Coefficient Inequalities For Classes of Univalent
Coefficient Inequalities For Classes of Univalent
Coefficient Inequalities For Classes of Univalent
c
SPM –ISSN-2175-1188 on line ISSN-0037-8712 in press
SPM: www.spm.uem.br/bspm doi:10.5269/bspm.43211
abstract: Using the principal of subordination and the q−derivative, we obtain sharp bounds for some
classes of univalent functions.
Contents
1 Introduction 1
2 Main results 2
1. Introduction
Denote by A the class of analytic functions:
∞
X
f (z) = z + an z n (z ∈ U = {z : z ∈ C , |z| < 1}). (1.1)
n=2
f (qz) − f (z)
Dq f (z) = , z 6= 0
(q − 1)z
∞
X
= 1+ [n]q an z n−1 , (1.2)
n=2
qn − 1
where, [n]q = , as q → 1− , [n]q → n, Dq f (0) = f ′ (0) and Dq (Dq f (z)) = Dq2 f (z). If η(z) = z n ,
q−1
then
q n − 1 n−1
Dq η(z) = Dq (z n ) = z = [n]q z n−1 ,
q−1
Denote by P the class of analytic functions φ of positive real part on U with φ(0) = 1, R{φ(z)} > 0.
Using the q-derivative Dq f (z), f ∈ A, κ ∈ P, 0 ≤ λ ≤ 1, b ∈ C∗ = C/{0}, let
λ 1 zDq f (z) Dq (zDq f (z))
Hq,b (κ) = f :1+ (1 − λ) +λ − 1 ≺ κ(z) , (1.3)
b f (z) Dq f (z)
search Foundation of the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology
(No. 2019R1I1A3A01050861).
2010 Mathematics Subject Classification: 30C45.
Submitted June 09, 2018. Published November 07, 2018
Typeset by BSP
M
style.
1 c Soc. Paran. de Mat.
2 M. K. Aouf, A. O. Mostafa and N. E. Cho
λ,α
and Ramachandran et al. [8], with α = 0 and β = 1. Also, we obtain the new class Hq,θ (κ) for
−iθ π
b = e (1 − α) cos θ, 0 ≤ α < 1, |θ| < 2 , where
h i
zDq f (z) Dq (zDq f (z))
eiθ (1 − λ) f (z) + λ Dq f (z) − α cos θ − i sin θ
λ,α
Hq,θ (κ) = f : ≺ κ(z) .
(1 − α) cos θ
1 + z2 1+z
p(z) = and p(z) = .
1 − z2 1−z
when ξ < 0 or ξ > 1, the equality holds if and only if p(z) is (1 + z)/(1 − z) or one of its rotations. If
0 < ξ < 1, then the equality holds if and only if p(z) is (1 + z 2 )/(1 − z 2 ) or one of its rotations. If ξ = 0,
the equality holds if and only if
1+γ 1+z 1−γ 1−z
p (z) = + (0 ≤ γ ≤ 1)
2 1−z 2 1+z
or one of its rotations. If γ = 1, the equality holds if and only if p is the reciprocal of one of the functions
such that equality holds in the case of ξ = 0.
Also the above upper bound is sharp, and it can be improved as follows when 0 < ξ < 1:
r2 − ξr12 + ξ |r1 |2 ≤ 2
1
0≤ξ≤
2
and
r2 − ξr2 + (1 − ξ) |r1 |2 ≤ 2
1
1 ≤ξ≤1 .
2
2. Main results
a3 − µa22
|b| |d1 |
≤ max {1,
2([3]q − 1)[1 + λ([3]q − 1)]
i
d2 bd1
h
2 ([3]q −1)[1+λ([3]q −1)
+
d1 ([2]q −1)[1+λ([2]q −1)]2 [1 + λ([2]q − 1) − µ ([2]q −1) .
(2.2)
λ
Proof: If f ∈ Hq,b (κ), then there is a function ω, analytic in U with ω(0) = 0 and |ω(z)| < 1 such that
1 + ω (z)
p (z) = = 1 + r1 z + r2 z 2 + ... . (2.4)
1 − ω (z)
We see that ℜ {p (z)} > 0 and p (0) = 1, since ω (z) is a Schwarz function.. Therefore,
p (z) − 1
κ (ω (z)) = κ
p (z) + 1
r12 r13
1 2 3
= κ r1 z + r2 − z + r3 − r1 r2 + z + ...
2 2 4
r2
1 1 1
= 1 + d1 r1 z + d1 r2 − 1 + d2 r12 z 2 + ..... (2.5)
2 2 2 4
1
[[2]q − 1 + λ([2]q − 1)2 ]a2 = bd1 r1 ,
2
and
1 zDq f (z) Dq (zDq f (z))
1+ (1 − λ) +λ − 1 = κ (z) .
b f (z) Dq f (z)
The proof of Theorem 1 is completed.
Remark 2.1. (i) Putting λ = 0 in Theorem 1, we obtain the result of Seoudy and Aouf [10, Theorem
1];
(ii) Putting λ = 1 in Theorem 1, we obtain the result of Seoudy and Aouf [10, Theorem 2];
(iii) Theorem 1 for b = 1, corrects the result of Ramachandram et al. [8, Theorem 2, α = 0, β = 1].
