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volume 6, issue 3, article 86, 2005.

Received 03 March, 2005;

accepted 26 July, 2005.

Communicated by:G. Kohr

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Journal of Inequalities in Pure and Applied Mathematics

A SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS ASSOCIATED WITH CERTAIN FRACTIONAL CALCULUS OPERATORS

G. MURUGUSUNDARAMOORTHY, THOMAS ROSY AND MASLINA DARUS

Department of Mathematics

Vellore Institute of Technology, Deemed University Vellore - 632014, India.

EMail:gmsmoorthy@yahoo.com Department of Mathematics Madras Christian College, Chennai - 600059, India.

School of Mathematical Sciences Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor, Malaysia.

EMail:maslina@pkrisc.cc.ukm.my

c

2000Victoria University ISSN (electronic): 1443-5756 062-05

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A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus

Operators

G. Murugusundaramoorthy, Thomas Rosy and Maslina Darus

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Abstract

In this paper, we introduce a new class of functions which are analytic and univalent with negative coefficients defined by using a certain fractional calculus and fractional calculus integral operators. Characterization property,the results on modified Hadamard product and integrals transforms are discussed.

Further, distortion theorem and radii of starlikeness and convexity are also de- termined here.

2000 Mathematics Subject Classification:30C45.

Key words: Univalent, Convex, Starlike, Uniformly convex , Fractional calculus inte- gral operator.

1. Introduction and Preliminaries

Fractional calculus operators have recently found interesting applications in the theory of analytic functions. The classical definition of fractional calculus and its other generalizations have fruitfully been applied in obtaining, the characterization properties, coefficient estimates and distortion inequalities for various subclasses of analytic functions.

Denote byAthe class of functions of the form

(1.1) f(z) =z+

∞

X

n=2

a_nzⁿ

which are analytic and univalent in the open discE ={z :z ∈ C and |z|<1}.

Also denote byT [11] the subclass ofAconsisting of functions of the form

(1.2) f(z) = z−

∞

X

n=2

anzⁿ, (an ≥0).

A functionf ∈Ais said to be in the class of uniformly convex functions of orderα,denoted byU CV(α)[9] if

(1.3) Re

1 + zf⁰⁰(z) f⁰(z) −α

≥β

zf⁰⁰(z) f⁰(z) −1

,

and is said to be in a corresponding subclass ofU CV(α)denote byS_p(α)if

(1.4) Re

zf⁰(z) f(z) −α

≥β

zf⁰(z) f(z) −1

,

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where−1≤α ≤1andz ∈E.

The class of uniformly convex and uniformly starlike functions has been extensively studied by Goodman [3,4] and Ma and Minda [6].

If f of the form (1.1) andg(z) = z +P∞

n=2b_nzⁿ are two functions inA, then the Hadamard product (or convolution) off andg is denoted byf ∗g and is given by

(1.5) (f∗g)(z) =z+

∞

X

n=2

a_nb_nzⁿ.

Letφ(a, c;z)be the incomplete beta function defined by (1.6) φ(a, c;z) = z+

∞

X

n=2

(a)_n

(c)_nzⁿ, c6= 0,−1,−2, . . . ,

where (λ)_n is the Pochhammer symbol defined in terms of the Gamma functions, by

(λ)n= Γ(λ+n) Γ(λ) =

( 1 n = 0

λ(λ+ 1)(λ+ 2)· · ·(λ+n−1), n ∈N} )

Further suppose

L(a, c)f(z) =φ(a, c;z)∗f(z), for f ∈A whereL(a, c)is called Carlson - Shaffer operator [2].

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For real numberµ(−∞< µ <1)andγ(−∞< γ <1)and a positive real numberη,we define the operator

U_0,z^µ,γ,η :A −→A by

(1.7) U_0,z^µ,γ,η =z+

∞

X

n=2

(2−γ+η)n−1(2)n−1

(2−γ)n−1(2−µ+η)n−1

a_nzⁿ, which forf(z)6= 0may be written as

(1.8) U_0,z^µ,γ,ηf(z) =







Γ(2−γ)Γ(2−µ+γ)

Γ(2−γ+η) z^γJ_0,z^µ,γ,ηf(z); 0≤µ <1

Γ(2−γ)Γ(2−µ+γ)

Γ(2−γ+η) z^γI_0,z^−µ,γ,ηf(z); −∞ ≤µ <0 whereJ_0,z^µ,γ,η andI_0,z^−µ,γ,η are fractional differential and fractional integral operators [12] respectively.

