Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 259054, 10 pages
doi:10.1155/2012/259054
Research Article
On an Integral Transform of
a Class of Analytic Functions
Sarika Verma, Sushma Gupta, and Sukhjit Singh
Department of Mathematics, Sant Longowal Institute of Engineering and Technology,
Longowal 148106, India
Correspondence should be addressed to Sarika Verma, sarika.16984@gmail.com
Received 5 June 2012; Accepted 14 September 2012
Academic Editor: Josip E. Pecaric
Copyright q 2012 Sarika Verma et al. 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.
For α, γ ≥ 0 and β < 1, let Wβ α, γ denote the class of all normalized analytic functions f in the
open unit disc E {z : |z| < 1} such that eiφ 1 − α 2γfz/z α − 2γf z γzf z − β >
0, z ∈ E for some φ ∈ R. It is known Noshiro 1934 and Warschawski 1935 that functions
in Wβ 1, 0 are close-to-convex and hence univalent for 0 ≤ β < 1. For f ∈ Wβ α, γ, we consider
1
the integral transform Fz Vλ fz : 0 λtftz/tdt, where λ is a nonnegative real-valued
1
integrable function satisfying the condition 0 λtdt 1. The aim of present paper is, for given
δ < 1, to find sharp values of β such that i Vλ f ∈ Wδ 1, 0 whenever f ∈ Wβ α, γ and ii
Vλ f ∈ Wδ α, γ whenever f ∈ Wβ α, γ.
1. Introduction
Let A denote the class of analytic functions f defined in the open unit disc E {z : |z| < 1}
with the normalizations f0 f 0 − 1 0, and let S be the subclass of A consisting of
∞
n
n
functions univalent in E. For any two functions fz z ∞
n2 an z and gz z n2 bn z
in A, the Hadamard product or convolution of f and g is the function f ∗ g defined by
∞
f ∗ g z z an bn zn .
1.1
n2
For f ∈ A, Fournier and Ruscheweyh 1 introduced the integral operator
Fz Vλ f z :
1
λt
0
ftz
dt,
t
1.2
2
Abstract and Applied Analysis
1
where λ is a nonnegative real-valued integrable function satisfying the condition 0 λtdt 1.
This operator contains some well-known operators such as Libera, Bernardi, and Komatu as
its special cases. Fournier and Ruscheweyh 1 applied the famous duality theory to show
that for a function f in the class
P β : f ∈ A : ∃φ ∈ R | eiφ f z − β > 0, z ∈ E ,
1.3
the linear integral operator Vλ f is univalent in E. Since then, this operator has been studied
by a number of authors for various choices of λt. In another remarkable paper, Barnard et
al. in 2 obtained conditions such that Vλ f ∈ P1 β whenever f is in the class
Pγ β :
f ∈ A : ∃φ ∈ R | eiφ
1−γ
fz
γf z − β
z
> 0, z ∈ E ,
1.4
with β < 1, γ ≥ 0. Note that for 0 ≤ β < 1, functions in P1 β ≡ Pβ satisfy the condition
f z > β in E and thus are close-to-convex in E. A domain D in C is close-to-convex if its
compliment in C can be written as union of nonintersecting half lines.
In 2008, Ponnusamy and Rønning 3 discussed the univalence of Vλ f for the
functions in the class
Rγ β : f ∈ A : ∃φ ∈ R | eiφ f z γzf z − β > 0, z ∈ E .
1.5
In a very recent paper, Ali et al. 4 studied the class
Wβ α, γ
fz α − 2γ f z γzf z − β > 0, z ∈ E ,
: f ∈ A : ∃φ ∈ R | eiφ 1 − α 2γ
z
1.6
where α, γ ≥ 0 and β < 1. In this paper, they obtained sufficient conditions so that the integral
transform Vλ f maps normalized analytic functions f ∈ Wβ α, γ into the class of starlike
functions. It is evident that Wβ 1, 0 ≡ Pβ, Wβ α, 0 ≡ Pα β and Wβ 1 2γ, γ ≡ Rγ β.
In the present paper, we shall mainly tackle the following problems.
1 For given δ < 1, find sharp values of β βδ, α such that Vλ f ∈ Wδ 1, 0
whenever f ∈ Wβ α, γ.
2 For given δ < 1, find sharp values of β βδ such that Vλ f ∈ Wδ α, γ whenever
f ∈ Wβ α, γ.
