# On an Integral Transform of  a Class of Analytic Functions

Authors: Sarika Verma; Sushma Gupta; Sukhjit Singh
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Forα,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 ℜeiϕ((1-α+2γ)(f(z)/z)+(α-2γ)f′(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 the integral transform F(z)=Vλ(f)(z):=∫01λ(t)(f(tz)/t)dt, where λ is a nonnegative real-valued integrable function satisfying the condition ∫01λ(t)dt=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).


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## Body

## 1. Introduction

Let𝒜 denote the class of analytic functions f defined in the open unit disc E={z:|z|<1} with the normalizations f(0)=f′(0)-1=0, and let S be the subclass of 𝒜 consisting of functions univalent in E. For any two functions f(z)=z+∑n=2∞‍anzn and g(z)=z+∑n=2∞‍bnzn in 𝒜, the Hadamard product (or convolution) of f and g is the function f*g defined by (1.1)(f*g)(z)=z+∑n=2∞‍anbnzn.
For f∈𝒜, Fournier and Ruscheweyh [1] introduced the integral operator
(1.2)F(z)=Vλ(f)(z):=∫01‍λ(t)f(tz)tdt,
where λ is a nonnegative real-valued integrable function satisfying the condition ∫01‍λ(t)dt=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
(1.3)𝒫(β):={f∈𝒜:∃ϕ∈ℝ∣ℜeiϕ(f'(z)-β)>0,z∈E},
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 λ(t). In another remarkable paper, Barnard et al. in [2] obtained conditions such that Vλ(f)∈𝒫1(β) whenever f is in the class
(1.4)𝒫γ(β):={f∈𝒜:∃ϕ∈ℝ∣ℜeiϕ((1-γ)f(z)z+γf′(z)-β)>0,z∈E},
with β<1,γ≥0. Note that for 0≤β<1, functions in 𝒫1(β)≡𝒫(β) satisfy the condition ℜf′(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
(1.5)ℛγ(β):={f∈𝒜:∃ϕ∈ℝ∣ℜeiϕ(f'(z)+γzf′′(z)-β)>0,z∈E}.
In a very recent paper, Ali et al. [4] studied the class
(1.6)𝒲β(α,γ):={f∈𝒜:∃ϕ∈ℝ∣ℜeiϕ((1-α+2γ)f(z)z+(α-2γ)f′(z)+γzf′′(z)-β)>0,z∈E},
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)≡𝒫(β), 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 givenδ<1, find sharp values of β=β(δ,α) such that Vλ(f)∈𝒲δ(1,0) whenever f∈𝒲β(α,γ).(2)
For givenδ<1, find sharp values of β=β(δ) 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 pFq, so we define it here.Letαj(j=1,2,…,p) and βj(j=1,2,…,q) be complex numbers with βj≠0,-1,-2,…(j=1,2,…,q). Then the generalized hypergeometric function pFq is defined by
(1.7)pFq(z)=pFq(α1,…,αp;β1,…,βq;z)=∑n=0∞‍(α1)n⋯(αp)n(β1)n⋯(βq)nznn!(p≤q+1),
where (a)n is the Pochhammer symbol, defined in terms of the Gamma function, by
(1.8)(a)n:=Γ(a+n)Γ(a)={1,n=0,a(a+1)⋯(a+n-1),n∈ℕ.
In particular, 2F1 is called the Gaussian hypergeometric function. We note that the pFq 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]).
Letβ1<1, β2<1, and η∈ℝ. Then, for p,q analytic in E with p(0)=q(0)=1, the conditions ℜp(z)>β1 and ℜeiη(q(z)-β2)>0 imply ℜeiη((p*q)(z)-δ)>0, where 1-δ=2(1-β1)(1-β2).