It is interesting to observe that

U_z^µf(z) = Γ(2−µ)z^µD_z^µf(z), − ∞< µ <1

= Ω^µ_zf(z) (1.9)

and D_z^µ is due to Owa [7]. U_z^µ is called a fractional integral operator of order µ, if−∞ < µ < 0and is called fractional differential operator of order µif 0≤µ <1.

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Further note that

U_0,z^µ,γ,ηf(z) = f(z) if µ=γ = 0 U_0,z^µ,γ,ηf(z) = zf⁰(z) if µ=γ = 1.

For−1≤α <1,a functionf ∈Ais said to be inS_µ,γ,η^∗ (α),if and only if

(1.10) Re

z(U_0,z^µ,γ,ηf(z))⁰ U_0,z^µ,γ,ηf(z) −α

≥

z(U_0,z^µ,γ,ηf(z))⁰ U_0,z^µ,γ,ηf(z) −1

, z ∈∆.

where−∞< µ <1,−∞< γ < 1,andη ∈R+. Now let us writeT R(µ, γ, η, α) =S_µ,γ,η^∗ (α)∩T.

It follows from the statement, that forµ=γ = 0, we have S_µ,γ,η^∗ (α) = S_p(α)

and forµ=γ −→1, we have

S_µ,γ,η^∗ (α) = U CV(α).

The classesS_p(α)andU CV(α)are introduced and studied by various authors including [8], [9] and [1].

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2. Characterization Property

We now investigate the characterization property for the functionfto belong to the classS_µ,γ,η^∗ (α),by obtaining the coefficient bounds.

Definition 2.1. A functionfis inT R(µ, γ, η, α)iff satisfies the analytic char- acterization

(2.1) Re

>

, where0≤α <1,−∞< µ <1,−∞< γ < 1,andη ∈R.

Theorem 2.1 (Coefficient Bounds). A functionfdefined by (1.2) is in the class T R(µ, γ, η, α)if and only if

(2.2)

∞

X

n=2

(2−γ+η)n−1(2)n−1

(2−γ)n−1(2−µ+η)n−1

· 2n−1−α

1−α |a_n| ≤1 where0≤α <1, −∞< µ <1, −∞< γ < 1,andη ∈R. Proof. It suffices to show that

≤Re

, and we have

≤Re

+ (1−α).

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That is

−Re

≤2

≤ P∞

n=2(n−1)ψ(n)|an| 1−P∞

n=2ψ(n)|a_n| where

ψ(n) = (2−γ+η)n−1(2)n−1

(2−γ)_n−1(2−µ+η)_n−1.

The above expression is bounded by (1 −α) and hence the assertion of the result.

Now we need to show thatf ∈ T R(µ, γ, η, α)satisfies the coefficient inequality. Iff ∈T R(µ, γ, η, α)andzis real then (2.1) yields

1−P∞

n=2nψ(n)a_nzⁿ⁻¹ 1−P∞

n=2ψ(n)a_nzⁿ⁻¹ −α≥ 1−P∞

n=2(n−1)ψ(n)a_nzⁿ⁻¹ 1−P∞

n=2ψ(n)a_nzⁿ⁻¹ . Lettingz →1along the real axis leads to the desired inequality

∞

X

n=2

(2n−1−α)ψ(n)a_n ≤1−α.

Corollary 2.2. Let a functionfdefined by (1.2) belong to the classT R(µ, γ, η, α).

Then

a_n≤ (2−γ)n−1(2−µ+η)n−1

(2−γ+η)n−1(2)n−1

· 1−α

2n−1−α, n ≥2.

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Next we consider the growth and distortion theorem for the classT R(µ, γ, η, α).

We shall omit the proof as the techniques are similar to various other papers.

Theorem 2.3. Let the functionf defined by (1.2) be in the classT R(µ, γ, η, α).

Then

|z| − |z|²(2−γ)(2−µ+η)(1−α) 2(2−γ+η)(3−α) (2.3)

≤ |U_0,z^µ,γ,ηf(z)|

≤ |z|+|z|²(2−γ)(2−µ+η)(1−α) 2(2−γ+η)(3−α) and

1− |z|(2−γ)(2−µ+η)(1−α) (2−γ+η)(3−α) (2.4)

≤ |(U_0,z^µ,γ,ηf(z))⁰|

≤1 +|z|(2−γ)(2−µ+η)(1−α) (2−γ+η)(3−α) . The bounds (2.3) and (2.4) are attained for functions given by (2.5) f(z) = z−(2−γ)(2−µ+η)(1−α)z²

2(2−γ+η)(3−α) . Theorem 2.4. Let a functionf be defined by (1.2) and

(2.6) g(z) =z−

∞

X

n=2

bnzⁿ

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be in the classT R(µ, γ, η, α).Then the functionhdefined by (2.7) h(z) = (1−λ)f(z) +λg(z) = z−

∞

X

n=2

q_nzⁿ

whereq_n= (1−λ)a_n+λb_n, 0≤λ ≤1is also in the classT R(µ, γ, η, α).

