Coefficient Bounds for Certain Subclasses of Analytic Functions Defined by Komatu Integral Operator

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Volume 2012, Article ID 309465,9pages doi:10.1155/2012/309465

Research Article

Coefficient Bounds for Certain

Subclasses of Analytic Functions Defined by

Komatu Integral Operator

Serap Bulut

School of Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Izmit-Kocaeli, Turkey

Correspondence should be addressed to Serap Bulut,[email protected]

Received 23 March 2012; Accepted 11 July 2012

Academic Editor: Yuri Latushkin

Copyrightq2012 Serap Bulut. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We determine the coeﬀcient bounds for functions in certain subclasses of analytic functions of complex order, which are introduced here by means of a certain non-homogeneous Cauchy–Euler type diﬀerential equation of order m. Relevant connections of some of the results obtained with those in earlier works are also provided.

1. Introduction, Definitions and Preliminaries

LetR −∞,∞be the set of real numbers, letCbe the set of complex numbers,

N:{1,2,3, . . .}N0\ {0} 1.1

be the set of positive integers and

N∗_:_N_{\ {}₁_}_{₂_,₃_,₄_{, . . .}_}_. _1.2

LetAdenote the class of functions of the form

fz z

∞

n2

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which are analytic in the unit disk:

U{z∈C:|z|<1}. 1.4

Recently, Komatu1introduced a certain integral operatorLδ

adefined by

Lδ_afz a

δ

Γδ

1

0

ta−2

log1

t

δ−1

fztdt, z∈U; a >0; δ≥0; fz∈ A. 1.5

Thus, iff∈ Ais of the form1.3, then it is easily seen from1.5thatsee1

Lδ_afz z

∞

n2

a an−1

δ

anzn, a >0; δ≥0. 1.6

Using the relation1.6, it is easily verfied that

zLδ_a 1fzaLδ_afz−a−1Lδ_a 1fz,

Lδ_azfz zLδ_afz.

1.7

We note that:

ifora1 andδkkis any integer, the multiplier transformationLk

1fz Ikfz was studied by Flett2and S˘alage˘an3;

iifor a 1 and δ −k k ∈ N0, the diﬀerential operatorL−k₁ fz Dkfzwas studied by S˘alage˘an3;

iiifora2 andδkkis any integer, the operatorLk

2fz Lkfzwas studied by Uralegaddi and Somanatha4;

ivfora2, the multiplier transformationLδ

2fz Iδfzwas studied by Jung et al.

5.

Using the operatorLδ

a, we now introduce the following classes.

Definition 1.1. One says that a functionf ∈ Ais in the classSa,δb, βif

Re

11

b

zLδ

afz

Lδafz

−1

> β, 1.8

wherez∈U; a >0; δ≥0; 0≤β <1; b∈C\ {0}.

Definition 1.2. One says that a functionf ∈ Ais in the classCa,δb, βif

Re

⎧ ⎪ ⎨ ⎪

⎩1

1

b zLδ

afz

Lδafz

⎫ ⎪ ⎬ ⎪

⎭> β, 1.9

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Note that

f∈ Ca,δ

b, β ⇔zf∈ Sa,δ

b, β . 1.10

In particular, the classes

Sa,δb,0≡ Sa,δb, Ca,δb,0≡ Ca,δb 1.11

introduced by Bulut6.

Making use of the Komatu integral operator Lδ

a, we now introduce each of the

following subclasses of analytic functions.

Definition 1.3. One denotes bySa,δλ, b, A, Bthe class of functionsf∈ Asatisfying

1 1

b

⎛ ⎜

⎝z

λzLδ

afz

1−λLδ

afz

λzLδafz

1−λLδafz

−1

⎞ ⎟

⎠≺ 1Az

1Bz, 1.12

wherez∈U;a >0; δ≥0; −1≤B < A≤1; 0≤λ≤1; b∈C\ {0}.

Definition 1.4. A functionf ∈ Ais said to be in the classBa,δλ, b, A, B, m;uif it satisfies the

following non-homogenous Cauchy-Euler diﬀerential equation:

zmd

m_w dzm m 1

um−1zm−1dm−

1_w

dzm−1 · · ·

m m w m−1 j0

uj gz

m−1

j0

uj1

wfz∈ A; g ∈ Sa,δλ, b, A, B; m∈N∗; u∈−1,∞ .

1.13

Remark 1.5. If we setδ0 in the classesSa,δλ, b, A, BandBa,δλ, b, A, B, m;u, then we have

the classes

Sλ, b, A, B, Kλ, b, A, B, m;u 1.14

introduced by Srivastava et al.7, respectively.

