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volume 5, issue 2, article 28, 2004.

Received 25 July, 2003;

accepted 09 February, 2004.

Communicated by:N.E. Cho

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Journal of Inequalities in Pure and Applied Mathematics

INEQUALITIES DEFINING CERTAIN SUBCLASSES OF ANALYTIC AND MULTIVALENT FUNCTIONS INVOLVING FRACTIONAL

CALCULUS OPERATORS

R.K. RAINA AND I.B. BAPNA

Department Of Mathematics M.P. University Of Agri. & Technology College Of Technology And Engineering Udaipur 313001, Rajasthan, India.

EMail:rainark_7@hotmail.com Department Of Mathematics, Govt. Postgraduate College Bhilwara 311001

Rajasthan, India.

EMail:bapnain@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 102-03

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Inequalities Defining Certain Subclasses of Analytic and Multivalent Functions Involving

Fractional Calculus Operators R.K. Raina and I.B. Bapna

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J. Ineq. Pure and Appl. Math. 5(2) Art. 28, 2004

Abstract

Making use of a certain fractional calculus operator, we introduce two new sub- classesM_δ(p;λ, µ, η)andT_δ(p;λ, µ, η)of analytic andp−valent functions in the open unit disk. The results investigated exhibit the sufficiency conditions for a function to belong to each of these classes. Several geometric properties of such multivalent functions follow, and these consequences are also mentioned.

2000 Mathematics Subject Classification:30C45, 26A33.

Key words: Analytic functions, Multivalent functions, Starlike functions, Convex func- tions, Fractional calculus operators.

1. Introduction and Definitions

LetA_p denote the class of functions of the form (1.1) f(z) = z^p+

∞

X

n=1

an+pz^n+p (p∈N={1,2,3, . . .}),

which are analytic and p−valent in the open unit disk U = {z : z ∈ C and

|z|<1}.

A functionf(z)∈ A_p is said to bep−valently starlike inU, if

(1.2) <

zf⁰(z) f(z)

>0 (z ∈ U),

and the functionf(z)∈ Apis said to bep−valently convex inU, if

(1.3) <

1 + zf⁰⁰(z) f⁰(z)

>0 (z ∈ U).

Further, a functionf(z)∈ A_pis said to bep−valently close-to-convex inU, if

(1.4) <

f⁰(z) z^p−1

>0 (z∈ U).

One may refer to [1], [2] and [9] for above definitions and other related details.

The operatorJ_0,z^λ,µ,η occurring in this paper is the Saigo type fractional calculus operator defined as follows ([8]):

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Definition 1.1. Let0≤λ <1andµ,η∈R, then (1.5) J_0,z^λ,µ,ηf(z)

= d dz

z^λ−µ Γ(1−λ)

Z z 0

(z−t)^−λF₁

µ−λ,1−η; 1−λ; 1− t z

f(t)dt

, where the functionf(z)is analytic in a simply-connected region of thez-plane containing the origin, with the order

f(z) =O(|z|^ε) (z →0), where ε >max{0, µ−η} −1.

It being understood that(z−t)^−λdenotes the principal value for0≤arg(z−

t)<2π.The₂F₁function occurring in the right-hand side of (1.5) is the famil- iar Gaussian hypergeometric function (see [9] for its definition).

Definition 1.2. Under the hypotheses of Definition 1.1, a fractional calculus operatorJλ+m,µ+m,η+m

0,z is defined by ([7]) (1.6) Jλ+m,µ+m,η+m

0,z f(z) = d^m

dz^mJ_0,z^λ,µ,ηf(z) (z ∈ U;m∈N0 ={0} ∪N). We observe that

(1.7) D^λ_zf(z) =J_0,z^λ,λ,ηf(z) (0≤λ <1), and

(1.8) D^λ+m_z f(z) = Jλ+m,λ+m,η+m

0,z f(z) (0≤λ <1;m∈N0), whereD_z^λ+m is the well known fractional derivative operator ([6], [9]).

We introduce here two subclasses of functionsM_δ(p;λ, µ, η)andT_δ(p;λ, µ, η) which are defined as follows.