## 2. Main Results

We use the notations introduced in [4]. Let μ≥0 and ν≥0 satisfy
(2.1)μ+ν=α-γ,μν=γ.
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-1)(1-ν)=0.(i)
Forγ>0, then choosing μ=1 gives ν=γ.(ii)
Forγ=0, then μ=0 and ν=α=1.Theorem 2.1.
Letμ≥0, ν≥0 satisfy (2.1). Further, let δ<1 be given, and define β=β(δ,μ,ν) by
(2.2)1-1-δ2{1-1ν∫01‍λ(t)(∫01‍ds1+tsμ)dt+(1ν-1)∫01‍λ(t)(∬01dηdζ1+tηνζμ)dt}-1,γ≠0,1-1-δ2{1-1α∫01‍λ(t)1+tdt+(1α-1)∫01‍λ(t)(∫01‍dη1+tηα)dt}-1,γ=0(μ=0,ν=α>0).
If f∈𝒲β(α,γ), then F=Vλ(f)∈𝒲δ(1,0)⊂S. The value of β is sharp.Proof.
The caseγ=0(μ=0,ν=α>0) corresponds to Theorem  1.5 in [2]. So we assume that γ>0.
Define(2.3)(1-α+2γ)f(z)z+(α-2γ)f′(z)+γzf′′(z)=H(z).
Writing f(z)=z+∑n=2∞‍anzn, it follows that
(2.4)H(z)=1+∑n=1∞‍an+1(nν+1)(nμ+1)zn.
It is a simple exercise to see that(2.5)f′(z)=H(z)*F23(2,1ν,1μ;1ν+1,1μ+1;z).
Let F(z)=Vλ(f)(z), where Vλ(f) is defined by (1.2). Then for γ≠0, we can write
(2.6)F′(z)=f′(z)*∫01‍λ(t)1-tzdt=H(z)*F23(2,1ν,1μ;1ν+1,1μ+1;z)*∫01‍λ(t)1-tzdt=H(z)*∫01‍λ(t)F23(2,1ν,1μ;1ν+1,1μ+1;tz)dt.
Sincef∈𝒲β(α,γ), it follows that ℜ{eiϕ(H(z)-β)}>0 for some ϕ∈ℝ. Now, for each γ>0, we first claim that
(2.7)ℜ[∫01‍λ(t)F23(2,1ν,1μ;1ν+1,1μ+1;tz)dt]>1-1-δ2(1-β),z∈E,
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)
(2.8)3F2(2,b,c;d,e;z)=(d-1)F23(1,b,c;d-1,e;z)-(d-2)F23(1,b,c;d,e;z),
it follows that
(2.9)3F2(2,1ν,1μ;1ν+1,1μ+1;z)=1ν∫01‍ds1-zsμ+(1-1ν)∬01dηdζ1-zηνζμ.
Thus,
(2.10)∫01‍λ(t)F23(2,1ν,1μ;1ν+1,1μ+1;tz)dt=∫01‍λ(t){1ν∫01‍ds1-tzsμ+(1-1ν)∬01dηdζ1-tzηνζμ}dt.
Therefore, for γ>0, we have
(2.11)ℜ[∫01‍λ(t)F23(2,1ν,1μ;1ν+1,1μ+1;tz)dt]>1ν∫01‍λ(t)(∫01‍ds1+tsμ)dt+(1ν-1)∫01‍λ(t)(∬01dηdζ1+tηνζμ)dt=1-1-δ2(1-β),
in the view of .
To prove the sharpness, letf∈𝒲β(α,γ) be the function determined by
(2.12)(1-α+2γ)f(z)z+(α-2γ)f′(z)+γzf′′(z)=β+(1-β)1+z1-z.
Using a series expansion, we see that we can write
(2.13)f(z)=z+∑n=2∞‍2(1-β)(nν+1-ν)(nμ+1-μ)zn.
Then,