If we takeA1−2β0≤β <1andB−1 in the classSa,δλ, b, A, B, then we have

a new class consisting of functionsf∈ Awhich satisfy the condition

Re ⎧ ⎪ ⎨ ⎪ ⎩1 1 b ⎛ ⎜ ⎝z

λzLδ

afz 1−λLδafz

λzLδafz

1−λLδafz

−1 ⎞ ⎟ ⎠ ⎫ ⎪ ⎬ ⎪

⎭> β, z∈U. 1.15

We denote this class bySa,δλ, b, β. Also we denote byBa,δλ, b, β, m;ufor corresponding

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Note that takingλ0 andλ1 for the classSa,δλ, b, β, we have the classesSa,δb, β

andCa,δb, β, respectively. In particular, the classes

Sa,0

λ, b, β ≡ SCb, λ, β , Ba,0

λ, b, β,2;u ≡ Bb, λ, β;u 1.16

are studied by Altıntas¸ et al.8.

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

Sa,δλ, b, A, B, Ba,δλ, b, A, B, m;u 1.17

of analytic functions of complex order, which we have introduced here. Our results would unify and extend the corresponding results obtained earlier by Robertson9, Nasr and Aouf

10, Altıntas¸ et al.8and Srivastava et al.7.

In our investigation, we will make use of the principle of subordination between analytic functions, which is explained in Definition1.6belowsee11.

Definition 1.6. For two functionsf and g, analytic inU, one says that the functionfzis

subordinate togzinU, and write

fz≺gz, z∈U, 1.18

if there exists a Schwarz functionwz, analytic inU, with

w0 0, |wz|<1, z∈U, 1.19

such that

fz gwz, z∈U. 1.20

In particular, if the functiongis univalent inU, the above subordination is equivalent to

f0 g0, fU⊂gU. 1.21

In order to prove our main resultsTheorems2.1and2.2in Section2, we first recall the following lemma due to Rogosinski12.

Lemma 1.7. Let the functionggiven by

gz

∞

k1

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be convex inU. Also let the functionfgiven by

fz

∞

k1

akzk, z∈U, 1.23

be holomorphic inU. If

fz≺gz, z∈U, 1.24

then

|ak| ≤ |b1|, k∈N. 1.25

2. The Main Results and Their Demonstration

We now state and prove each of our main results given by Theorems2.1and2.2below.

Theorem 2.1. Let the function f ∈ A be defined by 1.3. If the function f is in the class

Sa,δλ, b, A, B, then

|an| ≤

an−1

a

δn−2

j0

j|b|A−B

n−1!1λn−1, n∈N

∗_. _2.1

Proof. Let the functionf ∈ Abe given by1.3. Define a function

hz λzLδ

afz

1−λLδ

afz, z∈U. 2.2

We note that the functionhis of the form

hz z

∞

n2

Anzn, z∈U, 2.3

where, for convenience,

An 1λn−1

a an−1

δ

an, n∈N∗. 2.4

From Definition1.3and2.2, we obtain that

1 1

b

zhz hz −1

≺ 1Az

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Let us set

gz 1Az

1Bz 2.6

and define the functionpzby

pz 1 1

b

zhz hz −1

, z∈U. 2.7

Therefore, we have

pz≺gz, z∈U. 2.8

Hence, by Definition1.6, we deduce that

pz 1Awz

1Bwz w0 0; |wz|<1. 2.9

Note that

p0 g0 1, pz∈gU, z∈U. 2.10

Also from2.7, we find

zhz 1bpz−1 hz. 2.11

Let

pz 1c1zc2z2· · ·, z∈U. 2.12

SinceA1 1, in view of2.3,2.11and2.12, we obtain

n−1Anb{cn−1cn−2A2· · ·c1An−1} 2.13

forn∈N∗. On the other hand, according to the Lemma1.7, we obtain

pm0

m!

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By combining2.14and2.13, forn2,3,4, we obtain

|A2| ≤ |b|A−B,

|A3| ≤ |

b|A−B1|b|A−B

2! ,

|A4| ≤ |b|A−B1|b|A−B2|b|A−B

3! ,

2.15

respectively. Using the principle of mathematical induction, we obtain

|An| ≤

_n−₂

j0

j|b|A−B

n−1! , n∈N

∗_. 2.16

Now from2.4, it is clear that

|an| ≤

an−1

a

δn−2

j0

j|b|A−B

n−1!1λn−1, n∈N

∗_. _2.17

This evidently completes the proof of Theorem2.1.