Proof. The result follows easily by using (2.2) and (2.7).

We prove the following theorem by defining functionsf_j(z) (j = 1,2, . . . , m) of the form

(2.8) f_j(z) = z−

∞

X

n=2

a_n,jzⁿ for a_n,j ≥0, z ∈U.

Theorem 2.5 (Closure theorem). Let the functions fj(z) (j = 1,2. . . , m) defined by (2.8) be in the classesT R(µ, γ, η, α_j) (j = 1,2, . . . , m)respectively.

Then the functionh(z)defined by h(z) =z− 1

m

∞

X

n=2 m

X

j=1

! a_n,jzⁿ

is in the classT R(µ, γ, η, α)where

(2.9) α= min

1≤j≤m{αj} with 0≤αj <1.

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Proof. Since f_j ∈ T R(µ, γ, η, α_j) (j = 2, . . . , m)by applying Theorem 2.1, we observe that

∞

X

n=2

ψ(n)(2n−1−α) 1 m

m

X

j=1

an,j

!

= 1 m

m

X

j=1

∞

X

n=2

ψ(n)(2n−1−α)an,j

!

≤ 1 m

m

X

j=1

(1−α_j)≤1−α,

which in view of Theorem 2.1, again implies that h ∈ T R(µ, γ, η, α)and the proof is complete.

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3. Results Involving Modified Hadamard Products

We let

(f ∗g)(z) = z−

∞

X

n=2

a_nb_nzⁿ

be the modified Hadamard product of functions f and g defined by (1.2) and (2.6) respectively. The following results are proved using the techniques of Schild and Silverman [10].

Theorem 3.1. For functions f_j(z) (j = 1,2) defined by (2.8), let f₁(z) ∈ T R(µ, γ, η, α)andf2(z)∈T R(µ, γ, η, β).Thenf1∗f2 ∈T R(µ, γ, η, ξ)where

(3.1) ξ =ξ(µ, γ, η, β) = 1− 2(1−α)(1−β)

(3−α)(3−β)ψ(2)−(1−α)(1−β), whereψ(2) = (2−γ)(2−µ+η)^{(2−γ+η)(2)} .The result is the best possible for

f₁(z) =z− 1−α (3−α)ψ(2)z², f2(z) =z− 1−β

(3−β)ψ(2)z², whereψ(2) = (2−γ)(2−µ+η)^{(2−γ+η)(2)} .

Proof. In the view of Theorem2.1, it suffices to prove that

∞

X

n=2

2n−1−ξ

1−ξ ψ(n)an,1an,2 ≤1,

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whereξ is defined by (3.1) under the hypothesis, it follows from (2.1) and the Cauchy-Schwarz inequality that

(3.2)

∞

X

n=2

[2n−1−α]^1/2[2n−1−β]^1/2

p(1−α)(1−β) ψ(n)√

a_n,1a_n,2 ≤1.

Thus we need to find largestξsuch that

∞

X

n=2

2n−1−ξ

1−ξ ψ(n)a_n,1a_n,2

≤

∞

X

n=2

[2n−1−α]^1/2[2n−1−β]^1/2

p(1−α)(1−β) ψ(n)√

a_n,1a_n,2 ≤1 or, equivalently that

√a_n,1a_n,2 ≤ [2n−1−α]^1/2[2n−1−β]^1/2

p(1−α)(1−β) · 1−ξ

2n−1−ξ for n≥2.

By virtue of (3.2) it is sufficient to find the largestψ such that p(1−α)(1−β)

[2n−1−α]^1/2[2n−1−β]^1/2ψ(n)

≤ [2n−1−α]^1/2[2n−1−β]^1/2

p(1−α)(1−β) · 1−ξ

2n−1−ξ for n ≥2

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which yields

ξ≤1− 2(n−1)(1−α)(1−β)

(2n−1−α)(2n−1−β)ψ(n)−(1−α)(1−β), where

(3.3) ψ(n) = (2−γ+η)n−1(2)n−1

(2−γ)n−1(2−µ+η)n−1

for n≥2.

