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

Authors: Serap Bulut
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

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 orderm. Relevant connections of some of the results obtained with those in earlier works are also provided.


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## Body

## 1. Introduction, Definitions and Preliminaries

Letℝ=(-∞,∞) be the set of real numbers, let ℂ be the set of complex numbers,
(1.1)ℕ:={1,2,3,…}=ℕ0∖{0}
be the set of positive integers and
(1.2)ℕ*:=ℕ∖{1}={2,3,4,…}.Let𝒜 denote the class of functions of the form
(1.3)f(z)=z+∑n=2∞anzn
which are analytic in the unit disk:
(1.4)𝕌={z∈ℂ:|z|<1}.Recently, Komatu [1] introduced a certain integral operator Laδ defined by
(1.5)Laδf(z)=aδΓ(δ)∫01ta-2(log1t)δ-1f(zt)dt,z∈𝕌;a>0;δ≥0;f(z)∈𝒜.Thus, iff∈𝒜 is of the form (1.3), then it is easily seen from (1.5) that (see [1])
(1.6)Laδf(z)=z+∑n=2∞(aa+n-1)δanzn,a>0;δ≥0.Using the relation (1.6), it is easily verfied that
(1.7)z(Laδ+1f(z))′=aLaδf(z)-(a-1)Laδ+1f(z),Laδ(zf′(z))=z(Laδf(z))′.We note that:(i)
fora=1 and δ=k (k is any integer), the multiplier transformation L1kf(z)=Ikf(z) was studied by Flett [2] and  Sălageăn [3];(ii)
fora=1 and δ=-k (k∈ℕ0), the differential operator L1-kf(z)=Dkf(z) was studied by Sălageăn [3];(iii)
fora=2 and δ=k (k is any integer), the operator L2kf(z)=Lkf(z) was studied by Uralegaddi and Somanatha [4];(iv)
fora=2, the multiplier transformation L2δf(z)=Iδf(z) was studied by Jung et al. [5].Using the operatorLaδ, we now introduce the following classes.Definition 1.1.
One  says that a functionf∈𝒜 is in the class 𝒮a,δ(b,β) if
(1.8)Re{1+1b(z(Laδf(z))′Laδf(z)-1)}>β,
where z∈𝕌;a>0;δ≥0;0≤β<1;b∈ℂ∖{0}.Definition 1.2.
One says that a functionf∈𝒜 is in the class 𝒞a,δ(b,β) if
(1.9)Re{1+1bz(Laδf(z))′′(Laδf(z))′}>β,
where z∈𝕌;a>0;δ≥0;0≤β<1;b∈ℂ∖{0}.
Note that(1.10)f∈𝒞a,δ(b,β)⇔zf′∈𝒮a,δ(b,β).
In particular, the classes
(1.11)𝒮a,δ(b,0)≡𝒮a,δ(b),𝒞a,δ(b,0)≡𝒞a,δ(b)
introduced by Bulut [6].
Making use of the Komatu integral operatorLaδ, we now introduce each of the following subclasses of analytic functions.Definition 1.3.
One  denotes by𝒮a,δ(λ,b,A,B) the class of functions f∈𝒜 satisfying
(1.12)1+1b(z(λz(Laδf(z))′+(1-λ)Laδf(z))′λz(Laδf(z))′+(1-λ)Laδf(z)-1)≺1+Az1+Bz,
where z∈𝕌;a>0;δ≥0;-1≤B<A≤1;0≤λ≤1;b∈ℂ∖{0}.Definition 1.4.
