Gen. Math. Notes, Vol. 3, No. 2, April 2011, pp. 88-96 ISSN 2219-7184; Copyright © ICSRS Publication, 2011 www.i-csrs.org Available free online at http://www.geman.in Coefficient Inequality for Certain Sub-Classes of Analytic Functions with Respect to Symmetric Points B.S. Mehrok1, Harjinder Singh2 and Deepak Gupta3 1 Ex. Prof, Department of Mathematics, Khalsa College, Amritsar (Punjab) E-mail: beantsingh.mehrok@gmail.com 2 Department of Mathematics, Govt. Rajindra College, Bathinda (Punjab) E-mail: harjindpreet@gmail.com 3 Department of Mathematics, MM Engg. College, Mullana (Haryana) E-mail: guptadeepak2003@yahoo.co.in (Received: 8-2-11/ Accepted: 23-3-11) Abstract Let be the class of analytic functions = + ∑ in the unit disc E = z: |z| < 1. Denote by Sα; δ, A, B the class of analytic functions in satisfying the ! condition 1 − ! #!# + " $ ! " % " &! #!# ' " ≺ $)*+ % , (0 ≤ α ≤ 1 and 0 < 4 ≤ 1. We )*, - obtain sharp upper bounds for the functional |5 − 6 |. Keywords: Analytic functions, Starlike functions with respect to symmetrical points, Subordination. 1. Introduction Let denote the class of functions = + 7 1.1 Coefficient Inequality for Certain… 89 which are analytic in the unit disc E = z: |z| < 1. Denote by U, the class of functions wz = 7 c; z ; 1.2 ;) which are analytic in the unit disc E = z: |z| < 1 and satisfying there the conditions w0 = 0 and |wz| < 1. It is known [5] that |c) | ≤ 1, |c | ≤ 1 − |c) | . 1.3 Let z and Fz be two analytic functions in E. Then z is subordinate to Fz if there is an analytic function wz ∈ U such that z = F&wz'. Symbolically, we write z ≺ Fz. 1.4 2 F CD E G > 0. − − 1.5 To avoid repetitions, it is admitted once that 0 ≤ α ≤ 1 , −1 ≤ B < A ≤ 1 , 0 < 4 ≤ 1 and E = z: |z| < 1. Sakaguchi [6] introduced the class SA ∗ of functions in satisfying the condition These functions are called starlike with respect to symmetrical points in E. The class K K of convex functions with respect to symmetrical points in E was defined and studied by Das and Singh [2]. A function ∈ K K if and only if CD L &2 F ' F & − −' FM > 0. Denote by OK the class of functions in satisfying the following condition CD E 2z F z G > 0, gz − g−z g ∈ SA ∗ . 1.6 1.7 Goel and Mehrok[3] introduced and studied the class SA ∗ A, B of functions of the form (1.1) satisfying the condition 2 F 1 + Az ≺ . − − 1 + Bz 1.8 Janteng and Halim [4] considered the class K A A, B of functions of the form (1.1) satisfying the condition 2 F F 1 + Az ≺ . 1.9 F − − 1 + Bz 90 B. S. Mehrok et al. Denote by CA A, B and CA F A, B the classes of functions of the form (1.1) satisfying respectively the conditions and 1 + Az 2z F z ≺ , gz − g−z 1 + Bz 2z F zF 1 + Az ≺ , F gz − g−z 1 + Bz g ∈ SA ∗ g ∈ SA ∗ . 1.10 1.11 Let FA A, B and FA F A, B denote the classes of functions of the form (1.1) satisfying respectively the conditions and 2z F z 1 + Az ≺ , hz − h−z 1 + Bz 2z F zF 1 + Az ≺ , hz − h−zF 1 + Bz h ∈ KK h ∈ KK . 