EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 6, No. 4, 2013, 460-468
ISSN 1307-5543 – www.ejpam.com

Coefficient Estimates for Certain Subclasses of Analytic
Functions of Complex Order

Li Zhou, Qing-hua Xu∗

College of Mathematics and Information Science, JiangXi Normal University, NanChang 330022,
China

Abstract. In this paper, we introduce and investigate two interesting subclasses Hg(n, b,λ,α,δ) and
Hg(n, b,λ,α,δ; u) of analytic functions of complex order in the open unit disk U, which are defined by
means of the familiar multiplier operator. Formfunctions belonging to the each of these subclasses, we
obtain several results involving (for example) coefficient bounds. Then results presented here would
generalize many known results.

2010 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Aanalytic functions of complex order, coefficient bounds, multiplier operator,
Sălăgean derivative operator, Cauchy-Euler differential equation, principle of subordination

1. Introduce

Let R= (−∞,+∞) be the set of real numbers, C be the set of complex numbers,

N= {1,2, 3, . . .}

be the set of positive integers,
N2 = {2, 3,4, . . .}

and
N0 = N∪ {0}.

We also letA denote the class of functions f of the form

f (z) = z+
∞
∑

j=2

a jz
j , (1)

∗Corresponding author.

Email addresses: zlghhf@yahoo.com.cn (L. Zhou), xuqh@mail.ustc.edu.cn (Q-H Xu)

http://www.ejpam.com 460 c© 2013 EJPAM All rights reserved.



L. Zhou, Q-h Xu / Eur. J. Pure Appl. Math, 6 (2013), 460-468 461

which are analytic in the open unit disc

U= {z : z ∈ C and |z|< 1}.

A function f (z) ∈A is said to belong to the class S∗(α) of starlike functions of order α in
U if it satisfies the following inequality:

ℜ
�

z f ′(z)
f (z)

�

> α (z ∈ U; 0≤ α < 1).

For functions f (z) in the class S∗(α) given by (1), Robertson [11] proved some coefficient
bounds which we recall here as Lemma 1 below.

Lemma 1. If

f (z) = z+
∞
∑

j=2

a jz
j ∈ S∗(α),

then

|a j| ≤

j−2
∏

k=0
[k+ 2(1−α)]

j!
( j ∈ N2). (2)

Nasr and Aouf [10] and Altintaş et. al [1–8] have extended the coefficient bounds (2)
for the class of S∗(α) to hold true for various interesting subclasses of analytic functions of
complex order.

For a function f (z) in A , the multiplier operator Dn
α,δ f (z) was extended by Deniz and

Orhan in [17] as follows:

D0
α,δ f (z) = f (z)

D1
α,δ f (z) =Dα,δ f (z) = αδz2 f ′′(z) + (α−δ)z f ′(z) + (1−α+δ) f (z)

. . .

Dn
α,δ f (z) =Dα,δ(D

n−1
α,δ f (z))

where α≥ δ ≥ 0 and n ∈ N0. If f ∈A is given by (1) then from the definition of the operator

Dn
α,δ f (z), it is easy verity that Dn

α,δ f (z) = z+
∞
∑

k=2
φn

k akzk, where

φk = [1+ (αδk+α−δ)(k− 1)], (φn
k = [φk]n); α≥ δ ≥ 0 and n ∈ N0.

Remark 1. Dn
α,δ f (z) is a generalization of many other linear operators considered earlier. In

particular, for f (z) inA we have the following :

• Dn
1,0 f (z)≡ Dn f (z) the operator defined by Sălăgean (see [13]).

• Dn
α,0 f (z)≡ Dn

α f (z) (see [16]).



L. Zhou, Q-h Xu / Eur. J. Pure Appl. Math, 6 (2013), 460-468 462

Recently, several authors have obtained many interesting results for various subclasses of
analytic functions involving the Sălăgean derivative operator Dn f (z). For example, Deng [9]
defines a function classB(n,λ,α, b) by

ℜ(1+
1

b
[

z[(1−λ)Dn f (z) +λDn+1 f (z)]′

(1−λ)Dn f (z) +λDn+1 f (z)
− 1])> α

(0≤ α < 1;0≤ λ≤ 1; n ∈ N0; b ∈ C\{0})

and also investigated the subclass T (n,λ,α, b; u) of the analytic function class A , which
consists of functions f (z) ∈ A satisfying the following nonhomogenous Cauchy-differential
equation:

z2 d2w

dz2 + 2(1+ u)z
dw

dz
+ u(1+ u)w = (1+ u)(2+ u)h(z),

where
w = f (z) ∈A , h(z) ∈B(n,λ,α, b) and u ∈ R\(−∞,−1].