(2.14)F(z)=Vλ(f)(z)=z+2(1-β)∑n=2∞‍ψn(nν+1-ν)(nμ+1-μ)zn,
where ψn=∫01‍λ(t)tn-1dt. Equation  can be restated as
(2.15)11-β=21-δ{1-1ν∫01‍λ(t)(∫01‍ds1+tsμ)dt+(1ν-1)∫01‍λ(t)(∬01dηdζ1+tηνζμ)dt}=21-δ{1+∫01‍λ(t)(-1ν∫01‍ds1+tsμ+(1ν-1)∬01dηdζ1+tηνζμ)dt}=21-δ∫01‍λ(t){∑n=2∞‍(-1)n-1tn-1(nμ+1-μ)(-1ν+(1ν-1)1(nν+1-ν))}dt=-21-δ∑n=2∞‍(-1)n-1nψn(nν+1-ν)(nμ+1-μ).
Finally,
(2.16)F′(z)=1+2(1-β)∑n=2∞‍nψn(nν+1-ν)(nμ+1-μ)zn-1,
which for z=-1 takes the value
(2.17)F′(-1)=1+2(1-β)∑n=2∞‍(-1)n-1nψn(nν+1-ν)(nμ+1-μ)=1+2(1-β){-(1-δ)2(1-β)}=δ.
This shows that the result is sharp.Lettingγ=0 and α=1 in Theorem  1.1, we obtain the following result of Ruscheweyh [6].Corollary 2.2.
Letδ<1, and define β=β(δ,1)<1 by
(2.18)β(δ)=1-1-δ2{1-∫01‍λ(t)1+tdt}-1.
If f∈𝒲β(1,0)≡𝒫1(β), then F=Vλ(f)∈𝒲δ(1,0)⊂S. The value of β is sharp.Theorem 2.3.
Letδ<1 and α,γ≥0, and define β=β(δ)<1 by
(2.19)β1-β=-∫01‍λ(t)(1-((1+δ)/(1-δ))t)(1+t)dt.
If f∈𝒲β(α,γ), then Vλ(f)∈𝒲δ(α,γ). The value of β is sharp.Proof.
The idea of the proof is similar to the one used to prove Theorem  2 in [1].
LetF(z)=Vλ(f)(z)=∫01‍λ(t)(f(tz)/t)dt. Clearly,
(2.20)F′(z)=∫01‍λ(t)1-tzdt*f′(z).
Since, f∈𝒲β(α,γ), so with
(2.21)g(z)=(1-α+2γ)(f(z)/z)+(α-2γ)f′(z)+γzf′′(z)-β1-β,
we have ℜ[eiϕg(z)]>0, where ϕ∈ℝ.
Forγ≠α/2,
(2.22)f′(z)=1α-2γ(β+(1-β)g(z))-1-α+2γα-2γf(z)z-γα-2γzf′′(z).
Putting this value in (2.20),
(2.23)F′(z)=∫01‍λ(t)1-tzdt*(1α-2γ(β+(1-β)g(z))-1-α+2γα-2γf(z)z-γα-2γzf′′(z)).
Equivalently,
(2.24)F′(z)=1α-2γg(z)*[β+(1-β)∫01‍λ(t)1-tzdt]-1-α+2γα-2γF(z)z-γα-2γzF′′(z).
Thus
(2.25)(1-α+2γ)(F(z)/z)+(α-2γ)F'(z)+γzF′′(z)=g(z)*[β+(1-β)∫01‍λ(t)1-tzdt].
In the case when γ=α/2,
(2.26)g(z)=f(z)/z+γzf′′(z)-β1-β.
Since
(2.27)f(z)z=β+(1-β)g(z)-γzf′′(z),This leads to,(2.28)F(z)z+γzF′′(z)=g(z)*[β+(1-β)∫01‍λ(t)1-tzdt],
which is clearly (2.25) with γ=α/2.
FurtherF∈𝒲δ(α,γ) if and only if G(z):=(F(z)-δz)/(1-δ)∈𝒲0(α,γ). Now using (2.25), we obtain
(2.29)(1-α+2γ)G(z)z+(α-γ)G′(z)+γzG′′(z)=g(z)*[β-δ1-δ+1-β1-δ∫01‍λ(t)1-tzdt].
