Introduction LetAdenote the class of all analytic functions of the form f(z) =z+ P∞ m=2 amzm (1.1) defined in the unit discU ={z:|z|<1}

64, 1 (2012), 17–23 March 2012

research paper

RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE WITH VARYING ARGUMENTS

N. Ravikumar and S. Latha

Abstract. In this paper, we define the subclassesVδ(A, B) andKδ(A, B) of analytic func- tions by using Ω^δf(z). For functions belonging to these classes, we obtain coefficient estimates, distortion bounds and many more properties.

1. Introduction

LetAdenote the class of all analytic functions of the form f(z) =z+ P^∞

m=2

amz^m (1.1)

defined in the unit discU ={z:|z|<1}. LetN denote the subclass ofAconsisting of functions normalized byf(0) = 0 andf⁰(0) = 1 which are univalent in U.

Silverman [8] defined the class V(θm) as the class of all functions inN such that argam = θm for all m. If further there exists a real number β such that θm+ (m−1)β≡π (mod 2π), thenf is said to be in the classV(θm, β). The union ofV(θm, β) taken over all possible sequences{θm}and all possible real numbersβ is denoted byV.

The classAis closed under convolution or Hadamard product (f∗g)(z) =z+ P^∞

m=2

ambmz^m, z∈ U, (1.2) wheref is given by (1.1) andg(z) =z+P_∞

m=2b_mz^m.

Fractional derivative of orderδ of an analytic functionf is defined by D^δ_zf(z) = 1

Γ(1−δ) d dz

Z _z

0

f(t)

(z−t)^δ dt, 0≤δ≤1.

2010 AMS Subject Classification: 30C45.

Keywords and phrases: Univalent functions; Komato operator; fractional derivative; linear operator.

17

f is an analytic function in a simply connected region of thez-plane containing the origin and the multiplicity of (z −t)^−δ is removed by requiring log(z −t) to be real when (z −t) is greater than 0. Clearly f(z) = limδ→0D^δ_zf(z) and f⁰(z) = limδ→1D_z^δf(z).

For the analytic functionf of the form (1.1) we put Ω^δf(z) = Γ(2−δ)z^δD^δ_zf(z) =z+ P^∞

m=2

K(m, δ)amz^m,

whereK(m.δ) = Γ(m+1)Γ(2−δ) Γ(m+1−δ) .

Now we define the classVδ(A, B) consisting of functions f ∈ V such that z(Ω^δf(z))⁰

Ω^δf(z) = 1 +Aω(z)

1 +Bω(z), −1≤A < B≤1. (1.3) Hereω(z) is analytic,ω(0) = 0 and|ω(z)|<1,z∈ U. The following basic result is well known.

Lemma 1.1[Schwarz’s Lemma]Letωbe analytic withω(0) = 0, and|ω(z)|<1 forz ∈ U. Then |ω(z)| <|z|. The equality holds if and only if ω(z) =λz, where

|λ|= 1.

LetKδ(A, B) denote the class of functions f ∈ V such thatzf⁰∈ Vδ(A, B).

2. Main Results

Theorem 2.1. A function f ∈ V is inVδ(A, B)if and only if P∞

m=2

[(B+ 1)m−(A+ 1)]K(m, δ)|am| ≤(B−A), (2.1) where−1≤A < B ≤1,m≥2.

Proof. Supposef ∈ Vδ(A, B). Then z(Ω^δf(z))⁰

Ωδf(z) = 1 +Aω(z)

1 +Bω(z), −1≤A < B≤1.

From this we get,

ω(z) = z(Ω^δf(z))⁰−Ω^δf(z) Ω^δf(z)A−z(Ω^δf(z))⁰B. By Schwarz’s Lemma, we get

<









P∞ m=2

(1−m)K(m, δ)amz^m−1 (B−A) + P^∞

m=2

(mB−A)K(m, δ)amz^m−1









<1. (2.2)

Sincef ∈ V,f lies inV(θm, β) for some sequence{θm}and a real numberβ, such thatθm+ (m−1)β ≡π (mod 2π). Settingz=re^iβ, we get

<







 P∞ m=2

(1−m)K(m, δ)|am|r^m−1e^{i(θm+(m−1)β)} (B−A) + P^∞

m=2

(mB−A)|am|r^m−1e^{i(θm+(m−1)β)}









<1. (2.3)

P∞

m=2(m−1)K(m, δ)|am|r^m−1<(B−A)− P^∞

m=2(mB−A)K(m, δ)|am|r^m−1, P∞

m=2

[(B+ 1)m−(A+ 1)K(m, δ)|am|r^m−1<(B−A). (2.4) Lettingr→1, we get (2.1).