4 M. K. Aouf, A. O. Mostafa and N. E. Cho
Theorem 2.2. Let κ (z) in the form (2.1), with d1 > 0 and d2 ≥ 0. Let
(d2 −d1 )([2]q −1)2 [1+λ([2]q −1)]2 +([2]q −1)[1+λ([2]2q −1)]bd21
α1 = ([3]q −1)[1+λ([3]q −1)]bd21
, (2.8)
(d2 +d1 )([2]q −1) [1+λ([2]q −1)] +([2]q −1)[1+λ([2]2q −1)]bd21
2 2
α2 = ([3]q −1)[1+λ([3]q −1)]bd21
, (2.9)
d2 ([2]q −1) [1+λ([2]q −1)]2 +([2]q −1)[1+λ([2]2q −1)]bd21
2
α3 = ([3]q −1)[1+λ([3]q −1)]bd21
. (2.10)
λ
If f (z)∈Hq,b (κ) with b > 0, then
bd2
([3]q −1)[1+λ([3] q −1)]
+
[1+λ([2]2q −1)]
2 2
b d1 1
+ 2 − µ ([2]q −1) , µ ≤ α1 ,
([2]q −1)[1+λ([2]q −1)] ([3]qbd−1)[1+λ([3]
q −1)]
a3 − µa22 ≤
([3]q −1)[1+λ([3]q −1)] , α1 ≤ µ ≤ α2 , (2.11)
1
bd2
− ([3]q−1)[1+λ([3]q −1)] −
[1+λ([2]2q −1)]
b2 d21
− 1
− µ ([2]q −1) , µ ≥ α2 .
([2]q −1)[1+λ([2]q −1)] 2 ([3]q −1)[1+λ([3]q −1)]
Further, if α1 ≤ µ ≤ α3 , then
h
a3 − µa22 +
([2]q −1)2 [1+λ([2]q −1)]2 bd21
([3]q −1)[1+λ([3]q −1)]d21 b
d1 − d2 − ([2]q −1)[1+λ([2]q −1)]2
i
([3] −1)[1+λ([3] −1)] 2
× [1 + λ([2]2q − 1)] − µ q ([2]q −1) q ) |a2 | ≤ bd1
([3]q −1)[1+λ([3]q −1)] , (2.12)
and if α3 ≤ µ ≤ α2 , then
bd21
h
a3 − µa22 +
([2]q −1)2 [1+λ([2]q −1)]2
([3]q −1)[1+λ([3]q −1)]d21 b
d1 + d2 + ([2]q −1)[1+λ([2]q −1)]2
i
([3] −1)[1+λ([3] −1)] 2
× [1 + λ([2]2q − 1)] − µ q ([2]q −1) q ) |a2 | ≤ bd1
([3]q −1)[1+λ([3]q −1)] . (2.13)
The result is sharp.
Proof: The proof follows by applying (1.5) to (2.6) and (2.7). To show that the bounds are sharp, we
define the functions Kκk (k = 2, 3, 4, ...) by
1 zDq Kκk (z) Dq (zDq Kκk (z))
− 1 = κ z n−1 ,
1+ (1 − λ) +λ
b Kκk (z) Dq Kκk (z)
Fτ (0) = 0 = Fτ (0) − 1
and
1 zDq Gτ (z) Dq (zDq Gτ (z)) 1 + τz
1+ (1 − λ) +λ −1 =κ ,
b Gτ (z) Dq Gτ (z) z (z + τ )
′
Gτ (0) = 0 = Gτ (0) − 1.
λ
The functions Kκk , Fλ and Gλ ∈ Hq,b (κ). If µ < α1 or µ > α2 , then the equality holds if and only if f
is Kκ2 , or one of its rotations. When α1 < µ < α2 , the equality holds if and only if f is Kκ3 , or one of
its rotations. If µ = α1 , then the equality holds if and only if f is Fτ , or one of its rotations. If µ = α2 ,
then the equality holds if and only if f is Gτ , or one of its rotations.
Coefficient Inequalities for Classes of Certain Univalent Functions 5
Remark 2.2 (i) Taking q → 1− and λ = α, in the above results, we obtain the results of [12, with λ = 0];
(ii) Theorem 2 for b = 1, corrects the result of Ramachandram et al. [8, Theorem 1, α = 0, β = 1];
(iii) Putting λ = 0 in Theorem 2, we obtain the result of Seoudy and Aouf [10, Theorem 3];
(iv) Putting λ = 1 in Theorem 2, we obtain the result of Seoudy and Aouf [10, Theorem 3];
λ,α
(v) Taking b = e−iθ (1 − α) cos θ in the above results, we obtain results for the class Hq,θ (κ).
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M. K. Aouf,
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura, 35516, EGYPT.
E-mail address: mkaouf127@yahoo.com
and
A. O. Mostafa,
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura, 35516, EGYPT.
E-mail address: adelaeg254@yahoo.com
and