To prove one of our results, we shall need the generalized hypergeometric function
we define it here.
p Fq ,
so
Abstract and Applied Analysis
3
Let αj j 1, 2, . . . , p and βj j 1, 2, . . . , q be complex numbers with
0, −1, −2, . . . j 1, 2, . . . , q. Then the generalized hypergeometric function p Fq is defined
βj /
by
∞ α · · · α
1 n
p
zn
n
p Fq z p Fq α1 , . . . , αp ; β1 , . . . , βq ; z n!
n0 β1 n · · · βq n
p ≤q1 ,
1.7
where an is the Pochhammer symbol, defined in terms of the Gamma function, by
Γa n
an :
Γa
1,
n 0,
aa 1 · · · a n − 1,
n ∈ N.
1.8
In particular, 2 F1 is called the Gaussian hypergeometric function. We note that the p Fq series
in 1.7 converges absolutely for |z| < ∞ if p < q 1 and for z ∈ E if p q 1.
We shall also need the following lemma.
Lemma 1.1 see 5. Let β1 < 1, β2 < 1, and η ∈ R. Then, for p, q analytic in E with p0 q0 1, the conditions pz > β1 and eiη qz − β2 > 0 imply eiη p ∗ qz − δ > 0, where
1 − δ 21 − β1 1 − β2 .
2. Main Results
We use the notations introduced in 4. Let μ ≥ 0 and ν ≥ 0 satisfy
μ ν α − γ,
μν γ.
2.1
When γ 0, then μ is chosen to be 0, in which case, ν α ≥ 0. When α 1 2γ, 2.1 yields
μ ν 1 γ 1 μν or μ − 11 − ν 0.
i For γ > 0, then choosing μ 1 gives ν γ.
ii For γ 0, then μ 0 and ν α 1.
Theorem 2.1. Let μ ≥ 0, ν ≥ 0 satisfy 2.1. Further, let δ < 1 be given, and define β βδ, μ, ν by
1
1
−1
1
dη dζ
1−δ
1
1 1
ds
1−
dt dt
1−
−1
λt
λt
,
μ
ν μ
2
ν 0
ν
0 1 ts
0
0 1 tη ζ
−1
1
1
dη
1−δ
1 1 λt
1
dt
1−
1−
dt −1
λt
,
α
2
α 0 1t
α
0
0 1 tη
γ/
0,
γ 0 μ 0, ν α > 0 .
2.2
If f ∈ Wβ α, γ, then F Vλ f ∈ Wδ 1, 0 ⊂ S. The value of β is sharp.
Proof. The case γ 0 μ 0, ν α > 0 corresponds to Theorem 1.5 in 2. So we assume
that γ > 0.
4
Abstract and Applied Analysis
Define
1 − α 2γ
Writing fz z ∞
n2
fz α − 2γ f z γzf z Hz.
z
2.3
an zn , it follows that
Hz 1 ∞
an1 nν 1 nμ 1 zn .
2.4
n1
It is a simple exercise to see that
1
1
1 1
,
;z .
f z Hz ∗ 3 F2 2, , ;
ν μ ν1 μ1
2.5
Let Fz Vλ fz, where Vλ f is defined by 1.2. Then for γ /
0, we can write
1
λt
dt
0 1 − tz
1
1
1
λt
1 1
Hz ∗ 3 F2 2, , ;
,
;z ∗
dt
ν μ ν1 μ1
1
− tz
0
1
1
1
1 1
Hz ∗ λt 3 F2 2, , ;
,
; tz dt.
ν μ ν1 μ1
0
F z f z ∗
2.6
Since f ∈ Wβ α, γ, it follows that {eiφ Hz − β} > 0 for some φ ∈ R. Now, for each
γ > 0, we first claim that
1
λt 3 F2
0
1
1
1−δ
1 1
,
; tz dt > 1 − 2, , ;
,
ν μ ν1 μ1
2 1−β
z ∈ E,
2.7
which, by Lemma 1.1, implies that F ∈ Wδ 1, 0. Therefore, it suffices to verify the inequality
2.7. Using the identity which can be checked by comparing the coefficients of zn on both
sides
d − 1 3 F2 1, b, c; d − 1, e; z − d − 2 3 F2 1, b, c; d, e; z,
2.8
1
dη dζ
1
1
1 1 ds
1 1
1
,
;z 1−
.
