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Journal of Inequalities in Pure and Applied Mathematics A NEW SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS AND A CORRESPONDING SUBCLASS OF STARLIKE FUNCTIONS WITH FIXED SECOND COEFFICIENT volume 5, issue 4, article 85, 2004. G. MURUGUSUNDARAMOORTHY AND N. MAGESH Department of Mathematics Vellore Institute of Technology, Deemed University Vellore - 632014, India. EMail: gmsmoorthy@yahoo.com Department of Mathematics Adhiyamaan College of Engineering Hosur - 635109, India. EMail: nmagi_2000@yahoo.co.in Received 22 June, 2004; accepted 25 August, 2004. Communicated by: A. Sofo Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 125-04 Quit Abstract Making use of Linear operator theory, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients. The main object of this paper is to obtain coefficient estimates distortion bounds, closure theorems and extreme points for functions belonging to this new class. The results are generalized to families with fixed finitely many coefficients. 2000 Mathematics Subject Classification: 30C45. Key words: Univalent, Convex, Starlike, Uniformly convex, Uniformly starlike, Linear operator. The authors would like to thank the referee for his insightful suggestions. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 The Class S(α, β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Distortion Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 The Class T Sbn ,k (α, β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 References A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page Contents JJ J II I Go Back Close Quit Page 2 of 20 1. Introduction Denoted by S the class of functions of the form (1.1) f (z) = z + ∞ X an z n n=2 that are analytic and univalent in the unit disc 4 = {z : |z| < 1} and by ST and CV the subclasses of S that are respectively, starlike and convex. Goodman [2, 3] introduced and defined the following subclasses of CV and ST. A function f (z) is uniformly convex (uniformly starlike) in 4 if f (z) is in CV (ST ) and has the property that for every circular arc γ contained in 4, with center ξ also in 4, the arc f (γ) is convex (starlike) with respect to f (ξ). The class of uniformly convex functions is denoted by U CV and the class of uniformly starlike functions by U ST (for details see [2]). It is well known from [4, 5] that 00 00 zf (z) zf (z) ≤ Re 1 + 0 f ∈ U CV ⇔ 0 . f (z) f (z) In [5], Rønning introduced a new class of starlike functions related to U CV defined as 0 0 zf (z) zf (z) f ∈ Sp ⇔ − 1 ≤ Re . f (z) f (z) Note that f (z) ∈ U CV ⇔ zf 0 (z) ∈ Sp . Further, Rønning generalized the class Sp by introducing a parameter α, − 1 ≤ α < 1, 0 0 zf (z) zf (z) f ∈ Sp (α) ⇔ − 1 ≤ Re −α . f (z) f (z) A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page Contents JJ J II I Go Back Close Quit Page 3 of 20 Now we define the function φ(a, c; z) by φ(a, c; z) = z + (1.2) ∞ X (a)n−1 n=2 (c)n−1 zn, for c 6= 0, −1, −2, . . . , a 