Conversely, supposef ∈ Vand satisfies (2.1). In view of (2.4), which is implied by (2.1), sincer^m−1<1, we have

¯¯

¯¯ P∞ m=2

(1−m)K(m, δ)amz^m−1

¯¯

¯¯≤ P^∞

m=2

(m−1)K(m, δ)|am|r^m−1

<(B−A)− P^∞

m=2

(mB−A)K(m, δ)|am|r^m−1

≤

¯¯

¯¯(B−A)− P^∞

m=2

(mB−A)K(m, δ)amz^m−1

¯¯

¯¯

which gives (2.2) and hence it follows thatf ∈ Vδ(A, B).

Corollary 2.2. If f ∈ V is inVδ(A, B)then

|am| ≤ (B−A)

[(B+ 1)m−(A+ 1)]K(m, δ), for m≥2, −1≤A < B≤1.

The equality holds for the functionf given by f(z) =z+ (B−A)

[(B+ 1)m−(A+ 1)]K(m, δ)e^iθmz^m, z∈ U.

For parametric valuesa=n+ 1,c= 1, we get the following result proved by Padmanabhan and Jayamala [4] as corollaries to the above theorem.

Corollary 2.3. Let f ∈ V. Thenf ∈ Vn(A, B) if and only if P∞

m=2

(n+m−1)!

(n+ 1)! (m−1)!C_m|a_m|<(B−A), whereCm= (B+ 1)(n+m)−(A+ 1)(n+ 1).

The equality holds for the functionf given by f(z) =z+Γ(c+m−1)Γ(a+ 1)

Γ(a+m−1)Γ(c)

(B−A)

Dm e^iθmz^m, z∈ U.

Theorem 2.4. Letf ∈ V. Thenf(z) =z+P_∞

m=2amz^m is inK(A, B, a, c)if and only if

P∞ m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)mD_ma_m< B−A,

whereDm= [(B+ 1)(a+m−1)−(A+ 1)a],−1≤A < B≤1,a, c∈R\Z⁻₀. Now we examine the extreme points of the classV(A, B, a, c).

Theorem 2.5. Let f(z)∈ V(A, B, a, c) withargam=θm, where [θm+ (m−1)β]≡π (mod 2π). Define f1(z) =z and

fm(z) =z+Γ(c+m−1)Γ(a+ 1) Γ(a+m−1)Γ(c)

(B−A) Dm

e^iθmz^m, m= 2,3, . . . ,

−1≤A < B ≤1, a, c∈R\Z⁻₀, z ∈ U. f ∈ V(A, B, a, c) if and only if f can be expressed asf(z) =P_∞

m=1µ_mf_m(z)whereµ_m≥0 andP_∞

m=1µ_m= 1.

Proof. Iff(z) =P_∞

m=1µmfm(z) withP_∞

m=1µm= 1,µm≥0, then P∞

m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)Dmµm·Γ(c+m−1)Γ(a+ 1) Γ(a+m−1)Γ(c)

(B−A) Dm

= P^∞

m=2

µm(B−A) = (1−µ1)(B−A)≤(B−A).

Hencef ∈ V(A, B, a, c).

Conversely, letf(z) =z+P_∞

m=2amz^m∈ V(A, B, a, c), define µm= Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)

|am|Dm

(B−A), m= 2,3, . . . and define µ1= 1−P_∞

m=2µm. From Theorem 2.1,P_∞

m=2µm≤1 and soµ1 ≥0.

Sinceµmfm(z) =µmf+amz^m,P_∞

m=1µmfm(z) =z+P_∞

m=2amz^m=f(z).

Theorem 2.6. Define f1(z) =z and fm(z) =z+Γ(c+m−1)Γ(a+ 1)

Γ(a+m−1)Γ(c)

(B−A)

Dm z^m, m= 2,3, . . . ,

−1≤A < B≤1,a, c∈R\Z⁻₀,z∈ U. Thenf ∈ K(A, B, a, c)if and only iff can be expressed as f(z) =P_∞

m=1µmfm(z)whereµm≥0andP_∞

m=1µm= 1.

Theorem 2.7. The class V(A, B, a, c)is closed under convex linear combina- tion.

Proof. Letf, g∈ V(A, B, a, c) and let f(z) =z+ P^∞

m=2

amz^m, g(z) =z+ P^∞

m=2

bmz^m.

For η such that 0 ≤ η ≤ 1, it suffices to show that the function defined by h(z) = (1−η)f(z) +ηg(z),z∈ U belongs toV(A, B, a, c). Now

h(z) =z+ P^∞

m=2

[(1−η)am+ηbm]z^m. Applying Theorem 2.1 tof, g∈ V(A, B, a, c), we have

P∞ m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)Dm[(1−η)am+ηbm]

= (1−η) X∞

m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)Dmam+η X∞

m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)Dmbm

≤(1−η)(B−A) +η(B−A) =B−A.