3 F2 2, , ;
ν μ ν1 μ1
ν 0 1 − zsμ
ν
1
−
zην ζμ
0
2.9
3 F2 2, b, c; d, e; z
it follows that
Abstract and Applied Analysis
5
Thus,
1
0
1
1
1 1
,
; tz dt
λt 3 F2 2, , ;
ν μ ν1 μ1
1
1
dη dζ
1 1 ds
1
dt.
λt
1−
ν μ
ν 0 1 − tzsμ
ν
0
0 1 − tzη ζ
2.10
Therefore, for γ > 0, we have
1
1
1
1 1
,
; tz dt
λt 3 F2 2, , ;
ν μ ν1 μ1
0
1
1
1
dη dζ
1
ds
1 1
dt dt
−1
λt
λt
>
μ
ν μ
ν 0
ν
0 1 ts
0
0 1 tη ζ
2.11
1−δ
1− ,
2 1−β
in the view of 2.2.
To prove the sharpness, let f ∈ Wβ α, γ be the function determined by
1 − α 2γ
fz 1 z
α − 2γ f z γzf z β 1 − β
.
z
1−z
2.12
Using a series expansion, we see that we can write
2 1−β
fz z zn .
n2 nν 1 − ν nμ 1 − μ
∞
2.13
Then,
∞
Fz Vλ f z z 2 1 − β
ψn
zn ,
1
−
ν
nμ
1
−
μ
nν
n2
2.14
6
Abstract and Applied Analysis
where ψn 1
0
λttn−1 dt. Equation 2.2 can be restated as
1
1
1
dη dζ
1
2
ds
1
1 1
−1
λt
λt
dt dt
1−
μ
ν μ
1−β 1−δ
ν 0
ν
0 1 ts
0
0 1 tη ζ
1
1
dη dζ
1 1 ds
1
2
dt
1 λt −
−1
ν μ
1−δ
ν 0 1 tsμ
ν
0
0 1 tη ζ
2.15
1
∞
1
1
1
2
−1n−1 tn−1
dt
−1
λt
− 1−δ 0
ν
ν
nν 1 − ν
n2 nμ 1 − μ
−
∞
−1n−1 nψn
2 .
1 − δ n2 nν 1 − ν nμ 1 − μ
Finally,
∞
F z 1 2 1 − β
nψn
zn−1 ,
n2 nν 1 − ν nμ 1 − μ
2.16
which for z −1 takes the value
−1 − δ
−1n−1 nψn
δ.
12 1−β
2 1−β
n2 nν 1 − ν nμ 1 − μ
∞
F −1 1 2 1 − β
2.17
This shows that the result is sharp.
6.
Letting γ 0 and α 1 in Theorem 1.1, we obtain the following result of Ruscheweyh
Corollary 2.2. Let δ < 1, and define β βδ, 1 < 1 by
−1
1
1−δ
λt
βδ 1 −
1−
dt
.
2
0 1t
2.18
If f ∈ Wβ 1, 0 ≡ P1 β, then F Vλ f ∈ Wδ 1, 0 ⊂ S. The value of β is sharp.
Theorem 2.3. Let δ < 1 and α, γ ≥ 0, and define β βδ < 1 by
β
−
1−β
1
λt
0
1 − 1 δ/1 − δt
dt.
1 t
If f ∈ Wβ α, γ, then Vλ f ∈ Wδ α, γ. The value of β is sharp.
2.19
Abstract and Applied Analysis
7
Proof. The idea of the proof is similar to the one used to prove Theorem 2 in 1.
1
Let Fz Vλ fz 0 λtftz/tdt. Clearly,
F z 1
0
λt
dt ∗ f z.
1 − tz
2.20
Since, f ∈ Wβ α, γ, so with
1 − α 2γ
gz fz/z α − 2γ f z γzf z − β
,
1−β
2.21
we have eiφ gz > 0, where φ ∈ R.
For γ / α/2,
f z 1 − α 2γ fz
γ
1 −
zf z.
β 1 − β gz −
α − 2γ
α − 2γ
z
α − 2γ
2.22
Putting this value in 2.20,
F z 1
0
λt
dt ∗
1 − tz
1 − α 2γ fz
γ
1 −
zf z .
β 1 − β gz −
α − 2γ
α − 2γ
z
α − 2γ
2.23
Equivalently,
1 λt
1 − α 2γ Fz
γ
1
gz ∗ β 1 − β
dt −
−
zF z.
F z α − 2γ
1
−
tz
α
−
2γ
z
α
−
2γ
0
2.24
Thus
1 λt
dt . 2.25
1 − α 2γ Fz/z α − 2γ F z γzF z gz ∗ β 1 − β
0 1 − tz
In the case when γ α/2,
gz fz/z γzf z − β
.