In the same paper [9], coefficient bounds for the subclassB(n,λ,α, b) and T (n,λ,α, b, u)
of analytic functions of complex order were obtained.

By using the multiplier differential operator Dn
α,δ, we now define the following new sub-

classes of functions belonging to the classA .

Definition 1. Let g : U→ C be a convex function such that

g(0) = 1 and ℜ(g(z))> 0 (z ∈ U),

and f be an analytic function in U defined by (1). We say that f ∈Hg(n, b,λ,α,δ) if it satisfies
the following condition:

1+
1

b
[

z[F n
λ,α,δ(z)]

′

F n
λ,α,δ(z)

− 1] ∈ g(U) (z ∈ U),

where

F n
λ,α,δ(z) = (1−λ)D

n
α,δ f (z) +λDn+1

α,δ f (z) (α≥ δ ≥ 0,0≤ λ≤ 1; n ∈ N0; b ∈ C\{0}).

Definition 2. A function f (z) ∈A is said to be in the classHg(n, b,λ,α,δ; u), if it satisfies the
following nonhomogenous Cauchy-Euler differential equation:

z2 d2w

dz2 + 2(1+ u)z
dw

dz
+ u(1+ u)w = (1+ u)(2+ u)h(z) (3)

(w = f (z) ∈A , h(z) ∈Hg(n, b,λ,α,δ) and u ∈ R\(−∞,−1]).

Remark 2. Their are many choices of the function g and the values of α, δ which would provide
interesting subclasses of analytic functions of complex order. In particular, if we let

g(z) =
1+ (1− 2β)z

1− z
(0≤ β < 1; z ∈ U),α= 1, andδ = 0,



L. Zhou, Q-h Xu / Eur. J. Pure Appl. Math, 6 (2013), 460-468 463

it is easy to see g is a convex function in U and satisfies the hypotheses of Definition 1. If
f ∈Hg(n, b,λ,α,δ), then

ℜ(1+
1

b
[

z[(1−λ)Dn f (z) +λDn+1 f (z)]′

(1−λ)Dn f (z) +λDn+1 f (z)
− 1])> β(z ∈ U),

that is
f ∈B(n,λ,β , b).

Remark 3. In view of Remark 2, ifwe take

g(z) =
1+ (1− 2β)z

1− z
(0≤ β < 1; z ∈ U),α= 1, and δ = 0

in Definitions 1 and 2, it is easy to observe that the function classes

Hg(n, b,λ,α,δ) andHg(n, b,λ,α,δ; u)

become the aforementioned function classes

B(n,λ,α, b) and T (n,λ,α, b; u),

respectively.

In our investigation, we shall use the principle of subordination between analytic func-
tions, which is explained in Definition 3 below (see also [14, 15]).

Definition 3. For two functions f and g analytic in U, we say that the function f (z) is subordi-
nate to g(z) in U (written f ≺ g (z ∈ U)), if there exists a Schwarz function ω(z) analytic in U
with

ω(0) = 0 and |ω(z)|< 1(z ∈ U),

such that
f (z) = g(ω(z)) (z ∈ U).

In particular, if the function g is univalent in U, the above subordination is equivalent to

f (0) = g(0) and f (U)⊂ g(U).

In this paper, by use of the principle of subordination, we obtain coefficient bounds for
functions in the subclasses

Hg(n, b,λ,α,δ) andHg(n, b,λ,α,δ; u)

of analytic functions of complex order, which we have introduce here. Our results would unify
and extend the corresponding results obtained earlier by Nasr and Aouf [10], Altintaş et. al
[1–8] and Deng [9].



L. Zhou, Q-h Xu / Eur. J. Pure Appl. Math, 6 (2013), 460-468 464

2. Main Results and Their Proofs

In order to prove our main results(Theorems 1 and 2 below), we first recall the following
lemma due to Rogosinski [12].

Lemma 2. Let the function g given by

g(z) = z+
∞
∑

k=1

gkzk

be convex U. Also let the function f given by

f (z) = z+
∞
∑

k=1

akzk

be holomorphic in U. If
f (z)≺ g(z) (z ∈ U),

then
|ak| ≤ |g1| (k ∈ N).

