MATEMATIQKI VESNIK
originalni nauqni rad
research paper
63, 1 (2011), 73–78
March 2011
COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES
OF ANALYTIC FUNCTIONS OF COMPLEX ORDER
B. A. Frasin
Abstract. Let Qb (Φ, Ψ; α) be the class of normalized analytic functions defined in the open
unit disk and satisfying
n
³
´o
1 f (z) ∗ Φ(z)
Re 1 +
−1
>α
b f (z) ∗ Ψ(z)
for nonzero complex number b and for 0 ≤ α < 1. Sufficient condition, involving coefficient
inequalities, for f (z) to be in the class Qb (Φ, Ψ; α) is obtained. Our main result contains some
interesting corollaries as special cases.
1. Introduction and definitions
Let A denote the class of functions f (z) of the form
∞
P
f (z) = z +
an z n ,
n=2
which are analytic in the open unit disk U = {z : z ∈ C and |z|
< 1}. Furthermore,
P∞
let P be the class of functions p(z) of the form p(z) = 1 + n=1 pn z n , which are
analytic in U .
A function f (z) ∈ A is said to be starlike of complex order b (b ∈ C∗ := C\{0})
and type α (0 ≤ α < 1), that is f (z) ∈ Sb∗ (α), if and only if
½
µ
¶¾
1 zf 0 (z)
Re 1 +
−1
>α
(z ∈ U ; b ∈ C∗ ),
b
f (z)
and is said to be convex of complex order b (b ∈ C∗ ) and type α (0 ≤ α < 1),
denoted by Cb (α) if and only if
½
¾
1 zf 00 (z)
Re 1 +
>α
(z ∈ U ; b ∈ C∗ ).
b f 0 (z)
Note that Sb∗ (0) = Sb∗ and Cb (0) = Cb are the classes considered earlier by Nasr and
Auof [6] and Wiatrowski [10]. Also, S1∗ (α) = S ∗ (α) and C1 (α) = C(α) which are,
2010 AMS Subject Classification: 30C45.
Keywords and phrases: Analytic functions; Starlike and convex functions of complex order;
Hadamard product; Coefficient inequalities.
74
B.A. Frasin
respectively, the familiar classes of starlike functions of order α (0 ≤ α < 1) and
convex functions of order α (0 ≤ α < 1).
Further, let Pb (α) denote the class of functions f (z) ∈ A such that
½
¾
1 0
Re 1 + (f (z) − 1) > α
(0 ≤ α < 1, z ∈ U ; b ∈ C∗ ).
b
When b = 1, the class P1 (α) reduces to the class P(α) of analytic functions studied
in [5, 7, 9].
P∞
P∞
Given two analytic functions f (z) = z + n=2 an z n and g(z) = z + n=2 cn z n
their convolution or Hadamard product f (z) ∗ g(z) is defined by
∞
P
f (z) ∗ g(z) = z +
an cn z n
(z ∈ U ).
n=2
By using this product we introduce the class of prestarlike functions of complex
order b (b ∈ C∗ ) and type α (0 ≤ α < 1), which is denoted by Rb (α). Thus f (z) ∈ A
is said to be prestarlike function of complex order b (b ∈ C∗ and type
α (0 ≤ α < 1),
P∞
if and only if f (z)∗sα (z) ∈ Sb∗ (α) where sα (z) = z(1−z)2α−2 = z+ n=2 C(α, n)z n ;
Qn j−2α
C(α, n) = j=2 (n−1)!
(n ≥ 2). It may be noted that Rb (0) = Cb (0) and Rb (1/2) =
∗
Sb (1/2). When b = 1, the class R1 (α) reduces to the class R(α) of prestarlike
functions of order α (0 ≤ α < 1) (see [8]).
Making use of the Hadamard product, Frasin [1] introduced and studied the
following class of analytic functions:
Definition 1.1. Let b (b ∈ C∗ ) and α (0 ≤ α < 1) be given. Let the functions
∞
∞
P
P
Φ(z) = z +
λn z n and Ψ(z) = z +
µn z n
n=2
n=2
be analytic in U , such that λn ≥ 0, µn ≥ 0 and λn ≥ µn for n ≥ 2, we say that
f (z) ∈ A is in Qb (Φ, Ψ; α) if f (z) ∗ Ψ(z) 6= 0 and
µ
¶¾
½
1 f (z) ∗ Φ(z)
−1
>α
(z ∈ U).