This implies thath∈ V(A, B, a, c).

Corollary 2.8. If f1(z), f2(z)are inV(A, B, a, c) then the function defined byg(z) =¹₂[f1(z) +f2(z)]is also inV(A, B, a, c).

Theorem 2.9. The classK(A, B, a, c)is closed under convex linear combina- tion.

Theorem 2.10. Let for j = 1,2, . . . , m, fj(z) = z +P_∞

m=2am,jz^m ∈ V(A, B, a, c) and 0 < λj < 1 such that P_m

j=1λj = 1. Then the function F(z) defined byF(z) =P_m

j=1λjfj(z) is also inV(A, B, a, c).

Proof. For eachj∈ {1,2, . . . , m}we obtain P∞

m=2

Dm Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)|am|< B−A.

SinceF(z) =P_m

j=1λj(z−P_∞

m=2am,jz^m) =z−P_∞

m=2(P_m

j=1λjam,j)z^m, P∞

m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)Dm[P^m

j=1

λjam,j]

= P^m

j=1

λj

· _∞ P

m=2

Dm Γ(a+m−1)Γ(c) Γ(c+m−1)Γ(a+ 1)

¸

< P^m

j=1

λj(B−A)< B−A.

ThereforeF(z)∈ V(A, B, a, c).

Theorem 2.11. Letf(z)∈ V(A, B, a, c)and Komato operator off is defined by

k(z) = Z ₁

0

(c+ 1)^γ Γ(γ) t^c

µ log1

t

¶_γ−1 f(tz)

t dt, c >−1,γ≥0. Then k(z)∈ V(A, B, a, c).

Proof. We have Z ₁

0

t^c µ

log1 t

¶_γ−1

dt= Γ(γ) (c+ 1)^γ Z ₁

0

t^m+c−1 µ

log1 t

¶_γ−1

dt= Γ(γ)

(c+ 1)^γ, m= 2,3, . . . ,

k(z) = (c+ 1)^γ Γ(γ)

"Z ₁

0

t^c µ

log1 t

¶_γ−1

z dt+ P^∞

m=2

z^m Z ₁

0

amt^m+c−1 µ

log1 t

¶_γ−1 dt

#

=z+ P^∞

m=2

µc+ 1 c+m

¶_γ amz^m. Sincef ∈ V(A, B, a, c) and since¡_c+1

c+m

¢_γ

<1, we have P∞

m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)[(1 +A)−m(1 +B)]

µc+ 1 c+m

¶_γ

am< B−A.

In the next theorem we will find distortion bound forL(a, c)f(z).

Theorem 2.12. Iff ∈ V(A, B, a, c), then

|z| −(B−A)Γ(c+ 1)

D2Γ(c) |z|²≤ |L(a, c)f(z)| ≤ |z|+(B−A)Γ(c+ 1) D2Γ(c) |z|². Proof. Letf(z)∈ V(A, B, a, c). Using Theorem 2.1,

P∞ m=2

am≤(B−A)Γ(c+ 1) D2Γ(c) . Therefore

|L(a, c)f(z)| ≤ |z|+|z|² P^∞

m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)am<|z|+(B−A)Γ(c+ 1) D2Γ(c) |z|² and

|L(a, c)f(z)| ≥ |z|−|z|² P^∞

m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)a_m>|z|−(B−A)Γ(c+ 1) D2Γ(c) |z|². Remark 2.13. (i) For parametric values ofa= 1 and c= 1 we get

|z| −(B−A)Γ(c+ 1)

D2Γ(c) |z|²≤f(z)≤ |z|+(B−A)Γ(c+ 1) D2Γ(c) |z|². (ii) For parametric values ofa= 2 andc= 1 we get

1−(B−A)Γ(c+ 1)

D2Γ(c) |z| ≤f⁰(z)≤1 +(B−A)Γ(c+ 1) D2Γ(c) |z|.

Theorem 2.14. Let f ∈ V(A, B, a, c). Then for every 0≤δ <1 the function Hδ(z) = (1−δ)f(z) +δ

Z _z

0

f(t) t dt.

Proof. We haveHδ(z) =z+P_∞

m=2

¡1 + _m^δ −δ¢

amz^m. Since¡

1 + _m^δ −δ¢

<1, m≥2, so by Theorem 2.1,

P∞ m=2

µ 1 + δ

m−δ

¶

Dmam Γ(a+m−1)Γ(c) Γ(c+m−1)Γ(a+ 1)

< P^∞

m=2

Γ(a+m−1)Γ(c)

Γ(c+m−1)Γ(a+ 1)Dmam< B−A.

ThereforeHδ(z)∈ V(A, B, a, c).

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(received 30.07.2010, revised 14.01.2011; available online 20.02.2011)

Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore - 570 005, INDIA.

E-mail:[email protected], [email protected]