We now state and prove each of our main results given by Theorems 1 and 2 below.

Theorem 1. Let the function f ∈A be given by (1). If f ∈Hg(n, b,λ,α,δ), then

|a j| ≤

j−2
∏

k=0
(k+ |g ′(0)||b|)

φn
j [1−λ+λφ j]( j− 1)!

( j ∈ N2).

Proof. By definition of Dn
α,δ f (z) and F n

λ,α,δ(z), we can write

F n
λ,α,δ(z) = z+

∞
∑

j=2

A jz
j (z ∈ U), (4)

where
A j = φ

n
j (1−λ+λφ j) ( j ∈ N2). (5)

From Definition 1, we thus have

1+
1

b





z[F n
λ,α,δ(z)]

′

F n
λ,α,δ(z)

− 1



 ∈ g(U).

By setting

p(z) = 1+
1

b





z[F n
λ,α,δ(z)]

′

F n
λ,α,δ(z)

− 1



 , (6)



L. Zhou, Q-h Xu / Eur. J. Pure Appl. Math, 6 (2013), 460-468 465

we also deduce that
p(0) = g(0) = 1andp(z) ∈ g(U) (z ∈ U).

Therefore, we have
p(z)≺ g(z) (z ∈ U).

According to Lemma 2, we obtain

|pm|=

�

�

�

�

�

p(m)(0)
m!

�

�

�

�

�

≤ |g ′(0)|= |g1|. (7)

On the other hand, we find from (6) that

z[F n
λ,α,δ(z)]

′ = [1+ b(p(z)− 1)]F n
λ,α,δ(z) (z ∈ U). (8)

Next, we suppose that
p(z) = 1+ p1z+ p2z2+ . . . (z ∈ U). (9)

Since A1 = 1, in view of (4), (8), (9), we deduce that

( j− 1)A j = (p1A j−1+ p2A j−2+ . . .+ p j−1)b ( j ∈ N2). (10)

In view of (7) and (10), for j = 2, 3,4, we obtain

|A2| ≤|g ′(0)||b|,

|A3| ≤
|g ′(0)||b|(1+ |g ′(0)||b|)

2!

|A4| ≤
|g ′(0)||b|(1+ |g ′(0)||b|)(2+ |g ′(0)||b|)

3!
,

respectively. Also, making use of the principle of mathematical induction, we can obtain

|A j| ≤

j−2
∏

k=0
(k+ |g ′(0)||b|))

( j− 1)!
( j ∈ N2).

From (5), we can easily obtain

|a j| ≤

j−2
∏

k=0
(k+ |g ′(0)||b|)

φn
j [1−λ+λφ j]( j− 1)!

( j ∈ N2),

as asserted by Theorem 1. This completes the proof of Theorem 1.



L. Zhou, Q-h Xu / Eur. J. Pure Appl. Math, 6 (2013), 460-468 466

Theorem 2. Let the function f ∈A be given by (1). If f ∈Hg(n, b,λ,α,δ; u), then

|a j| ≤
(1+ u)(2+ u)

j−2
∏

k=0
(k+ |g ′(0)||b|)

( j+ u)( j+ u+ 1)φn
j [1−λ+λφ j]( j− 1)!

( j ∈ N2; u ∈ R\(−∞,−1]).

Proof. Let the function f ∈A be given by (1). Also let

h(z) = z+
∞
∑

j=2

h jz
j ∈Hg(n, b,λ,α,δ).

Thus, from (3), we deduce that

a j =
(1+ u)(2+ u)h j

( j+ u)( j+ u+ 1)
( j ∈ N2; u ∈ R\(−∞,−1]).

Using Theorem 1, we obtain

|a j| ≤
(1+ u)(2+ u)

j−2
∏

k=0
(k+ |g ′(0)||b|)

( j+ u)( j+ u+ 1)φn
j [1−λ+λφ j]( j− 1)!

( j ∈ N2; u ∈ R\(−∞,−1]),

as claimed in Theorem 2. This completes the proof of Theorem 2.

3. Corollaries and Consequences

In view of Remark 2, if we set

g(z) =
1+ (1− 2β)z

1− z
(0≤ β < 1; z ∈ U), α= 1, and δ = 0

in Theorems 1 and 2, respectively, we can easily deduce the following two corollaries, which
we merely state here without proofs.