Sinceψ(n)is a decreasing function ofn (n ≥2),we have ξ =ξ(µ, γ, η, α, β) = 1− 2(1−α)(1−β)

(3−α)(3−β)ψ(2)−(1−α)(1−β), whereψ(2) = (2−γ)(2−µ+η)^{(2−γ+η)(2)} .Thus completes the proof.

Theorem 3.2. Let the functionsf_j(z) (j = 1,2)defined by (2.8) be in the class T R(µ, γ, η, α).Then(f₁∗f₂)(z)∈T R(µ, γ, η, δ),where

δ = 1− 2(1−α)²

(3−α)²ψ(2)−(1−α)², withψ(2) = (2−γ)(2−µ+η)^{(2−γ+η)(2)} .

Proof. By takingβ =αin the above theorem, the results follows.

Theorem 3.3. Let the functionf defined by (1.2) be in the classT R(µ, γ, η, α).

Also let

g(z) = z−

∞

X

n=2

b_nzⁿ for |b_n| ≤1.

Then(f∗g)(z)∈T R(µ, γ, η, α).

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Proof. Since

∞

X

n=2

ψ(n)(2n−1−α)|a_nb_n|=

∞

X

n=2

ψ(n)(2n−1−α)a_n|b_n|

≤

∞

X

n=2

ψ(n)(2n−1−α)a_n

≤1−α (by Theorem2.1),

whereψ(n)is defined by (3.3). Hence it follows that(f∗g)(z)∈T R(µ, γ, η, α).

Corollary 3.4. Let the functionfdefined by (1.2) be in the classT R(µ, γ, η, α).

Also letg(z) =z−P∞

n=2b_nzⁿfor0≤b_n≤1.Then(f∗g)(z)∈T R(µ, γ, η, α).

For functions in the class T R(µ, γ, η, α) we can prove the following inclu- sion property also.

Theorem 3.5. Let the functionsf_j(z) (j = 1,2)defined by (2.5) be in the class T R(µ, γ, η, α).Then the functionhdefined by

h(z) = z−

∞

X

n=2

(a²_n,1+a²_n,2)zⁿ is in the classT R(µ, γ, η,∆)where

∆ = 1− 4(1−α)²

(3−α)²ψ(2)−2(1−α)² with ψ(2) = 2(2−γ+η)

(2−γ)(2−µ+η).

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Proof. In view of Theorem2.1, it is sufficient to prove that (3.4)

∞

X

n=2

ψ(n)2n−1−∆

1−∆ (a²_n,1+a²_n,2)≤1

wherefj(z)∈T R(µ, γ, η, α) (j = 1,2). We find from (2.8) and Theorem2.1, that

(3.5)

∞

X

n=2

ψ(n)2n−1−α 1−α

2

a²_n,j ≤

∞

X

n=2

ψ(n)2n−1−α 1−α a_n,j

2

≤1, which would yield

(3.6)

∞

X

n=2

1 2

ψ(n)2n−1−α 1−α

2

(a²_n,1+a²_n,2)≤1.

On comparing (3.5) and (3.6) it can be seen that inequality (3.4) will be satisfied if

ψ(n)2n−1−∆

1−∆ (a²_n,1+a²_n,2)≤ 1 2

ψ(n)2n−1−α 1−α

2

(a²_n,1+a²_n,2).

That is, if

(3.7) ∆≤1− 4(1−α)²

(2n−1−α)²ψ(n)−2(1−α)², whereψ(n)is given by (3.3). Hence we conclude from (3.7)

∆ = ∆(µ, γ, η, α) = 1− 4(1−α)²

(3−α)² ψ(2)−2(1−α)², whereψ(2) = (2−γ)(2−µ+η)^2(2−γ+η) which completes the proof.

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4. Integral Transform of the Class T R(µ, γ, η, α)

Forf ∈T R(µ, γ, η, α)we define the integral transform V_λ(f)(z) =

Z 1 0

λ(t)f(tz) t dt,

where λ is a real valued, non-negative weight function normalized so that R1

0 λ(t)dt = 1. Since special cases of λ(t) are particularly interesting such as λ(t) = (1 +c)t^c, c >−1,for whichVλ is known as the Bernardi operator, and

λ(t) = (c+ 1)^δ λ(δ) t^c

log 1

t ^δ−1

, c >−1, δ ≥0 which gives the Komatu operator. For more details see [5].