6= −1; z ∈ ∆ where (λ)n is the Pochhammer symbol defined by (1.3) (λ)n = = Γ(n + λ) Γ(λ) ( 1; n=0 λ(λ + 1)(λ + 2) . . . (λ + n − 1), n ∈ N = {1, 2, . . . } ) . Carlson and Shaffer [1] introduced a linear operator L(a, c), by (1.4) L(a, c)f (z) = φ(a, c; z) ∗ f (z) ∞ X (a)n−1 =z+ an z n , z ∈ 4, (c) n−1 n=2 where ∗ stands for the Hadamard product or convolution product of two power series ∞ ∞ X X n ϕ(z) = ϕn z and ψ(z) = ψn z n n=1 defined by (ϕ ∗ ψ)(z) = ϕ(z) ∗ ψ(z) = n=1 ∞ X n=1 ϕn ψn z n . A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page Contents JJ J II I Go Back Close Quit Page 4 of 20 We note that L(a, a)f (z) = f (z), L(2, 1)f (z) = zf 0 (z), L(m + 1, 1)f (z) = Dm f (z), where Dm f (z) is the Ruscheweyh derivative of f (z) defined by Ruscheweyh [6] as z (1.5) Dm f (z) = ∗ f (z), m > −1. (1 − z)m+1 Which is equivalently, Dm f (z) = z dm m−1 {z f (z)}. m! dz m For β ≥ 0 and −1 ≤ α < 1, we let S(α, β) denote the subclass of S consisting of functions f (z) of the form (1.1) and satisfying the analytic criterion z(L(a, c)f (z))0 z(L(a, c)f (z)) (1.6) Re − α > β − 1 , z ∈ 4. L(a, c)f (z) L(a, c)f (z) T We also let T S(α, β) = S(α, β) T where T, the subclass of S consisting of functions of the form (1.7) 0 f (z) = z − ∞ X an z n , an ≥ 0, ∀ n ≥ 2, n=2 was introduced and studied by Silverman [7]. The main object of this paper is to obtain necessary and sufficient conditions for the functions f (z) ∈ T S(α, β). Furthermore we obtain extreme points, distortion bounds and closure properties for f (z) ∈ T S(α, β) by fixing the second coefficient. A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page Contents JJ J II I Go Back Close Quit Page 5 of 20 2. The Class S(α, β) In this section we obtain necessary and sufficient conditions for functions f (z) in the classes T S(α, β). Theorem 2.1. A function f (z) of the form (1.1) is in S(α, β) if (2.1) ∞ X [n(1 + β) − (α + β)] n=2 (a)n−1 |an | ≤ 1 − α, (c)n−1 −1 ≤ α < 1, β ≥ 0. Proof. It suffices to show that z(L(a, c)f (z))0 z(L(a, c)f (z))0 β − 1 − Re − 1 ≤ 1 − α. L(a, c)f (z) L(a, c)f (z) A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh We have z(L(a, c)f (z))0 z(L(a, c)f (z))0 β − 1 − Re −1 L(a, c)f (z) L(a, c)f (z) z(L(a, c)f (z))0 ≤ (1 + β) − 1 L(a, c)f (z) P∞ n−1 (1 + β) n=2 (n − 1) (a) |a | (c)n−1 n ≤ . P∞ (a)n−1 1 − n=2 (c)n−1 |an | This last expression is bounded above by (1 − α) if ∞ X n=2 [n(1 + β) − (α + β)] (a)n−1 |an | ≤ 1 − α, (c)n−1 Title Page Contents JJ J II I Go Back Close Quit Page 6 of 20 and hence the proof is complete. Theorem 2.2. A necessary and sufficient condition for f (z) of the form (1.7) to be in the class T S(α, β), −1 ≤ α < 1, β ≥ 0 is that ∞ X (2.2) [n(1 + β) − (α + β)] n=2 (a)n−1 an ≤ 1 − α. (c)n−1 Proof. In view of Theorem 2.1, we need only to prove the necessity. If f (z) ∈ T S(α, β) and z is real then P∞ P (a)n−1 (a)n−1 n−1 n−1 (n − 1) a z a z 1− ∞ n n n n=2 (c)n−1 (c)n−1 n=2 −α≥β . P∞ (a)n−1 P∞ (a)n−1 n−1 n−1 1− an z an z 1− n=2 (c)n−1 