A functionf∈𝒜 is said to be in the class ℬa,δ(λ,b,A,B,m;u) if it satisfies the following non-homogenous Cauchy-Euler differential equation:
(1.13)zmdmwdzm+(m1)(u+m-1)zm-1dm-1wdzm-1+⋯+(mm)w∏j=0m-1‍(u+j)=g(z)∏j=0m-1‍(u+j+1)(w=f(z)∈𝒜;g∈𝒮a,δ(λ,b,A,B);m∈ℕ*;u∈(-1,∞)).Remark 1.5.
If we setδ=0 in the classes 𝒮a,δ(λ,b,A,B) and ℬa,δ(λ,b,A,B,m;u), then we have the classes
(1.14)𝒮(λ,b,A,B),𝒦(λ,b,A,B,m;u)
introduced by Srivastava et al. [7], respectively.
If we takeA=1-2β  (0≤β<1) and B=-1 in the class 𝒮a,δ(λ,b,A,B), then we have a new class consisting of functions f∈𝒜 which satisfy the condition
(1.15)Re{1+1b(z(λz(Laδf(z))′+(1-λ)Laδf(z))′λz(Laδf(z))′+(1-λ)Laδf(z)-1)}>β,z∈𝕌.
We denote this class by 𝒮a,δ(λ,b,β). Also we denote by ℬa,δ(λ,b,β,m;u) for corresponding class to ℬa,δ(λ,b,1-2β,-1,m;u).
Note that takingλ=0 and λ=1 for the class 𝒮a,δ(λ,b,β), we have the classes 𝒮a,δ(b,β) and 𝒞a,δ(b,β), respectively. In particular, the classes
(1.16)𝒮a,0(λ,b,β)≡𝒮𝒞(b,λ,β),ℬa,0(λ,b,β,2;u)≡ℬ(b,λ,β;u)
are studied by Altıntaş et al. [8].
In this work, by using the principle of subordination, we obtain coefficient bounds for functions in the subclasses(1.17)𝒮a,δ(λ,b,A,B),ℬa,δ(λ,b,A,B,m;u)
of analytic functions of complex order, which we have introduced here. Our results would unify and extend the corresponding results obtained earlier by Robertson [9], Nasr and Aouf [10], Altıntaş et al. [8] and Srivastava et al. [7].
In our investigation, we will make use of the principle of subordination between analytic functions, which is explained in Definition1.6 below (see [11]).Definition 1.6.
For two functionsf and g, analytic in 𝕌, one says that the function f(z) is subordinate to g(z) in 𝕌, and write
(1.18)f(z)≺g(z),z∈𝕌,
if there exists a Schwarz function w(z), analytic in 𝕌, with
(1.19)w(0)=0,|w(z)|<1,z∈𝕌,
such that
(1.20)f(z)=g(w(z)),z∈𝕌.
In particular, if the function g is univalent in 𝕌, the above subordination is equivalent to
(1.21)f(0)=g(0),f(𝕌)⊂g(𝕌).
In order to prove our main results (Theorems2.1 and 2.2 in Section 2), we first recall the following lemma due to Rogosinski [12].Lemma 1.7.
Let the functiong given by
(1.22)g(z)=∑k=1∞bkzk,z∈𝕌,
be convex in 𝕌. Also let the function f given by
(1.23)f(z)=∑k=1∞akzk,z∈𝕌,
be holomorphic in 𝕌. If
(1.24)f(z)≺g(z),z∈𝕌,
then
(1.25)|ak|≤|b1|,k∈ℕ.