Theorem 2.2. Let the function f ∈ A be defined by 1.3. If the function f is in the class

Ba,δλ, b, A, B, m;u, then

|an| ≤

an−1

a

δn−2

j0

j|b|A−B

n−1!1λn−1

_m−1

j0

uj1

_m−1

j0

ujn , n∈N

∗_. _2.18

Proof. Let the functionf ∈ Abe given by1.3. Also let

qz z

∞

n2

Bnzn∈ Sa,δλ, b, A, B, 2.19

so that

an

_m−1

j0

uj1

_m−1

j0

ujn Bn, n∈N

∗_, _u_∈₋₁_,_∞_. _2.20

Thus, by using Theorem2.1, we obtain

|an| ≤

an−1

a

δn−2

j0

j|b|A−B

n−1!1λn−1

_m−₁

j0

uj1

_m−₁

j0

ujn . 2.21

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3. Corollaries and Consequences

In this section, we apply our main resultsTheorems2.1and2.2in order to deduce each of the following corollaries and consequences.

It is easy to see that

j|b|A−B≤j2|b|A−B

1−B , j∈N

∗_, ₋₁_≤_{B < A}_≤₁_{, b}_∈_C_{\ {}₀_}_, _3.1

which would obviously yield significant improvements over Theorems 2.1 and 2.2 see Srivastava et al.7.

SettingA1−2β0≤β <1andB−1 in Theorems2.1and2.2, we have

Corollary 3.1. Let the functionf∈ Abe defined by1.3. If the functionfis in the classSa,δλ, b, β,

then

|an| ≤

an−1

a

δn−2

j0

j2|b|1−β

n−1!1λn−1, n∈N

∗_. _3.2

Remark 3.2. Takingδ0 in Corollary3.1, we have8, Theorem 1.

Corollary 3.3. Let the function f ∈ A be defined by 1.3. If the function f is in the class

Ba,δλ, b, β, m;u, then

|an| ≤

an−1

a

δn−2

j0

j2|b|1−β

n−1!1λn−1

_m−₁

j0

uj1

_m−₁

j0

ujn , n∈N

∗_. _3.3

Remark 3.4. Takingδ0 andm2 in Corollary3.3, we have8, Theorem 2.

Lettingλ0 andλ1 in Corollary3.1, we get following corollaries, respectively.

Corollary 3.5. Let the functionf∈ Abe defined by1.3. If the functionfis in the classSa,δb, β,

then

|an| ≤

an−1

a

δn−2

j0

j2|b|1−β

n−1! , n∈N

∗_. _3.4

Corollary 3.6. Let the functionf∈ Abe defined by1.3. If the functionfis in the classCa,δb, β,

then

|an| ≤

an−1

a

δn−2

j0

j2|b|1−β

n! , n∈N

∗_. _3.5

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References

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pp. 141–145, 1990.

2 T. M. Flett, “The dual of an inequality of Hardy and Littlewood and some related inequalities,” Journal

of Mathematical Analysis and Applications, vol. 38, pp. 746–765, 1972.

3 G. S. S˘al˘agean, “Subclasses of univalent functions,” in Complex Analysis-Fifth Romanian-Finnish

Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin,

Germany, 1983.

4 B. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in Current Topics in

Analytic Function Theory, pp. 371–374, World Scientific, River Edge, NJ, USA, 1992.

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certain one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications, vol. 176, no. 1, pp. 138–147, 1993.

6 S. Bulut, “Fekete-Szeg ¨o problem for subclasses of analytic functions defined by Komatu integral

operator,” submitted.

7 H. M. Srivastava, O. Altıntas¸, and S. K. Serenbay, “Coeﬃcient bounds for certain subclasses of starlike

functions of complex order,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1359–1363, 2011.

8 O. Altıntas¸, H. Irmak, S. Owa, and H. M. Srivastava, “Coeﬃcient bounds for some families of starlike

and convex functions of complex order,” Applied Mathematics Letters, vol. 20, no. 12, pp. 1218–1222, 2007.

9 M. S. Robertson, “On the theory of univalent functions,” Annals of Mathematics (2), vol. 37, no. 2, pp.

374–408, 1936.

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 Assiut University A, vol. 12, no. 1, pp. 153–159, 1983.

11 S. S. Miller and P. T. Mocanu, Diﬀerential Subordinations: Theory and Applications, vol. 225 of Monographs

and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.

12 W. Rogosinski, “On the coeﬃcients of subordinate functions,” Proceedings of the London Mathematical

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