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Definition 1.3. Letδ ∈R\{0},p∈N,0≤λ <1,µ <1, andη >max(λ, µ)−

p− 1. Then the function f(z) ∈ Ap is said to belong to Mδ(p;λ, µ, η) if it satisfies the inequality

(1.9)

zJλ+1,µ+1,η+1

0,z f(z)

J_0,z^λ,µ,ηf(z)

!δ

−(p−µ)^δ

<(p−µ)^δ (z ∈ U),

where the value of

z Jλ+1,µ+1,η+1

0,z f(z)/J_0,z^λ,µ,ηf(z)δ

is taken as its principal value.

Definition 1.4. Under the hypotheses of Definition1.3, the functionf(z)∈ A_p is said to belong toT_δ(p;λ, µ, η)if it satisfies the inequality

(1.10)

z^µ−pJ_0,z^λ,µ,ηf(z)δ

−

Γ(p+ 1)Γ(p+η−µ+ 1) Γ(p−µ+ 1)Γ(p+η−λ+ 1)

δ

<

Γ(p+ 1)Γ(p+η−µ+ 1) Γ(p−µ+ 1)Γ(p+η−λ+ 1)

δ

(z ∈ U),

where the value of

z^µ−pJ_0,z^λ,µ,ηf(z)δ

is considered to be its principal value.

Forλ=µ, we have

(1.11) M_δ(p;µ, µ, η) =M_δ(p;µ), and

(1.12) T_δ(p;µ, µ, η) =T_δ(p;µ).

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The classes M_δ(p;µ)and T_δ(p;µ) were studied recently in [4]. In view of the operational relation (1.8), it may be noted that the functions in M1(p; 0) are p−valently starlike in U,whereas, the functions inT₁(p; 1) arep−valently close-to-convex inU.

In this paper we investigate characterization properties giving sufficiency conditions for functions of the form (1.1) to belong to the classesM_δ(p;λ, µ, η) andT_δ(p;λ, µ, η)involving the fractional calculus operator (1.6). Several consequences of the main results and their relevance to known results are also pointed out.

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2. Results Required

We mention the following results which are used in the sequel:

Lemma 2.1. ([8]). If0≤λ <1;µ,η∈Randk > max{0, µ−η} −1, then (2.1) J_0,z^λ,µ,ηz^k= Γ(1 +k)Γ(1−µ+η+k)

Γ(1−µ+k)Γ(1−λ+η+k)z^k−µ.

Lemma 2.2. ([5]). Let w(z) be an analytic function in the unit disk U with w(0) = 0, and let0 < r < 1. If |w(z)|attains atz₀ its maximum value on the circle|z|=r, then

(2.2) z₀w⁰(z₀) =kw(z₀) (k≥1).

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3. Main Results

We begin by proving

Theorem 3.1. Letδ∈R\{0},p∈N,0≤λ <1,µ < 1,η >max(λ, µ)−p−1, and a > 0, b ≥ 0, such thata+ 2b ≤ 1. If a function f(z) ∈ A_p satisfies the inequality

(3.1) <

"

1 +z Jλ+2,µ+2,η+2

0,z f(z)

Jλ+1,µ+1,η+1

0,z f(z)− Jλ+1,µ+1,η+1

0,z f(z)

J_0,z^λ,µ,ηf(z)

!#











< a+b

δ(1 +a)(1−b) (δ >0)

> a+b

δ(1 +a)(1−b) (δ <0)

(z ∈ U),

thenf(z)∈ M_δ(p;λ, µ, η).

Proof. Letf(z)∈ A_p, and define a functionw(z)by

(3.2) zJλ+1,µ+1,η+1

0,z f(z)

J_0,z^λ,µ,ηf(z)

!δ

= (p−µ)^δ

1 +aw(z) 1−bw(z)

(z ∈ U).

Then it follows from (2.1) that w(z)is analytic function in U, and w(0) = 0.