## 2. The Main Results and Their Demonstration

We now state and prove each of our main results given by Theorems2.1 and 2.2 below.Theorem 2.1.
Let the functionf∈𝒜 be defined by (1.3). If the function f is in the class 𝒮a,δ(λ,b,A,B), then
(2.1)|an|≤(a+n-1a)δ∏j=0n-2[j+|b|(A-B)](n-1)!(1+λ(n-1)),n∈ℕ*.Proof.
Let the functionf∈𝒜 be given by (1.3). Define a function
(2.2)h(z)=λz(Laδf(z))′+(1-λ)Laδf(z),z∈𝕌.
We note that the function h is of the form
(2.3)h(z)=z+∑n=2∞‍Anzn,z∈𝕌,
where, for convenience,
(2.4)An=(1+λ(n-1))(aa+n-1)δan,n∈ℕ*.
From Definition 1.3 and (2.2), we obtain that
(2.5)1+1b(zh′(z)h(z)-1)≺1+Az1+Bz,z∈𝕌.
Let us set
(2.6)g(z)=1+Az1+Bz
and define the function p(z) by
(2.7)p(z)=1+1b(zh′(z)h(z)-1),z∈𝕌.
Therefore, we have
(2.8)p(z)≺g(z),z∈𝕌.
Hence, by Definition 1.6, we deduce that
(2.9)p(z)=1+Aw(z)1+Bw(z)(w(0)=0;|w(z)|<1).
Note that
(2.10)p(0)=g(0)=1,p(z)∈g(𝕌),z∈𝕌.
Also from (2.7), we find
(2.11)zh′(z)=[1+b(p(z)-1)]h(z).
Let
(2.12)p(z)=1+c1z+c2z2+⋯,z∈𝕌.
Since A1=1, in view of (2.3), (2.11) and (2.12), we obtain
(2.13)(n-1)An=b{cn-1+cn-2A2+⋯+c1An-1}
for n∈ℕ*. On the other hand, according to the Lemma 1.7, we obtain
(2.14)|p(m)(0)m!|≤A-B,m∈ℕ.
By combining (2.14) and (2.13), for n=2,3,4, we obtain
(2.15)|A2|≤|b|(A-B),|A3|≤|b|(A-B)(1+|b|(A-B))2!,|A4|≤|b|(A-B)(1+|b|(A-B))(2+|b|(A-B))3!,
respectively. Using the principle of mathematical induction, we obtain
(2.16)|An|≤∏j=0n-2[j+|b|(A-B)](n-1)!,n∈ℕ*.
Now from (2.4), it is clear that
(2.17)|an|≤(a+n-1a)δ∏j=0n-2[j+|b|(A-B)](n-1)!(1+λ(n-1)),n∈ℕ*.
This evidently completes the proof of Theorem 2.1.Theorem 2.2.
Let the functionf∈𝒜 be defined by (1.3). If the function f is in the class ℬa,δ(λ,b,A,B,m;u), then
(2.18)|an|≤(a+n-1a)δ∏j=0n-2[j+|b|(A-B)](n-1)!(1+λ(n-1))∏j=0m-1(u+j+1)∏j=0m-1(u+j+n),n∈ℕ*.Proof.
Let the functionf∈𝒜 be given by (1.3). Also let
(2.19)q(z)=z+∑n=2∞‍Bnzn∈𝒮a,δ(λ,b,A,B),
so that
(2.20)an=∏j=0m-1(u+j+1)∏j=0m-1(u+j+n)Bn,n∈ℕ*,u∈(-1,∞).
Thus, by using Theorem 2.1, we obtain
(2.21)|an|≤(a+n-1a)δ∏j=0n-2[j+|b|(A-B)](n-1)!(1+λ(n-1))∏j=0m-1(u+j+1)∏j=0m-1(u+j+n).
This completes the proof of Theorem 2.2.


## 3. Corollaries and Consequences

In this section, we apply our main results (Theorems2.1 and 2.2) in order to deduce each of the following corollaries and consequences.It is easy to see that(3.1)j+|b|(A-B)≤j+2|b|(A-B)1-B,j∈ℕ*,-1≤B<A≤1,b∈ℂ∖{0},
which would obviously yield significant improvements over Theorems 2.1 and 2.2 (see Srivastava et al. [7]).SettingA=1-2β  (0≤β<1) and B=-1 in Theorems 2.1 and 2.2, we haveCorollary 3.1.
Let the functionf∈𝒜 be defined by (1.3). If the function f is in the class 𝒮a,δ(λ,b,β), then
(3.2)|an|≤(a+n-1a)δ∏j=0n-2[j+2|b|(1-β)](n-1)!(1+λ(n-1)),n∈ℕ*.Remark 3.2.
Takingδ=0 in Corollary 3.1, we have [8, Theorem 1].Corollary 3.3.
Let the functionf∈𝒜 be defined by (1.3). If the function f is in the class ℬa,δ(λ,b,β,m;u), then
(3.3)|an|≤(a+n-1a)δ∏j=0n-2[j+2|b|(1-β)](n-1)!(1+λ(n-1))∏j=0m-1(u+j+1)∏j=0m-1(u+j+n),n∈ℕ*.Remark 3.4.
Takingδ=0 and m=2 in Corollary 3.3, we have [8, Theorem 2].Lettingλ=0 and λ=1 in Corollary 3.1, we get following corollaries, respectively.Corollary 3.5.
Let the functionf∈𝒜 be defined by (1.3). If the function f is in the class 𝒮a,δ(b,β), then
(3.4)|an|≤(a+n-1a)δ∏j=0n-2[j+2|b|(1-β)](n-1)!,n∈ℕ*.Corollary 3.6.
Let the functionf∈𝒜 be defined by (1.3). If the function f is in the class 𝒞a,δ(b,β), then
(3.5)|an|≤(a+n-1a)δ∏j=0n-2[j+2|b|(1-β)]n!,n∈ℕ*.For other related results, see also [9, 10].



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