1.12 1.13 It is easy to verify that SA ∗ 1, −1 ≡ SA ∗, K A 1, −1 ≡ K A and CA 1, −1 ≡ CA . Also SA ∗ 1 − 2β, −1 ≡ SA ∗ β is the class of starlike functions with respect to symmetrical points of order defined by Das and Singh [1]. Let us denote by SA ∗ δ; A , B, K A δ; A , B, CA δ; A , B, CA F δ; A, B, FA δ; A, B and FA F δ; A, B the subclasses of functions z in which satisfy respectively the following conditions 2 F 1 + Az Z ≺X Y , 1.14 − − 1 + Bz &2 F ' 1 + Az Z Y , F ≺X 1 + Bz & − −' F 2z F z 1 + Az Z ≺X Y ; g ∈ SA ∗ , gz − g−z 1 + Bz &2z F z' 1.16 1 + Az Z Y ; g ∈ SA ∗ , 1 + Bz 1.17 2z F zF 1 + Az Z ≺X Y ; h ∈ KK . hz − h−zF 1 + Bz 1.19 F &gz − g−z' and 1.15 F ≺X 2z F z 1 + Az Z ≺X Y ; h ∈ KK , hz − h−z 1 + Bz We also introduce the following classes 1.18 Coefficient Inequality for Certain… Sα; δ; A, B = L ∈ ; 1 − α 91 Z &2 F ' 2 F ≺ X1 + AzY M +α F − − 1 + Bz & − −' F 2z F z 1 + Az Z &2z F z' Tα; δ; A, B = L ∈ ; 1 − α +α Y , g ∈ SA ∗ M F ≺X gz − g−z 1 + Bz &gz − g−z' F 2z F z 1 + Az Z &2z F z' Hα; δ; A, B = L ∈ ; 1 − α +α ≺ X Y , h ∈ KAM hz − h−zF 1 + Bz hz − h−z F 2. 1.20 1.21 1.22 Coefficient Inequality ]^__` a. b [6]. eDf gz = z + 7 bh z h ∈ . iℎDk, lm k ≥ 2, h 1 if g ∈ SA ∗ |bh | ≤ o1 if g ∈ K A n Theorem 2.1. Let ∈ Sα; δ; A, B . Then where u δA − B , if |λ + μ| ≤ ν, s21 + 2α |5 − 6 | ≤ t δ A − B s 41 + α |λ + μ|, if |λ + μ| ≥ ν, r 1 + α [2B + 1 − δA − B] , δ1 + 2αA − B 21 + α ν= . δ1 + 2αA − B λ= Proof. Since ∈ Sα; δ; A, B, it follows that &2 F ' 1 + Awz Z 2 F 1 − α +α Y F =X − − 1 + Bwz & − −' F 2.1 2.3 2.4 2.5 2.6 2.7 By expanding (2.7), we obtain 1 + 21 + α z + 21 + 2α5 z + ⋯ 1 = 1 + δA − Bc) z + XδA − Bc − δA − B zB + 1 − δA − B{ c) Y z + ⋯ 2 Identifying the terms and solving, we get = δA − B c 21 + α ) 2.2 2.8 92 B. S. Mehrok et al. 5 = δA − B δA − B[2B + 1 − δA − B] c − c) 21 + 2α 41 + 2α From (2.8) and (2.9), we have 5 − 6 = where λ is defined by (2.5). δA − B δ A − B λ + μc) , c − 21 + 2α 41 + α Applying triangular inequality, |5 − 6 | ≤ δA − B δ A − B |c | + |λ + μ||c) | 21 + 2α 41 + α From (1.3) using |c | ≤ 1 − |c) | , (2.10) leads us to |5 − 6 | where ν is defined by (2.6). If |λ + μ| ≤ ν, then δA − B δ A − B |λ + μ| − ν|c) | , ≤ + 21 + 2α 41 + α |5 − 6 | ≤ The bound is sharp for wz = z . δA − B . 21 + 2α If |λ + μ| ≥ ν, then from (1.3) using |c) | ≤ 1, we get |5 − 6 | ≤ This bound is sharp for wz = z. δ A − B |λ + μ|. 41 + α Corollary 2.1. If ∈ SA ∗ δ; A , B then where δA − B if |λ) + μ| ≤ ν) , |5 − 6 | ≤ | 2 δ A − B |λ) + μ| if |λ) + μ| ≥ ν) , 4 λ) = 2.9 2B + 1 − δA − B δA − B Corollary 2.2. If ∈ SA ∗ A , B, then k} ν) = 2 . δA − B 2.10 Coefficient Inequality for Certain… where 93 A − B if |λ + μ| ≤ ν , 2 |5 − 6 | ≤ | A − B |λ + μ| if |λ + μ| ≥ ν , 4 λ = Corollary 2.3. If ∈ SA ∗ , then 2B A − B k} ν = 2 . A − B |5 − 6 | ≤ max. 1, |μ − 1|. Corollary 2.4. If ∈ K A δ; A , B , then δA − B 6 |5 − 6 | ≤ tδ A − B |λ5 + μ| r 16 u where λ5 = if |λ5 + μ| ≤ ν5 , if |λ5 + μ| ≥ ν5 , 42B + 1 − δA − B 8 and ν5 = . 3δA − B 3δA − B Corollary 2.5. If ∈ K A A, B, then A − B if |λ + μ| ≤ ν , 6 |5 − 6 | ≤ | A − B |λ + μ| if |λ + μ| ≥ ν , 16 where λ = Corollary 2.6. If ∈ K K , then 8B 8 k} ν = . 3A − B 3A − B 4 1 1 |5 − 6 | ≤ Max. , μ − . 3 4 3 Theorem 2.2. If f ∈ Tα; δ; A, B and µ is a complex number, then |5 − where 6 | ≤ u δA − B + 1 + 2α s 31 + 2α 1 tδ A − B s 41 + α |λ + μ| + 3 r if |λ + μ| ≤ ν, if |λ + μ| ≥ ν, 2.11 2.12 94 B. S. Mehrok et al. 21 + α [2B + 1 − δA − B] λ= , 3δ1 + 2αA − B 41 + α ν= . 