Corollary 1. Let the function ∈A be given by (1). If f ∈B(n,λ,β , b), then

|a j| ≤

j−2
∏

k=0
[k+ 2|b|(1− β)]

jn(1−λ+λ j)( j− 1)!
( j ∈ N2).

Corollary 2. Let the function f ∈A be given by (1). If f ∈ T (n,λ,β , b; u), then

|a j| ≤
(1+ u)(2+ u)

j−2
∏

k=0
[k+ 2|b|(1− βπ)]

jn(1−λ+λ j)( j− 1)!( j+ u)( j+ 1+ u)
( j ∈ N2; u ∈ R\(−∞,−1]).

Remark 4. Corollaries 1 and 2 were obtained by Deng [9]. However, by use of Theorems 1 and
2, we are able to derive these results much more easily.



REFERENCES 467

ACKNOWLEDGEMENTS This work was supported by NNSF of China (Grant Nos. 11261022,
11061015), the Jiangxi Provincial Natural Science Foundation of China (Grant No.
20132BAB201004), and the Natural Science Foundation of Department of Education of Jiangxi
Province, China (Grant No. GJJ12177).

References

[1] O Altintaş, H Irmak, S Owa, and H M Srivastava. Coefficients bounds for some families
of starlike and convex functions of complex order. Applied Mathematics Letters. 20: 1218-
1222, 2007.

[2] O Altintaş, H Irmak, and H M Srivastava. Fractional calculus and certain starlike func-
tions with negative coefficients. Computers & Mathematics with Applications. 30(2): 9-
15, 1995.

[3] O Altintaş and Ö Özkan. Starlike, convex and close-to-convex functions of complex or-
der. Hacettepe Bulletin of Natural Sciences and Engineering. Series B. 28: 37-46, 1991.

[4] O Altintaş and Ö Özkan. On the classes of starlike and convex functions of complex.
Hacettepe Bulletin of Natural Sciences and Engineering. Series B. 30: 63-68, 2001.

[5] O Altintaş, Ö Özkan, and H M Srivastava. Neighborhoods of a class of analytic functions
with negative coefficients. Applied Mathematics Letters. 13(3): 63-67, 1995.

[6] O Altintaş, Ö Özkan, and H M Srivastava. Majorization by starlike functions of complex
order. Complex Variables, Theory and Application. 46: 207-218, 2001.

[7] O Altintaş, Ö Özkan, and H M Srivastava. Neighborhoods of a certain family of multi-
valent functions with negative coefficient. Computers & Mathematics with Applications.
47:1667-1672, 2004.

[8] O Altintaş and H M Srivastava. Some majorization problems associated with p-valently
starlike and convex functions of complex order. East Asian Mathematical Journal. 17:
175-183, 2001.

[9] Q Deng. Certain subclass of analytic functions with complex order. Applied Mathematics
and Computation. 208: 359-362, 2009.

[10] M A Nasr and M K Aouf. Radius of convexity for the class of starlike functions of complex
order. Bulletin of the Faculty of science. A. Physics and Mathematics. 12: 153-159, 1983.

[11] M S Robertson. On the theory of univalent functions. Annals of Mathematics (Series 1).
37: 374-408, 1936.

[12] W Rogosinski. On the coefficients of subordinate functions. Proceedings of the London
Mathematical Society (Series 1). 48: 48-82, 1943.



REFERENCES 468

[13] G . Sălăgean. Subclass of univalent functions. Complex Analysis. 37: 374-408, 1936.

[14] H M Srivastava, S S Eker, and B Seker. A certain convolution approach for subclasses of
analytic functions with negative coefficients. Integral Transforms and Special Functions.
20: 687-699, 2009.

[15] H M Srivastava, D-G Yang, and N-E Xu. Subordinations for multivalent analytic functions
associated with the Dziok-Srivastava operator. Integral Transforms and Special Functions.
20: 581-606, 2009.

[16] F M Al-Oboudi. On univalent functions defined by a generalized Sălăgean operator.
International Journal of Mathematics and Mathematical Sciences. 27: 1429-1436, 2004.

[17] E Deniz and H Orhan. The Fekete-Szegö problem for a generalized subclass of analytic
functions. Kyungpook Mathematical Journal. 50: 37-47, 2010.