Re 1 +
b f (z) ∗ Ψ(z)
We note that, by suitably choosing of Φ(z) and Ψ(z)
we obtain´ the above
³
z
z
subclasses of A of complex order b and type α : Qb (1−z)2 , 1−z
; α = Sb∗ (α);
´
³
´
³
z
z
z+z 2
Qb (1−z)
= Cb (α); Qb (1−z)
= Pb (α) and
3 , (1−z)2 ; α
2 , z; α
³
´
2
z+(1−2α)z
z
Qb (1−z)3−2α , (1−z)2−2α ; α = Rb (α).
In fact many new subclasses of A of complex order b and type α can be defined
and studied by suitably choosing Φ(z) and Ψ(z). For example,
³
´
n
n
³
´o
o
z
Qb 1−z
, z; α := Tb (α) = f (z) ∈ A : Re 1 + 1b f (z)
−
1
>
α
,
z
and
³
Qb
z+z 2
(1−z)3 , z; α
and so on.
´
©
©
: Mb (α) = f (z) ∈ A : Re 1 +
1
b
ª
ª
((zf 0 (z))0 − 1) > α
Coefficient inequalities for certain classes of analytic functions
75
In this paper, we obtain sufficient condition, involving coefficient inequalities,
for f (z) to be in the class Qb (Φ, Ψ; α). Several special cases and consequences of
these coefficient inequalities are also pointed out.
In order to derive our main results, we have to recall here the following lemmas:
Lemma 1.1. [4] A function p(z) ∈ P satisfies Re p(z) > 0 (z ∈ U) if and only
if p(z) 6= x−1
x+1 (z ∈ U) for all |x| = 1.
1.2. A function f (z) ∈ A is in Qb (Φ, Ψ; α) if and only if 1 +
P∞ Lemma
n−1
A
z
6= 0 where
n=2 n
An =
λn + (2b − 1 − 2αb)µn + x(λn − µn )
an
2b(1 − α)
and λ1 = µ1 = 1.
Proof. Applying Lemma 1.1, we have
³
´
(z)∗Φ(z)
1 + 1b ff (z)∗Ψ(z)
−1 −α
x−1
6=
(z ∈ U ; x ∈ C; |x| = 1).
(1.1)
1−α
x+1
Then, we need not consider Lemma 1.1 for z = 0, because it follows that p(0) =
1 6= x−1
x+1 for all |x| = 1. From (1.1), it follows that
(x + 1)(f (z) ∗ Φ(z)) + (2b − 1 − 2αb − x)(f (z) ∗ Ψ(z)) 6= 0.
P∞
Thus, we have 2b(1 − α)z + n=2 [λn + (2b − 1 − 2αb)µn + x(λn − µn )] an z n 6= 0
(z ∈ U ; x ∈ C; |x| = 1), or, equivalently,
µ
¶
∞ λ + (2b − 1 − 2αb)µ + x(λ − µ )
P
n
n
n
n
2b(1 − α)z 1 +
an z n−1 6= 0
(1.2)
2b(1 − α)
n=2
(z ∈ U ; x ∈ C; |x| = 1). Now, dividing both sides of (1.2) by 2b(1 − α)z (z 6= 0),
we obtain
∞ λ + (2b − 1 − 2αb)µ + x(λ − µ )
P
n
n
n
n
1+
an z n−1 6= 0
2b(1 − α)
n=2
(z ∈ U ; x ∈ C; |x| = 1), which completes the proof of Lemma 1.2.
2. Coefficient conditions for functions in the class Qb (Φ, Ψ; α)
With the help of Lemma 1.2, we have
Theorem 2.1. If f (z) ∈ A satisfies the following condition:
µ¯ n ½ l
µ
¶ ¾µ
¶¯
∞
¯P P
¯
P
γ
δ
l−k
¯
¯
(λ
+
(2b
−
1
−
2αb)µ
)(−1)
a
k
k
k
¯
l−k
n−l ¯
n=2
l=1 k=1
¯n ½
µ
¶ ¾µ
¶¯¶
l
¯P P
¯
γ
δ
l−k
¯
¯ ≤ 2 |b| (1 − α)
+¯
(λk − µk )(−1)
ak
¯
l
−
k
n
−
l
l=1 k=1
(0 ≤ α < 1; b ∈ C∗ ; γ, δ ∈ R), then f (z) ∈ Qb (Φ, Ψ; α).
76
B.A. Frasin
Proof.