Since ℜeiϕg(z)>0 for some ϕ∈ℝ, it follows by duality principle [8, page 23] that
(2.30)(1-α+2γ)G(z)z+(α-2γ)G′(z)+γzG′′(z)≠0
if, and only if,
(2.31)ℜ[β-δ1-δ+1-β1-δ∫01‍λ(t)1-tzdt]>12.
Using ℜ(1/(1-tz))>1/(1+t), we get
(2.32)ℜ[β-δ1-δ+1-β1-δ∫01‍λ(t)1-tzdt]>1-β1-δ[β-δ1-β+∫01‍λ(t)1+tdt].
By using (2.19), we have
(2.33)β-(1+δ)/21-β=-∫01‍λ(t)(1+t)dt.
Thus,
(2.34)β-δ1-β+∫01‍λ(t)1+tdt=121-δ1-β,
which implies that
(2.35)ℜ[β-δ1-δ+1-β1-δ∫01‍λ(t)1-tzdt]>1-β1-δ[β-δ1-β+∫01‍λ(t)1+tdt]=12.
Thus, we deduce, using duality principle, that (1-α+2γ)(G(z)/z)+(α-γ)G′(z)+γzG′′(z) is contained in a half plane not containing the origin. So, G∈𝒲0(α,γ) and hence F∈𝒲δ(α,γ).
To prove the sharpness, letf(z)=z+2(1-β)∑n=2∞‍(zn/(nμ+1-μ)(nν+1-ν)).
(2.36)F(z)=Vλ(f)(z)=z+2(1-β)∑n=2∞‍znωn(nμ+1-μ)(nν+1-ν),whereωn=∫01‍λ(t)tn-1dt.
Further,
(2.37)β1-β=-∫01‍λ(t)(1-((1+δ)/(1-δ))t)(1+t)dt
gives
(2.38)β1-β=-1+∫01‍λ(t)(1+(1+δ)/(1-δ))(1+t)tdt,
or
(2.39)11-β=21-δ∫01‍tλ(t)1+tdt=21-δ∑n=2∞‍(-1)nωn.
Further, assume that
(2.40)H(z)=(1-α+2γ)F(z)z+(α-γ)F′(z)+γzF′′(z).
Since F(z)=z+2(1-β)∑n=2∞‍(ωnzn/(nμ+1-μ)(nν+1-ν)),
so,(2.41)H(z)=1+2(1-β)∑n=2∞‍ωnzn-1.
Therefore, for z=-1,
(2.42)H(-1)=1-2(1-β)∑n=2∞‍ωn(-1)n=1-2(1-β)1-δ2(1-β)=δ.
This shows that the result is sharp.Lettingγ=0 in Theorem 2.3 above, we obtain the following result of Kim and Rønning [9].Corollary 2.4.
Letδ<1 and α≥0, and define β=β(δ) by
(2.43)β1-β=-∫01‍λ(t)(1-((1+δ)/(1-δ))t)(1+t)dt.
If f∈𝒲β(α,0)≡𝒫α(β), then Vλ(f)∈𝒲δ(α,0)≡𝒫α(δ). The value of β is sharp.
Upon settingλ(t)=(1+c)tc with -1<c, we have the following corollary.Corollary 2.5.
Letδ<1, α,γ≥0, and -1<c≤0 be given, and let G(z) be defined by
(2.44)G(z)=(1+c)zc∫0z‍uc-1f(u)du.
Suppose that f∈𝒲β(α,γ), then G∈𝒲δ(α,0), where
(2.45)β=2(1+c)F12(1,2+c;3+c,-1)-(2+c)2(1+c)F12(1,2+c;3+c,-1).
The constant β is sharp.The special case of Corollary2.5 (with γ=0) has been obtained by Aghalary et al. [11].



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