First we show that the classT R(µ, γ, η, α)is closed underV_λ(f).

Theorem 4.1. Letf ∈T R(µ, γ, η, α).ThenV_λ(f)∈T R(µ, γ, η, α).

Proof. By definition, we have V_λ(f) = (c+ 1)^δ

λ(δ) Z 1

0

(−1)^δ−1t^c(logt)^δ−1 z−

∞

X

n=2

a_nzⁿtⁿ⁻¹

! dt

= (−1)^δ−1(c+ 1)^δ

λ(δ) lim

r→0⁺

"

Z 1 r

t^c(logt)^δ−1 z−

∞

X

n=2

a_nzⁿtⁿ⁻¹

! dt

# , and a simple calculation gives

Vλ(f)(z) = z−

∞

X

n=2

c+ 1 c+n

δ

anzⁿ.

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We need to prove that (4.1)

∞

X

n=2

2n−1−α

1−α · (2−γ+η)n−1(2)n−1

(2−γ)n−1(2−µ+η)n−1

c+ 1 c+n

δ

a_n<1.

On the other hand by Theorem2.1,f ∈T R(µ, γ, η, α)if and only if

∞

X

n=2

2n−1−α

1−α · (2−γ +η)n−1(2)n−1

(2−γ)n−1(2−µ+η)n−1

<1.

Hence _c+n^c+1 <1.Therefore (4.1) holds and the proof is complete.

Next we provide a starlike condition for functions in T R(µ, γ, η, α) and V_λ(f).

Theorem 4.2. Letf ∈T R(µ, γ, η, α).ThenV_λ(f)is starlike of order0≤γ <

1in|z|< R₁ where

R₁ = inf

n

"

c+n c+ 1

δ

· 1−γ(2n−1−α) (n−γ)(1−α) φ(n)

#_n−1¹ . Proof. It is sufficient to prove

(4.2)

z(V_λ(f)(z))⁰ V_λ(f)(z) −1

<1−γ.

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For the left hand side of (4.2) we have

z(V_λ(f)(z))⁰ V_λ(f)(z) −1

=

P∞

n=2(1−n)(_c+n^c+1)^δa_nzⁿ⁻¹ 1−P∞

n=2(^c+1_c+n)^δa_nzⁿ⁻¹

≤ P∞

n=2(n−1)(_c+n^c+1)^δa_n|z|ⁿ⁻¹ 1−P∞

n=2(_c+n^c+1)^δa_n|z|ⁿ⁻¹ . This last expression is less than(1−γ)since

|z|ⁿ⁻¹ <

c+ 1 c+n

δ

(1−γ)[2n−1−α]

(n−γ)(1−α) φ(n).

Therefore the proof is complete.

Using the fact thatf is convex if and only if zf⁰ is starlike, we obtain the following:

Theorem 4.3. Letf ∈T R(µ, γ, η, α).ThenV_λ(f)is convex of order0≤γ <1 in|z|< R₂where

R₂ = inf

n

"

c+n c+ 1

δ

(1−γ)[2n−1−α]

n(n−γ)(1−α) φ(n)

#_n−1¹ . We omit the proof as it is easily derived.

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References

[1] R. BHARATI, R. PARVATHAM ANDA. SWAMINATHAN, On subclass of uniformly convex functions and corresponding clss of starlike functions, Tamkang J. of Math., 28(1) (1997), 17–33.

[2] B.C. CARLSONANDS.B. SHAFFER, Starlike and prestarlike hypergro- metric functions, SIAM J. Math. Anal., 15 (2002), 737–745.

[3] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56 (1991), 87–92.

[4] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. &

Appl., 155 (1991), 364–370.

[5] Y.C. KIM ANDF. RØNNING, Integral transform of certain subclasses of analytic functions, J. Math. Anal. Appl., 258 (2001), 466–489.

[6] W. MAANDD. MINDA, Uniformly convex functions, Ann. Polon. Math., 57 (1992), 165–175.

[7] S. OWA, On the distribution theorem I, Kyungpook Math. J., 18 (1978), 53–59.

[8] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196.

[9] F. RØNNING, Integral representations for bounded starlike functions, An- nal. Polon. Math., 60 (1995), 289–297.

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[10] A. SCHILD AND H. SILVERMAN, Convolution of univalent functions with negative coefficients, Ann. Univ. Marie Curie-Sklodowska Sect. A, 29 (1975), 99–107.

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