Note that (1 − z)γ 6= 0, (1 + z)δ 6= 0 (γ, δ ∈ R; z ∈ U). Thus to prove
P∞
that 1 + n=2 An z n−1 6= 0, it is sufficient that
µ
¶
∞
P
1+
An z n−1 (1 − z)γ (1 + z)δ
n=2
·n ½
µ
¶¾ µ
¶¸
∞
P
P Pl
γ
δ
l−k
A
(−1)
z n−1 6= 0,
=1+
k=1 k
l
−
k
n
−
l
n=2 l=1
where A0 = 0 and A1 = 1. Therefore, if f (z) ∈ A satisfies
¯n ½ l
µ
¶¾ µ
¶¯
∞ ¯P
¯
P
P
γ
δ
l−k
¯ ≤ 1,
¯
Ak (−1)
¯
l−k
n−l ¯
n=2 l=1
k=1
that is, if
¯n ½
∞ ¯P
l
P
P
1
¯
(λk + (2b − 1 − 2αb)µk
2 |b| (1 − α) n=2 ¯l=1 k=1
µ
¶ ¾µ
¶¯
¯
γ
δ
l−k
¯
+x[λk − µk ])(−1)
ak
l−k
n−l ¯
µ¯ n ½ l
µ
¶ ¾
∞
¯P P
P
1
γ
l−k
¯
≤
(λ
+
(2b
−
1
−
2αb)µ
)(−1)
ak ×
k
k
2 |b| (1 − α) n=2 ¯l=1 k=1
l−k
¯ ½
µ
¶¯
µ
¶ ¾µ
¶¯¶
l
¯
¯n
¯
δ
γ
δ
¯ + |x| ¯ P P (λk − µk )(−1)l−k
¯
×
a
k
¯
¯
n−l
l−k
n−l ¯
l=1
≤1
k=1
(0 ≤ α < 1; b ∈ C∗ ; x ∈ C; |x| = 1; γ, δ ∈ R),
then f (z) ∈ Qb (Φ, Ψ; α) and so the proof is completed.
3. Particular cases
By considering some special cases of the analytic functions Φ(z) and Ψ(z), we
deduce the following coefficient conditions for functions f (z) to be in the subclasses
Sα∗ (b), Cα (b), Pα (b) and Rα (b) of analytic functions of complex order b (b ∈ C∗ )
and type α (0 ≤ α < 1) as defined in Section 1.
Letting Φ(z) = z/(1 − z)2 and Ψ(z) = z/(1 − z) in Theorem 2.1, we have
Corollary 3.1 If f (z) ∈ A satisfies the following condition:
µ¯ n ½ l
µ
¶ ¾µ
¶¯
¯P P
¯
γ
δ
l−k
¯
¯
(k
+
2b
−
1
−
2αb)(−1)
a
k
¯
l−k
n−l ¯
n=2
l=1 k=1
¯
µ
¶ ¾µ
¶¯¯!
n ½ l
¯X
P
γ
δ
¯
¯
l−k
+¯
(k − 1)(−1)
ak
¯ ≤ 2 |b| (1 − α)
¯
¯
l
−
k
n
−
l
k=1
∞
P
l=1
for some α (0 ≤ α < 1), b ∈ C∗ and γ, δ ∈ R, then f (z) ∈ Sα∗ (b). In particular,
for γ = δ = 0, if f (z) ∈ A satisfies the following condition:
∞
P
{|n + 2b − 1 − 2αb| + (n − 1)} |an | ≤ 2 |b| (1 − α)
n=2
for some α (0 ≤ α < 1) and b ∈ C∗ , then f (z) ∈ Sα∗ (b).
Coefficient inequalities for certain classes of analytic functions
77
Letting Φ(z) = (z + z 2 )/(1 − z)3 and Ψ(z) = z/(1 − z)2 in Theorem 2.1, we
have
Corollary 3.2. If f (z) ∈ A satisfies the following condition:
µ
¶ ¾µ
¶¯
µ¯ n ½ l
¯
¯P P 2
δ
γ
l−k
¯
¯
ak
(k + (2b − 1 − 2αb)k)(−1)
¯
n−l ¯
l−k
n=2
l=1 k=1
¯n ½ l
µ
¶ ¾µ
¶¯¶
¯
¯P P 2
γ
δ
l−k
¯
¯ ≤ 2 |b| (1 − α)
+¯
(k − k)(−1)
ak
l−k
n−l ¯
l=1 k=1
∞
P
for some α (0 ≤ α < 1), b ∈ C∗ and γ, δ ∈ R, then f (z) ∈ Cb (α). In particular,
for γ = δ = 0, if f (z) ∈ A satisfies the following condition:
∞
P
n{|n − 1 + 2b| + (n − 1)} |an | ≤ 2 |b| (1 − α)
n=2
for some α (0 ≤ α < 1) and b ∈ C∗ , then f (z) ∈ Cb (α).