n=2 (c)n−1 A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Letting z → 1 along the real axis, we obtain the desired inequality ∞ X (a)n−1 an ≤ 1 − α, −1 ≤ α < 1, β ≥ 0. [n(1 + β) − (α + β)] (c)n−1 n=2 Corollary 2.3. Let the function f (z) defined by (1.7) be in the class T S(α, β). Then an ≤ (1 − α)(c)n−1 , n ≥ 2, −1 ≤ α ≤ 1, β ≥ 0. [n(1 + β) − (α + β)](a)n−1 Title Page Contents JJ J II I Go Back Close Quit Page 7 of 20 Remark 1. In view of Theorem 2.2, we can see that if f (z) is of the form (1.7) and is in the class T S(α, β) then a2 = (2.3) (1 − α)(c) . (2 + β − α)(a) By fixing the second coefficient, we introduce a new subclass T Sb (α, β) of T S(α, β) and obtain the following theorems. Let T Sb (α, β) denote the class of functions f (z) in T S(α, β) and be of the form (2.4) ∞ b(1 − α)(c) 2 X f (z) = z − z − an z n (an ≥ 0), 0 ≤ b ≤ 1. (2 + β − α)(a) n=3 Theorem 2.4. Let function f (z) be defined by (2.4). Then f (z) ∈ T Sb (α, β) if and only if (2.5) A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page ∞ X (a)n−1 an ≤ (1 − b)(1 − α), [n(1 + β) − (α + β)] (c) n−1 n=3 −1 ≤ α < 1, β ≥ 0. Proof. Substituting Contents JJ J II I Go Back a2 = b(1 − α) (c) , 0 ≤ b ≤ 1. (2 + β − α) (a) in (2.2) and simple computation leads to the desired result. Close Quit Page 8 of 20 Corollary 2.5. Let the function f (z) defined by (2.4) be in the class T Sb (α, β). Then (2.6) an ≤ (1 − b)(1 − α)(c)n−1 , n ≥ 3, −1 ≤ α ≤ 1, β ≥ 0. [n(1 + β) − (α + β)](a)n−1 Theorem 2.6. The class T Sb (α, β) is closed under convex linear combination. Proof. Let the function f (z) be defined by (2.4) and g(z) defined by ∞ b(1 − α) (c) 2 X g(z) = z − z − dn z n , (2 + β − α) (a) n=3 (2.7) A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient where dn ≥ 0 and 0 ≤ b ≤ 1. Assuming that f (z) and g(z) are in the class T Sb (α, β), it is sufficient to prove that the function H(z) defined by G. Murugusundaramoorthy and N. Magesh H(z) = λf (z) + (1 − λ)g(z), (0 ≤ λ ≤ 1) Title Page (2.8) Contents is also in the class T Sb (α, β). Since H(z) = z − (2.9) ∞ X b(1 − α)(c) 2 z − {λan + (1 − λ)dn }z n , (2 + β − α)(a) n=3 an ≥ 0, dn ≥ 0, 0 ≤ b ≤ 1, we observe that (2.10) ∞ X n=3 [n(1 + β) − (α + β)] (a)n−1 (λan + (1 − λ)dn ) ≤ (1 − b)(1 − α) (c)n−1 JJ J II I Go Back Close Quit Page 9 of 20 which is, in view of Theorem 2.4, again, implies that H(z) ∈ T Sb (α, β) which completes the proof of the theorem. Theorem 2.7. Let the functions (2.11) ∞ b(1 − α)(c) 2 X z − fj (z) = z − an, j z n , an,j ≥ 0 (2 + β − α)(a) n=3 be in the class T Sb (α, β) for every j (j = 1, 2, . . . , m). Then the function F (z) defined by F (z) = (2.12) m X µj fj (z), A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient j=1 G. Murugusundaramoorthy and N. Magesh is also in the class T Sb (α, β), where m X (2.13) µj = 1. j=1 Contents Proof. Combining the definitions (2.11) and (2.12), further by (2.13) we have ! ∞ m b(1 − α)(c) 2 X X (2.14) F (z) = z − z − µj an,j z n . (2 + β − α)(a) n=3 j=1 Since fj (z) ∈ T Sb (α, β) for every j = 1, 2, . . . , m, Theorem 2.4 yields (2.15) ∞ X n=3 [n(1 + β) − (α + β)] Title Page (a)n−1 an,j ≤ (1 − b)(1 − α), (c)n−1 JJ J II I Go Back Close Quit Page 10 of 20 for j = 1, 2, . . . , m. Thus we obtain ∞ X [n(1 + β) − (α + β)] n=3 = m X (a)n−1 (c)n−1 m X ! µj an,j j=1 ∞ X (a)n−1 [n(1 + β) − (α + β)] an,j (c)n−1 n=3 µj j=1 ! ≤ (1 − b)(1 − α) in view of Theorem 2.4. So, F (z) ∈ T Sb (α, β). Theorem 2.8. Let f2 (z) = z − (2.16) b(1 − α)(c) 2 z (2 + β − α)(a) and (2.17) A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page b(1 − α)(c) 2 (1 − b)(1 − α)(c)n−1 fn (z) = z − z − zn (2 + β − α)(a) [n(1 + β) − (α + β)](a)n−1 for n = 3, 4, . . . . Then f (z) is in the class T Sb (α, β) if and only if it can be expressed in the form f (z) = (2.18) ∞ X Contents JJ J II I Go Back λn fn (z), Close n=2 where λn ≥ 0 and ∞ P n=2 Quit λn = 1. Page 11 of 20 Proof. We suppose that f (z) can be expressed in the form (2.18). Then we have ∞ (1 − b)(1 − α)(c)n−1 b(1 − α)(c) 2 X zn f (z) = z − z − λn (2 + β − α)(a) [n(1 + β) − (α + β)](a)n−1 n=3 (2.19) =z− ∞ X An z n , n=2 where A2 = (2.20) b(1 − α)(c) (2 + β − α)(a) and (2.21) An = λn (1 − b)(1 − α)(c)n−1 , n = 3, 4, . . . . [n(1 + β) − (α + β)](a)n−1 Therefore ∞ ∞ X X (a)n−1 An = b(1 − α) + λn (1 − b)(1 − α) [n(1 + β) − (α + β)] (c) n−1 n=3 n=2 (2.22) = (1 − α)[b + (1 − λ2 )(1 − b)] ≤ (1 − α), it follows from Theorem 2.2 and Theorem 2.4 that f (z) is in the class T Sb (α, β). Conversely, we suppose that f (z) defined by (2.4) is in the class T Sb (α, β). Then by using (2.6), we get (2.23) (1 − b)(1 − α)(c)n−1 , (n ≥ 3). an ≤ [n(1 + β) − (α + β)](a)n−1 A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page Contents JJ J II I Go Back Close Quit Page 12 of 20 Setting (2.24) λn = [n(1 + β) − (α + β)](a)n−1 an , (n ≥ 3) (1 − b)(1 − α)(c)n−1 and (2.25) λ2 = 1 − ∞ X λn , n=3 we have (2.18). This completes the proof of Theorem 2.8. Corollary 2.9. The extreme points of the class T Sb (α, β) are functions fn (z), n ≥ 2 given by Theorem 2.8. A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page Contents JJ J II I Go Back Close Quit Page 13 of 20 3. Distortion Theorems In order to obtain distortion bounds for the function f ∈ T Sb (α, β), we first prove the following lemmas. Lemma 3.1. Let the function f3 (z) be defined by (3.1) f3 (z) = z − b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3 z − z . (2 + β − α)(a) (3 + 2β − α)(a)2 Then, for 0 ≤ r < 1 and 0 ≤ b ≤ 1, (3.2) |f3 (reiθ )| ≥ r − b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3 r − r (2 + β − α)(a) (3 + 2β − α)(a)2 with equality for θ = 0. For either 0 ≤ b < b0 and 0 ≤ r ≤ r0 or b0 ≤ b ≤ 1, (3.3) |f3 (reiθ )| ≤ r + b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3 r − r (2 + β − α)(a) (3 + 2β − α)(a)2 with equality for θ = π, where 1 2(1 − α)(c)(c)2 × {−[(3 + 2β − α)(a)2 (c) + 4(2 + β − α)(a)(c)2 − (1 − α)(c)(c)2 ] + [((3 + 2β − α)(a)2 (c) + 4(2 + β − α)(a)(c)2 − (1 − α)(c)(c)22 (3.4) b0 = + 16(2 + β − α)(1 − α)(a)(c)(c)22 ]1/2 } A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page Contents JJ J II I Go Back Close Quit Page 14 of 20 and (3.5) r0 = 1 {−2(1 − b)(2 + β − α)(a)(c + 1) b(1 − b)(1 − α)(c)2 + [4(1 − b)2 (2 + β − α)2 (a)2 (c + 1)2 + b2 (1 − b)(3 + 2β − α)(1 − α)(a)2 (c)2 ]1/2 }. Proof. We employ the technique as used by Silverman and Silvia [8]. Since (3.6) ∂|f3 (reiθ )|2 ∂θ 3 = 2(1 − α)r sin θ b(c) 4(1 − b)(c)2 + r cos θ (2 + β − α)(a) (3 + 2β − α)(a)2 b(1 − b)(1 − α)(c)(c)2 2 − r (2 + β − α)(3 + 2β − α)(a)(a)2 we can see that (3.7) A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page ∂|f3 (reiθ )|2 =0 ∂θ for θ1 = 0, θ2 = π, and 1 b[(1 − b)(1 − α)(c)2 r2 − (3 + 2β − α)(a)2 ] −1 (3.8) θ3 = cos r 4(1 − b)(2 + β − α)(a)(c + 1) since θ3 is a valid root only when −1 ≤ cos θ3 ≤ 1. Hence we have a third root if and only if r0 ≤ r < 1 and 0 ≤ b ≤ b0 . Thus the results of the theorem follow from comparing the extremal values |f3 (reiθk )|, k = 1, 2, 3 on the appropriate intervals. Contents JJ J II I Go Back Close Quit Page 15 of 20 Lemma 3.2. Let the functions fn (z) be defined by (2.17) and n ≥ 4. Then |fn (reiθ )| ≤ |f4 (−r)|. (3.9) Proof. Since fn (z) = z − and rn n b(1 − α)(c) 2 (1 − b)(1 − α)(c)n−1 zn z − (2 + β − α)(a) [n(1 + β) − (α + β)](a)n−1 is a decreasing function of n, we have b(1 − α)(c) 2 (1 − b)(1 − α)(c)3 4 r − r (2 + β − α)(a) [4 + 3β − α](a)3 = −f4 (−r), |fn (reiθ | ≤ r + A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh which shows (3.9). Theorem 3.3. Let the function f (z) defined by (2.4) belong to the class T Sb (α, β), then for 0 ≤ r < 1, b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3 |f (reiθ )| ≥ r − r − r (2 + β − α)(a) [3 + 2β − α](a)2 (3.10) with equality for f3 (z) at z = r, and n o (3.11) |f (reiθ )| ≤ max max |f3 (reiθ )|, −f4 (−r) , θ where max |f3 (reiθ )| is given by Lemma 3.1. θ Title Page Contents JJ J II I Go Back Close Quit Page 16 of 20 Proof. The proof of Theorem 3.3 is obtained by comparing the bounds of Lemma 3.1 and Lemma 3.2. Remark 2. Taking b = 1 in Theorem 3.3 we obtain the following result. Corollary 3.4. Let the function f (z) defined by (1.7) be in the class T S(α, β). Then for |z| = r < 1, we have r− (3.12) (1 − α)(c) (1 − α)(c) r2 ≤ |f (z)| ≤ r + r2 . (2 + β − α)(a) (2 + β − α)(a) Lemma 3.5. Let the function f3 (z) be defined by (3.1). Then, for 0 ≤ r < 1, and 0 ≤ b ≤ 1, (3.13) |f30 (reiθ )| ≥ 1 − 2b(1 − α)(c) 3(1 − b)(1 − α)(c)2 2 r− r (2 + β − α)(a) (3 + 2β − α)(a)2 A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh with equality for θ = 0. For either 0 ≤ b < b1 and 0 ≤ r ≤ r1 or b1 ≤ b ≤ 1, (3.14) |f30 (reiθ )| 2b(1 − α)(c) 3(1 − b)(1 − α)(c)2 2 ≤1+ r− r (2 + β − α)(a) (3 + 2β − α)(a)2 with equality for θ = π, where 1 6(1 − α)(c)(c2 ) × {−[(3 + 2β − α)(a)2 (c) + 6(2 + β − α)(a)(c)2 − 3(1 − α)(c)(c)2 ] + {((3 + 2β − α)(a)2 (c) + 6(2 + β − α)(a)(c)2 − 3(1 − α)(c)(c)2 )2 (3.15) b1 = + 72(2 + β − α)(1 − α)(a)(c)(c22 )}1/2 } Title Page Contents JJ J II I Go