1−β
2.26
Since
fz
β 1 − β gz − γzf z,
z
2.27
8
Abstract and Applied Analysis
This leads to,
1 λt
Fz
γzF z gz ∗ β 1 − β
dt ,
z
0 1 − tz
2.28
which is clearly 2.25 with γ α/2.
Further F ∈ Wδ α, γ if and only if Gz : Fz − δz/1 − δ ∈ W0 α, γ. Now using
2.25, we obtain
Gz β − δ 1 − β 1 λt
α − γ G z γzG z gz ∗
dt .
1 − α 2γ
z
1 − δ 1 − δ 0 1 − tz
2.29
Since eiφ gz > 0 for some φ ∈ R, it follows by duality principle 8, page 23 that
1 − α 2γ
Gz α − 2γ G z γzG z /0
z
2.30
if, and only if,
1
λt
dt > .
1 − tz
2
2.31
1
1−β β−δ
λt
λt
dt >
dt .
1 − tz
1−δ 1−β
0 1t
2.32
β−δ 1−β
1−δ 1−δ
1
0
Using 1/1 − tz > 1/1 t, we get
β−δ 1−β
1−δ 1−δ
1
0
By using 2.19, we have
β − 1 δ/2
−
1−β
1
0
λt
dt.
1 t
2.33
Thus,
β−δ
1−β
1
0
11−δ
λt
dt ,
1t
21−β
2.34
which implies that
β−δ 1−β
1−δ 1−δ
1
0
1
1−β β−δ
1
λt
λt
dt >
dt .
1 − tz
1−δ 1−β
2
0 1t
2.35
Abstract and Applied Analysis
9
Thus, we deduce, using duality principle, that 1 − α 2γGz/z α − γG z γzG z is
contained in a half plane not containing the origin. So, G ∈ W0 α, γ and hence F ∈ Wδ α, γ.
n
To prove the sharpness, let fz z 21 − β ∞
n2 z /nμ 1 − μnν 1 − ν.
∞
Fz Vλ f z z 2 1 − β
zn ωn
,
nμ 1 − μ nν 1 − ν
n2
where ωn 1
λttn−1 dt.
0
2.36
Further,
β
−
1−β
1
λt
0
1 − 1 δ/1 − δt
dt
1 t
2.37
gives
β
−1 1−β
1
1 1 δ/1 − δ
t dt,
1 t
2.38
∞
2 tλt
dt −1n ωn .
1t
1 − δ n2
2.39
λt
0
or
1
2
1−β 1−δ
1
0
Further, assume that
Fz Hz 1 − α 2γ
α − γ F z γzF z.
z
Since Fz z 21 − β
so,
∞
n
n2 ωn z /nμ
2.40
1 − μnν 1 − ν,
∞
ωn zn−1 .
Hz 1 2 1 − β
2.41
n2
Therefore, for z −1,
∞
1−δ
ωn −1n 1 − 2 1 − β H−1 1 − 2 1 − β
δ.
2 1−β
n2
2.42
This shows that the result is sharp.
9.
Letting γ 0 in Theorem 2.3 above, we obtain the following result of Kim and Rønning
10
Abstract and Applied Analysis
Corollary 2.4. Let δ < 1 and α ≥ 0, and define β βδ by
β
−
1−β
1
λt
0
1 − 1 δ/1 − δt
dt.
1 t
2.43
If f ∈ Wβ α, 0 ≡ Pα β, then Vλ f ∈ Wδ α, 0 ≡ Pα δ. The value of β is sharp.
Upon setting λt 1 ctc with −1 < c, we have the following corollary.
Corollary 2.5. Let δ < 1, α, γ ≥ 0, and −1 < c ≤ 0 be given, and let Gz be defined by
Gz 1 c
zc
z
uc−1 fudu.
2.44
0
Suppose that f ∈ Wβ α, γ, then G ∈ Wδ α, 0, where
β
21 c 2 F1 1, 2 c; 3 c, −1 − 2 c
.
21 c 2 F1 1, 2 c; 3 c, −1
2.45
The constant β is sharp.
11.
The special case of Corollary 2.5 with γ 0 has been obtained by Aghalary et al.
References
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8 S. Ruscheweyh, Convolutions in Geometric Function Theory, vol. 83 of Séminaire de Mathématiques
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