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Differentiation of (3.2) gives (

1 +z Jλ+2,µ+2,η+2

0,z f(z)

Jλ+1,µ+1,η+1

0,z f(z)− Jλ+1,µ+1,η+1

0,z f(z)

J_0,z^λ,µ,ηf(z)

!) (3.3)

= 1 δ

(a+b)zw⁰(z) (1 +aw(z)) (1−bw(z))

=φ(z)(say).

Assume that there exists a pointz₀ ∈ U such that max

|z|≤|z0||w(z)|=|w(z₀)|= 1.

Then, applying Lemma2.2, we can write

z₀w⁰(z₀) = kw(z₀) (k ≥1), andw(z0) =e^iθ(θ∈[0,2π)), so that from (3.3) we have

<{φ(z₀)}= k(a+b) δ <

w(z0)

(1 +aw(z₀)) (1−bw(z₀))

= k δ<

1

1−bw(z₀) − 1 1 +aw(z₀)

= k δ<

1−be^−iθ

1 +b²−2bcosθ − 1 +ae^−iθ 1 +a²+ 2acosθ

= k δ

( 1

2 + _1−b^b²⁻¹_cos_θ − 1 2 + _1+a^a²⁻¹_cos_θ

)

= k∆

δ ,

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whereθ6= cos⁻¹(−1/a)andθ6= cos⁻¹(−1/b).

Simple calculations (under the constraints mentioned with the hypotheses for the parameters aandb) yield that ∆ ≥ _(1+a)(1−b)^(a+b) , and sincek ≥ 1,it follows that

(3.4) <{φ(z₀)}= k∆

δ







> δ(1+a)(1−b)^(a+b) (δ >0),

< δ(1+a)(1−b)^(a+b) (δ <0).

This contradicts our condition (3.1), and we conclude from (3.2) that

zJλ+1,µ+1,η+1

0,z f(z)

J_0,z^λ,µ,η

!δ

−(p−µ)^δ

= (p−µ)^δ

(a+b)w(z) 1−bw(z)

< (p−µ)^δ

a+b 1−b

≤(p−µ)^δ. This completes the proof of Theorem3.1.

Next we prove

Theorem 3.2. Letδ∈R\{0},p∈N,0≤λ <1,µ < 1,η >max(λ, µ)−p−1, and a > 0, b ≥ 0such thata+ 2b ≤ 1. If a functionf(z) ∈ A_p satisfies the inequality

(3.5) < zJλ+1,µ+1,η+1

0,z f(z)

J_0,z^λ,µ,η

!





< p−µ+δ(1+a)(1−b)^a+b (δ > 0)

> p−µ+δ(1+a)(1−b)^a+b (δ > 0)

(z ∈ U),

thenf(z)∈Tδ(p;λ, µ, η).

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Proof. Consider (3.6)

z^µ−pJ_0,z^λ,µ,ηf(z) δ

=

Γ(1 +p)Γ(1 +p+η−µ) Γ(1 +p−µ)Γ(1 +p+η−λ)

δ

1 +aw(z) 1−bw(z)

(z ∈ U).

Using the same method as elucidated in the proof of Theorem3.1, we arrive at the desired result.

Remark 3.1. If we setλ = µ, a= 1, b = 0,then Theorems3.1 and3.2 by ap- pealing to the operational relation (1.8) correspond to the recently established results due to Irmak et al. [4, pp. 271–272].

Theorems3.1and3.2would also yield various results involving analytic and multivalent functions by suitably choosing the values ofa, b,δ,µandp. Setting δ = 1in Theorems3.1and3.2, we have

Corollary 3.3. Let p ∈ N, 0 ≤ λ < 1, µ < 1, η > max(λ, µ)−p−1, and a > 0, b ≥ 0 such that a + 2b ≤ 1. If a function f(z) ∈ A_p satisfies the inequality

(3.7) <

(

1 +z Jλ+2,µ+2,η+2

0,z f(z)

Jλ+1,µ+1,η+1

0,z f(z)− Jλ+1,µ+1,η+1

0,z f(z)

J_0,z^λ,µ,ηf(z)

!)