Back Close Quit Page 17 of 20 and (3.16) r1 = 1 {−3(1 − b)(2 + β − α)(a)(c + 1) 3b(1 − b)(1 − α)(c2 ) + [8(1 − b)2 (2 + β − α)2 (a)2 (c + 1)2 + 3b2 (1 − b)(3 + 2β − α)(1 − α)(a)2 (c)2 ]1/2 }. Proof. The proof of Lemma 3.5 is much akin to the proof of Lemma 3.1. Theorem 3.6. Let the function f (z) defined by (2.4) belong to the class T Sb (α, β), then for 0 ≤ r < 1, |f 0 (reiθ )| ≥ 1 − (3.17) 2b(1 − α)(c) 3(1 − b)(1 − α)(c)2 2 r− r (2 + β − α)(a) [3 + 2β − α](a)2 with equality for f30 (z) at z = r, and n o (3.18) |f 0 (reiθ )| ≤ max max |f30 (reiθ )|, −f40 (−r) , θ where max |f30 (reiθ )| is given by Lemma 3.5. θ Remark 3. Putting b = 1 in Theorem 3.6 we obtain the following result. Corollary 3.7. Let the function f (z) defined by (1.2) be in the class T S(α, β). Then for |z| = r < 1, we have (3.19) 2(1 − α)(c) 2(1 − α)(c) 1− r ≤ |f 0 (z)| ≤ 1 + r. (2 + β − α)(a) (2 + β − α)(a) A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient G. Murugusundaramoorthy and N. Magesh Title Page Contents JJ J II I Go Back Close Quit Page 18 of 20 4. The Class T Sbn,k (α, β) Instead of fixing just the second coefficient, we can fix finitely many coefficients. Let T Sbn ,k (α, β) denote the class of functions in T Sb (α, β) of the form (4.1) f (z) = z − k X n=2 where 0 ≤ ∞ X bn (1 − α)(c)n−1 n z − an z n , [n(1 + β) − (α + β)](a)n−1 n=k+1 Pk n=2 bn = b ≤ 1. Note that T Sb2 ,2 (α, β) = T Sb (α, β). Theorem 4.1. The extreme points of the class T Sbn ,k (α, β) are fk (z) = z − k X n=2 bn (1 − α)(c)n−1 zn [n(1 + β) − (α + β)](a)n−1 G. Murugusundaramoorthy and N. Magesh Title Page and fn (z) = z − A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient k X n=2 bn (1 − α)(c)n−1 zn [n(1 + β) − (α + β)](a)n−1 − ∞ X (1 − b)(1 − α)(c)n−1 zn [n(1 + β) − (α + β)](a) n−1 n=k+1 The details of the proof are omitted, since the characterization of the extreme points enables us to solve the standard extremal problems in the same manner as was done for T Sb (α, β). Contents JJ J II I Go Back Close Quit Page 19 of 20 References [1] B.C. CARLSON AND S.B. SHAFFER, Starlike and prestarlike hypergrometric functions, SIAM J.Math. Anal., 15 (2002), 737–745. [2] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56 (1991), 87–92. [3] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. & Appl., 155 (1991), 364–370. [4] W. MA AND D. MINDA, Uniformly convex functions, Ann. Polon. Math., 57 (1992), 165–175. A New Subclass of Uniformly Convex Functions and a Corresponding Subclass of Starlike Functions with Fixed Second Coefficient [5] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196. G. Murugusundaramoorthy and N. Magesh [6] S. RUSCHEWEYH, New criteria for Univalent Functions, Proc. Amer. Math. Soc., 49 (1975), 109–115. Title Page [7] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109–116. [8] H. SILVERMAN AND E.M. SILVIA, Fixed coefficients for subclasses of starlike functions, Houston. J. Math., 7 (1997), 129–136. Contents JJ J II I Go Back Close Quit Page 20 of 20