Letting Φ(z) = (z + (1 − 2α)z 2 )/(1 − z)3−2α and Ψ(z) = z/(1 − z)2−2α in
Theorem 2.1, we have
Corollary 3.3. If f (z) ∈ A satisfies the following condition:
µ¯ n ½ l
µ
¶ ¾µ
¶¯
¯P P
¯
γ
δ
l−k
¯
¯
(C(α,
k)(k
+
(2b
−
1
−
2αb))(−1)
a
k
¯
l−k
n−l ¯
n=2
l=1 k=1
¯ ½
µ
¶ ¾µ
¶¯¶
l
n
¯P
¯
P
γ
δ
l−k
¯
¯ ≤ 2 |b| (1 − α)
+¯
C(α, k)(k − 1)(−1)
ak
l−k
n−l ¯
l=1 k=1
∞
P
for some α (0 ≤ α < 1), b ∈ C∗ and γ, δ ∈ R, then f (z) ∈ Rb (α). In particular,
for γ = δ = 0, if f (z) ∈ A satisfies the following condition:
∞
P
C(α, n)[|n + 2b − 1 − 2αb| + (n − 1)] |an | ≤ |b| (1 − α)
n=2
for some α (0 ≤ α < 1) and b ∈ C∗ , then f (z) ∈ Rb (α).
Letting Φ(z) = z/(1 − z)2 and Ψ(z) = z in Theorem 2.1, we have
Corollary 3.4. If f (z) ∈ A satisfies the following condition:
¯ ½
µ
¶ ¾µ
¶¯
∞ ¯P
l
n
¯
P
P
γ
δ
l−k
¯
¯ ≤ |b| (1 − α)
k(−1)
a
k
¯
l−k
n−l ¯
n=2 l=1
k=1
for some α (0 ≤ α < 1), b ∈ C∗ and γ, δ ∈ R, then f (z)P
∈ Pb (α). In particular,
∞
for γ = δ = 0, if f (z) ∈ A satisfies the following condition n=2 n |an | ≤ |b| (1 − α)
∗
for some α (0 ≤ α < 1) and b ∈ C , then f (z) ∈ Pb (α).
Letting Φ(z) = z/(1 − z) and Ψ(z) = z in Theorem 2.1, we have
Corollary 3.5. If f (z) ∈ A satisfies the following condition
¯n ½
µ
¶ ¾µ
¶¯
∞ ¯P
l
¯
P
P
γ
δ
l−k
¯
¯ ≤ |b| (1 − α)
(−1)
a
k
¯
l−k
n−l ¯
n=2 l=1
k=1
78
B.A. Frasin
for some α (0 ≤ α < 1), b ∈ C∗ and γ, δ ∈ R, then f (z) ∈
PT∞b (α). In particular,
for γ = δ = 0, if f (z) ∈ A satisfies the following condition n=2 |an | ≤ |b| (1 − α)
for some α (0 ≤ α < 1) and b ∈ C∗ , then f (z) ∈ Tb (α).
Remark 3.6. (i) If we set α = 0 in Corollary 3.1 and Corollary 3.2, we have
sufficient conditions for functions f (z) to be in the classes S ∗ (b) and C(b) obtained
by Hayami and Owa in [2].
(ii) If we set b = 1 in Corollary 3.1 and Corollary 3.2, we have sufficient
conditions for functions f (z) to be in the classes S ∗ (α) and C(α) obtained by
Hayami et al. in [4].
(iii) If we set b = 1 in Corollary 3.4 and Corollary 3.5, we have sufficient
conditions for functions f (z) to be in the classes P(α) and T (α) obtained by Hayami
and Owa in [3].
Acknowledgement. The author would like to thank the referees for their
helpful comments and suggestions.
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(received 27.04.2010; in revised form 13.06.2010)
Faculty of Science, Department of Mathematics, Al al-Bayt University, P.O. Box: 130095 Mafraq,
Jordan
E-mail: bafrasin@yahoo.com