< a+b

(1 +a)(1−b) (z ∈ U), thenf(z)∈ M1(p;λ, µ, η).

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Corollary 3.4. Let p ∈ N, 0 ≤ λ < 1, µ < 1, η > max(λ, µ)−p−1, and a > 0, b ≥ 0 such that a + 2b ≤ 1. If a function f(z) ∈ Ap satisfies the inequality

(3.8) < zJλ+1,µ+1,η+1

0,z f(z)

J_0,z^λ,µ,ηf(z)

!

< p−µ+ a+b

(1 +a)(1−b) (z ∈ U), thenf(z)∈ T₁(p;λ, µ, η).

Corollaries 3.3 and 3.4 on putting λ = µ = 0, and using (1.8) give the following results:

Corollary 3.5. Let p ∈ N, a > 0, b ≥ 0such that a+ 2b ≤ 1.If a function f(z)∈ A_psatisfies the inequality

(3.9) <

1 + zf⁰⁰(z)

f⁰(z) −zf⁰(z) f(z)

< a+b

(1 +a)(1−b) (z ∈ U), thenf(z)isp-valently starlike inU.

Corollary 3.6. Let p ∈ N, a > 0, b ≥ 0such that a+ 2b ≤ 1.If a function f(z)∈ Apsatisfies the inequality

(3.10) <

zf0(z) f(z)

< p+ a+b

(1 +a)(1−b) (z ∈ U), then <n

f(z) z^p

o

>0, (z ∈ U).

Lastly, Corollaries3.3and3.4on puttingλ=µ= 1,and using (1.8) give

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Corollary 3.7. Let p ∈ N, a > 0, b ≥ 0such that a+ 2b ≤ 1.If a function f(z)∈ Apsatisfies the inequality

(3.11) <

1 + zf⁰⁰⁰(z)

f⁰⁰(z) − zf⁰⁰(z) f⁰(z)

< a+b

(1 +a)(1−b) (z ∈ U), thenf(z)isp−valently convex inU.

Corollary 3.8. Let p ∈ N, a > 0, b ≥ 0such that a+ 2b ≤ 1.If a function f(z)∈ A_p satisfies the inequality

(3.12) <

zf⁰⁰(z) f⁰(z)

< p−1 + a+b

(1 +a)(1−b) (z ∈ U), thenf(z)isp−valently close-to -convex inU.

Remark 3.2. When a = 1, b = 0, then the Corollaries 3.5 – 3.8 correspond to the known results [3, pp. 457–458] involving inequalities onp−valent func- tions.

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References

[1] P.L. DUREN, Univalent Functions, Grundlehren der Mathematischen Wis- senschaffen 259, Springer-Verlag, NewYork, Berlin, Heidelberg and Tokyo (1983).

[2] A.W. GOODMAN, Univalent Functions, Vols. I and II, Polygonal Publish- ing House, Washington, New Jersy, 1983.

[3] H. IRMAK AND O.F. CETIN, Some theorems involving inequalities on p−valent functions, Turkish J. Math., 23 (1999), 453–459.

[4] H. IRMAK, G. TINAZTEPE, Y.C. KIM AND J.H. CHOI, Certain classes and inequalities involving fractional calculus and multivalent functions, Fr- acl.Cal. Appl. Anal., 3 (2002), 267–274.

[5] I.S. JACK, Functions starlike and convex of orderα, J. London Math. Soc., 3 (1971), 469–474.

[6] S. OWA, On the distortion theorems. I, Kyungpook Math. J., 18 (1978), 53–59

[7] R.K. RAINA ANDJAE HO CHOI, Some results connected with a subclass of analytic functions involving certain fractional calculus operators, J. Fr- acl. Cal., 23 (2003), 19–25.

[8] R.K. RAINAANDH.M. SRIVASTAVA, A certain subclass of analytic func- tions associated with operators of fractional calculus, Comput. Math. Appl., 32 (1996), 13–19.

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[9] H.M. SRIVASTAVAANDS. OWA (Eds.), Current Topics in Analytic Func- tion Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992.