ, α, β) and obtain sharp lower bounds for the ratios of real part of f (z) to fk (z) and f ′ (z) to fk′ (z). Theorem 3.1. Let f (z) ∈ T S(λ, α, β) be given by (1.1) and define the partial sums f1 (z) and fk (z), by (3.1) f1 (z) = z; and fk (z) = z + k X an z n , (k ∈ N/1) n=2 Suppose also that ∞ X dn |an | ≤ 1, n=2 where (3.2) dn := (1 + λ(n − 1))[n(α + β) − (α + β)] (a)n−1 . (1 − α) (c)n−1 Then f ∈ T S(λ, α, β). Furthermore, ¾ ½ 1 f (z) z ∈ U, k ∈ N >1− (3.3) Re fk (z) dk+1 and (3.4) Re ½ fk (z) f (z) ¾ > dk+1 . 1 + dk+1 Proof. For the coefficients dn given by (3.2) it is not difficult to verify that (3.5) dn+1 > dn > 1. Therefore we have (3.6) k X n=2 |an | + dk+1 ∞ X n=k+1 |an | ≤ ∞ X n=2 dn |an | ≤ 1 140 G.Murugusundaramoorthy, T. Rosy and K.Muthunagai by using the hypothesis (3.2). By setting ½ µ ¶¾ f (z) 1 g1 (z) = dk+1 − 1− fk (z) dk+1 ∞ P dk+1 an z n−1 n=k+1 = 1+ (3.7) k P 1+ an z n−1 n=2 and applying (3.6), we find that (3.8) ∞ P dk+1 ¯ ¯ ¯ g1 (z) − 1 ¯ ¯ ¯ ¯ g1 (z) + 1 ¯ ≤ |an | n=k+1 2−2 n P ∞ P |an | − dk+1 n=2 |an | n=k+1 ≤ 1, z ∈ U, which readily yields the assertion (3.3) of Theorem 3.1. In order to see that (3.9) f (z) = z + z k+1 dk+1 gives sharp result, we observe that for z = reiπ/k that 1 1 − dk+1 as z → 1− . Similarly, if we take ½ (3.10) fk (z) dk+1 g2 (z) = (1 + dk+1 ) − f (z) 1 + dk+1 ∞ P (1 + dn+1 ) an z n−1 n=k+1 = 1− ∞ P 1+ an z n−1 f (z) fk (z) ¾ n=2 and making use of (3.6), we can deduce that (3.11) ¯ ¯ ¯ g2 (z) − 1 ¯ ¯ ¯ ¯ g2 (z) + 1 ¯ ≤ (1 + dk+1 ) ∞ P |an | n=k+1 2−2 k P n=2 |an | − (1 − dk+1 ) ∞ P n=k+1 |an | = 1+ zk dk+1 → Carlson - Shaffer operator... 141 which leads us immediately to the assertion (3.4) of Theorem 3.1. The bound in (3.4) is sharp for each k ∈ N with the extremal function f (z) given by (3.9). The proof of the Theorem 3.1, is thus complete. Theorem 3.2. If f (z) of the form (1.1) satisfies the condition (2.1). Then ½ ′ ¾ f (z) k+1 (3.12) Re . ≥1− ′ fk (z) dk+1 Proof. By setting µ ¶¾ k+1 f ′ (z) − 1− g(z) = dk+1 fk′ (z) dk+1 ∞ ∞ P P k+1 1 + dk+1 nan z n−1 nan z n−1 + n=2 n=k+1 = k P nan z n−1 1+ ½ n=2 dk+1 k+1 = 1+ ∞ P nan z n−1 n=k+1 1+ k P . nan z n−1 n=2 (3.13) Now ¯ ¯ ¯ g(z) − 1 ¯ ¯ ¯ ¯ g(z) + 1 ¯ ≤ if k X dk+1 k+1 2−2 k P ∞ P n|an | n=k+1 n|an | − n=2 dk+1 k+1 ∞ P . n|an | n=k+1 ¯ ¯ ¯ g(z) − 1 ¯ ¯ ¯ ¯ g(z) + 1 ¯ ≤ 1 ∞ dk+1 X n|an | + n|an | ≤ 1 k + 1 n=k+1 n=2 (3.14) since the left hand side of (3.14) is bounded above by k P dn |an | if n=2 (3.15) k X n=2 (dn − n)|an | + ∞ X n=k+1 dn − dk+1 n|an | ≥ 0, k+1 142 G.Murugusundaramoorthy, T. Rosy and K.Muthunagai and the proof is complete. The result is sharp for the extremal function f (z) = z + z k+1 . ck+1 Theorem 3.3. If f (z) of the form (1.1) satisfies the condition (2.1) then ½ ′ ¾ f (z) dk+1 (3.16) Re k′ . ≥ f (z) k + 1 + dk+1 Proof. By setting dk+1 f ′ (z) − g(z) = [(k + 1) + dk+1 ] k′ f (z) k + 1 + dk+1 ³ ´ P ∞ k+1 1 + dk+1 nan z n−1 n=k+1 = 1− k P 1+ nan z n−1 ½ ¾ n=2 and making use of (3.15), we deduce that ¯ ¯ ¯ g(z) − 1 ¯ ¯ ¯ ¯ g(z) + 1 ¯ ≤ 2−2 k P ³ n=2 1+ dk+1 k+1 ´ P ∞ n|an | n=k+1 ³ n|an | − 1 + dk+1 k+1 ´ P ∞ ≤ 1, n|an | n=k+1 which leads us immediately to the assertion of Theorem 3.3. References [1] E.Aqlan,J.M.Jahangiri and S.R.Kulkarni,Classes of k-uniformly convex and starlike functions, Tamkang Journal of Mathematics 35(3), (2004), 261–266. [2] B.C.Carlson and S.B.Shaffer, Starlike and prestarlike hypergrometric functions, SIAM J.Math. Anal., 15 (2002), 737 - 745. [3] A.W. Goodman, On uniformly convex functions, Ann. polon. Math., 56 (1991), 87-92. [4] A.W. Goodman, On uniformly starlike functions, J. Math. Anal. & Appl., 155 (1991),364-370. 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G.Murugusundaramoorthy, School of Science and Humanities, VIT University, Vellore - 632014, India. E-mail:gmsmoorthy@yahoo.com K.Muthunagai Department of Mathematics, Hindustan College of Engineering, Chennai, India. Thomas Rosy Department of Mathematics Madras Christian College, Chennai - 600059, India.