3δ1 + 2αA − B Proof. Since ∈ Tα; δ; A, B, by definition of subordination &2z F z' 2z F z 1 + Awz Z 1 − α +α = X Y F gz − g−z 1 + Bwz &gz − g−z' F 2.13 2.14 2.15 for some gz = z + 7 b; z ; ∈ SA∗ . ; By expanding (2.15), we obtain 1 + 21 + α z + 1 + 2α35 − b5 z + ⋯ 1 = 1 + δA − Bc) z + zδA − Bc − δA − B B + 1 − δA − B c) { z + ⋯ 2 After equating the terms and solving, we get = δA − B c 21 + α ) δA − B δA − B[2B + 1 − δA − B] 1 c − c) + b5 31 + 2α 61 + 2α 3 From (2.16) and (2.17), we have 5 = 5 − 6 = where λ is defined by (2.13). δA − B δ A − B 1 λ c − + μc + b , ) 41 + α 31 + 2α 3 5 2.16 2.17 Applying triangular inequality, we obtain |5 − 6 | ≤ δA − B δ A − B 1 |c | + |λ | |b |. + μ||c + ) 31 + 2α 41 + α 3 5 Using lemma 2.1 and |c | ≤ 1 − |c) |, (2.18) leads us to 1 δA − B δ A − B |5 − 6 | ≤ |λ + μ| − ν|c) | + , + 31 + 2α 41 + α 3 where ν is defined by (2.14). If |λ + μ| ≤ ν, then |5 − 6 | ≤ δA − B + 1 + 2α . 31 + 2α 2.18 Coefficient Inequality for Certain… 95 This is the sharp bound for wz = z . If |λ + μ| ≥ ν, then from (1.3) using |c) | ≤ 1, we get |5 − 6 | ≤ This bound is sharp for wz = z. δ A − B 1 |λ + μ| + . 41 + α 3 Corollary 2.7. If ∈ CA δ; A , B then δA − B + 1 3 |5 − 6 | ≤ A − B |λ + μ| + 4 3δ t r 12 u where λ = 22B + 1 − δA − B 3δA − B Corollary 2.8. If ∈ CA A , B, then where if |λ + μ| ≤ ν , if |λ + μ| ≥ ν , k} ν = 4 . 3δA − B A−B+1 if |λ + μ| ≤ ν , 3 |5 − 6 | ≤ | 3A − B |λ + μ| + 4 if |λ + μ| ≥ ν , 12 λ = Corollary 2.9. If ∈ CA , then 4B 4 k} ν = . 3A − B 3A − B 1 2 |5 − 6 | ≤ max. 1, + μ − . 3 3 Corollary 2.10. If ∈ CA F δ; A, B, then δA − B + 3 if |λ + μ| ≤ ν , 9 |5 − 6 | ≤ t3δ A − B |λ + μ| + 16 if |λ + μ| ≥ ν , r 48 u where λ = 82B + 1 − δA − B 16 k} ν = . 9δA − B 9δA − B 96 B. S. Mehrok et al. Corollary 2.11. If ∈ CA F A, B then A−B+3 if |λ + μ| ≤ ν , 9 |a5 − μa | ≤ | 3A − B |λ + μ| + 16 , if |λ + μ| ≥ ν , 48 16B 16 k} ν = . 9A − B 9A − B Corollary 2.12. If ∈ CA F , then where λ = 5 1 1 8 |5 − 6 | ≤ max. , + μ − . 9 3 4 9 On the same lines as in Theorem 2.2, we have the following: Theorem 2.3. If f ∈ Hα; δ; A, B and µ is a complex number, then where u 3δA − B + 2α + 1 , s 91 + 2α |5 − 6 | ≤ 1 t δ A − B s 41 + α |λ + μ| + 9 , r 21 + α [2B + 1 − δA − B] λ= 3δ1 + 2αA − B References [1] if |λ + μ| ≤ ν, if |λ + μ| ≥ ν, 41 + α k} ν = . 3δ1 + 2αA − B R.N. Das and P. Singh, On properties of certain subclasses of close-to-convex functions, Ann. Univ. Mariae Curie SklodowskaI, 2(1976), 15-22. [2] R.N. Das and P. Singh, Radius of convexity for certain subclass of close-to-convex functions, J. Indian Math. Soc., 41(1977), 363-369. [3] R.M. Goel and B.S. Mehrok, A subclass of starlike functions with respect to symmetric points, Tamkang J. of Math., 13(1) (1982), 11-24. [4] A. Janteng and S.A. Halim, A subclass of convex functions with respect to symmetric points, Proc. 16th National Symp. on Sc. Math., (2008). [5] Z. Nehari, Conformal Mapping, McGraw-Hill, Comp., Inc., New York, (1952). [6] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11(1959), 72-80.